SLM Vauxhall Tunbridge Wells are looking for an enthusiastic Apprentice Technician who is keen start a career within vehicle mechanics to join the team.
This opportunity will be run in conjunction with the Vauxhall Apprentice scheme and manufacturer training will be provided from Vauxhall. No experience is necessary as full training will be given, but the right candidate must have an interest in the motor industry and be willing to learn new skills.
You must also be hard working and enjoy problem solving whilst being a good team player, as well as possess a passion for delivering exceptional service. A clean driving licence is also required.
To apply, please send your CV to Robert at robert.taylor@slm.co.uk.


text:
Both 'G&M' official companies sign up to a strategic partnership
TOPICS:G&M Orient
In news of another excellent signing, G&M Orient has partnered with G&M Cosmetics to help expand their reach into the Australian Daigou markets. This distribution deal beginning on the 1st of July 2020 will see us help G& M Cosmetics achieve their goal of expanding their ranges into the market.
No relation to us (although we're proud to call them our newest brand), G&M Cosmetics is named for founder Zvonko Jordanov's children, George and Maria. A family business founded in 1996, G&M Cosmetics products are 100% Australian made and run by the Jordanovs to this day.
Over these past two decades, G&M Cosmetics has grown exponentially and become renowned for their skincare for the whole family. They are also widely regarded for their incredible value, and for having manufacturing standards of the highest level. They have won many prestigious awards, and been finalists for many others.
At G&M Orient, we love Australian brands that utilise the wonderful ingredients our country has to offer, and that's why we've made G&M Cosmetics our latest partner. They spend years developing and refining their products, with ingredients including botanical extracts and natural oils that Australia is famous for.
From Macadamia Oil, Manuka Honey, Goats Milk and Kakadu Plum to Lanolin Oil, Lemon Tea Tree and Emu Oil, only the highest quality ingredients are used in G&M Cosmetics' products. Their mission is to continue to keep making an impact on the skin health of people all over Australia, and the world.
G&M Cosmetics' products have a great track record for improving a variety of skin conditions, and they are passionate about making their effective, nourishing and organic skincare available to as many people as possible. Making people's skin look better improves their confidence, as well as their overall well-being – and that's what G&M Cosmetics loves to be a part of.
Whether it's their range of mineral oil-free and paraben-free Australian Creams MKII or their Native Australian brand, each product is also free from artificial colours and fragrances. They harness the power of natural Australian ingredients for sophisticated skincare that doesn't break the bank, while still being incredibly effective.
Australian pharmacies and supermarkets already stock this fast-growing brand, and now, thanks to signing with us here at G&M Orient, other countries will get to experience this remarkable skincare for themselves. We are very excited about this partnership and extend G&M Cosmetics a warm welcome into the G&M Orient family.
For up to date news about our latest signings and distribution deals, be sure to check our website.
About G&M Orient
G&M Orient Pty Ltd is an Australian and New Zealand distribution company offering well-known global brands in health & beauty products. We have over 80 renowned brands, stocking over 1,000+ products. Our range includes maternal products for mothers & their babies, skin care, personal care, honey & more. Connect with G&M Orient on Facebook, LinkedIn, and Instagram.
G&M Orient Pty Ltd
Email: salesmgr@gmorient.com
9/183 McCredie Road,
Smithfield NSW 2164
Website: https://www.gmorient.com.au
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Antoine Andrew Chung is currently a second year student at the University of Technology, Jamaica, studying Communication Arts and Technology. Over the year, he has received multiple awards for: Visual Arts, Sports and Science. As a well rounded, talented and artistic individual, he is very passionate about graphic designing, photography, as well as composing and singing songs. His enthusiasm and optimism towards individual and group assessment undergoes the fundamentals of work ethics and professionalism both inside and outside of class.
As an upcoming and ambitious photographer, he takes great pride in his work. His objective in life is to maximize the quality of the work being done through hard work and dedication. As said by the young photographer, "it is not about the quantity of the work being accomplished , but the quality of the job and the effect it has on its customers". His philosophy towards achieving his goals are to raise the standards of any brand he comes in contact with, giving it the effect that will only last for a lifetime. He believes that with hard work and dedication their is no need or place for mediocrity.
The young visionaries believes that all this and more can be achieved with great focus, and capacity to use critical and inferential thinking along with problem solving strategies. His ability to work under pressure is very commendable as he not only has excellent time management skills but exudes positivity and control over his environment and the work being done. Mr. Chung's vision as a humanitarian is to make a difference in society by impacting the life of students, his peers and his country men by helping others to reach their goals while changing the views of society through his lenses. Apart from his appearance he is a very sociable and humorous individual with a personality that oozes with exuberance.
This particular young man can easily adapt to any environment that he is placed in.
What began as doodling on the back of his books mutated into professional graphic artist and illustrator. He has work with some of the world's most vaunted brands… Red Stripe, Nestle, Flow, Carib Cement, Red Bull, Sterling, Yellow Page, just to name a few. As apart of the Red Stripe's Jamaica 50th TV advert, done by Prism Communications, ranked as one of the 'Best Ads' on the international advertising website Adforum.com.
Antoine Chung (Toiung), is a graphic, web and logo designer by profession but well rounding in the multimedia area. Although this is a full time job, he still managed to do freelance design on the side during this time — commitment or addiction, who is to know?. He's a positive role model and a team player. Failure is not an option. He only has one plan, plan A and it is to WIN.


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"Get out, Traitor!" – German People chases Minister of Justice Heiko Maas to his Armored Mercedes
Tue 3:36 pm +00:00, 3 May 2016 1
Face of a dying Nation
http://vidmax.com/news
Heiko Maas, the German Federal Minister of Justice, was unable to finish his Labor Day celebration speech on the 1st of May as he was loudly booed and chased off the stage by the German people. The people repeatedly shouted "Traitor", "Leftist Rat", "Get out!", "We are the People" and "Maas must go!", eventually getting him to cancel his speech and flee to his armored vehicle escorted by his armed bodyguards.
Maas is considered one of the biggest proponents of expanding censorship laws, demanding persecution, fines and jail-time for everybody posting "hate speech" on social media.
He also does not acknowledge the existence of the German people, backing his party, the Social Democrats, in the opinion that Europe was always made up of immigrants and Germany's only chance of redemption for the eternal guilt of World War 2 is inviting as many Muslim immigrants as possible.
His party took devastating losses in polls across the country, losing to the Alternative for Germany (AfD) by a landslide in the last state election of Saxony, where he held his speech.
Maas' party recently published plans to build 350,000 entirely new homes for "newcomers" to solve the "demographic crisis". In his speech he claimed that "the people shouting 'traitor' don't even know what's happening to them". But it appears that they know very well what is being done to them.
The German people are confused and angry about why they're told that they have to be frugal and avoid having children because of the immense cost while simultaneously working their fingers to the bones to fund a foreign invasion.
The people in the audience hence ridiculed him for claiming that the actual workers in the audience "hijack Labor Day". The hypocrisy of celebrating Labor and fair wages while his party supports the import of millions of unskilled workers is what got him chased off the stage.
German Justice Minister That Promised To Arrest Citizens Who Criticize "Refugee" Invasion Flees In Terror After HUNDREDS Of German Patriots Confront Him Screaming "Get Out Traitor!"
By Andrew Bieszad on May 2, 2016
Hundreds of German patriots protested and attacked Heiko Maaas yesterday, the German Minister of Justice who promised to arrest and prosecute any and all German citizens who would criticize the "refugee" invasion of Germany. The protests were so aggressive that he had to be escorted away by DOZENS of police officers for his safety.
From N-TV.de:
With whistles and booing protesters have disrupted the concert of German Justice Minister Heiko Maas at a rally May Day in Zwickau. The union there was a group from the right political spectrum.The speaker's platform had to be protected by the police. But there have be no physical altercations, said a police spokesman.
According MDR drowned about 100 demonstrators the Minister with whistles, "traitors" – and "Maas must go" chants. Accordingly, the SPD politician had shortened hisB speech and had the rally immediately thereafter, followed by "We are the people" chants.
The DGB chairman of Southwest Saxony region, Sabine Zimmermann, was horrified. "I have never experienced that at a union rally at a May 1, the stage will be protected by the police must," she said. yet it had been kept all the words. The DGB his right to an event on May 1 will not let take.
People correctly say that Germany is dying as nation right now, and that is true- to a point. The other side is that by the fact of these protests, Merkel's "neue Deutschland"is on life support right now. And things are not going to get better in Germany for the near future.
There is going to be a civil war in Germany in the next few years, and it will be this war that determine the nation's future.
But what will not endure is Chancellor Merkel's planned "vision" for what she had for Germany. After all is said and done, Merkel may consider herself very lucky if, as some major German politicians have already suggested, Merkel is able to flee to South America and able to live out the rest of her life instead of "hanging out" in Germany, like what Mussolini did in Italy after World War II:
Source: http://shoebat.com/2016/05/02/german-justice-minister-that-promised-to-arrest-citizens-who-criticize-refugee-invasion-flees-in-terror-after-hundreds-of-german-patriots-confront-him-screaming-get-out-traitor/
One Response to ""Get out, Traitor!" – German People chases Minister of Justice Heiko Maas to his Armored Mercedes"
Babushka says:
Kudos to these people in Germany.
They celebrated their Labour Day in a manner we had all hoped was in the past – by facing up to another lying, arrogant, dumb-cluck politician talking down with empty promises and implied disciplines as though the workers are naughty children.
I grew up in the shadows of WW ll horrors as depicted in the photo here.
Many on the internet today are well intentioned people who like me wanted a world of never-again-war.
It has been a horrendous trauma for me to realise that there is an enduring thread that runs through the political class that actually works to destroy nature-all of it, if only they can profit.
Thank you Tap for creating this 'green space'
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This is a call for submissions of poetry and art, and for pitches for essays about poetry, poetics, and social change.
The next issue of Geez, guest edited by Geez poetry editor Melanie Dennis Unrau, will explore the poetries and poetics of social change.
We're looking for poetry, art, and essays that fit with our goal of naming and responding to all forms of injustice (racial, economic, homophobic, transphobic, ecocidal, ableist, colonial, sexist, misogynist, etc.).
The poetic is a vital force in resisting oppressive systems and injustice. It provides imagination, inspiration, and a vocabulary that transcends rational thought and expresses our groaning and yearning.
We want all kinds of poems and reflections upon poetry that are mindful of joys, sorrows, interconnectedness, legacies of injustice, and struggles for new ways of living and relating.
We welcome complete poems for consideration. We welcome pitches for articles, short and long, on the ways in which poetry has affected you or changed your world, or the role of poetics in human communities, or how poetry can articulate new forms of cross-species relations.
Spoken over megaphones, stencilled on placards, printed on pamphlets, and sung in protest songs, poetry has played a role in recent protest movements from anti-war activism to Occupy, from the Arab Spring and the Indignados to Standing Rock. Poets all over the world—Anna Akhmatova, Ken Saro-Wiwa, Pablo Neruda, and more—have been imprisoned, exiled, and martyred as dissidents, subversives, and visionaries.
Writing about the role of poetry in feminism, Black lesbian poet Audre Lorde wrote that for women "poetry is not a luxury. It is a vital necessity of our existence. It forms the quality of the light within which we predicate our hopes and dreams toward survival and change, first made into language, then into idea, then into more tangible action. Poetry is the way we help give name to the nameless so it can be thought." How does poetry or the poetic inspire, articulate, and support movements for social change? Send us your socially engaged poetry and art, or pitch an essay about a poem or image that had a profound impact.
W.H. Auden famously wrote that "poetry makes nothing happen." Canadian work poet and anthologist Tom Wayman calls poetry "Songs Without Price," virtual non-commodities that function outside of the monetary economy and have subversive potential. Can poetry count as activism in itself? The debate rages on in academic and poetry circles. Meanwhile, recent poetry projects like The Enpipe Line (against Enbridge's now-cancelled Northern Gateway Pipeline) and Poets for the Peace (against the Site C Dam in British Columbia) blur the border (wherever it is) between creativity and activism. Geez has long had an ear to the ground seeking activist poetry: send us your activist poems or your poems about activism, or pitch a story or essay.
Poetics can be making, creation, labour, and even a way of life. Monastic and contemplative practices are meant to reshape our lives, our thinking, how we see things, and the way we inhabit the world. So are practices like nonviolence, permaculture, cycling, blockades, and intentional community. Phrases like "we make the road by walking" (from the Zapatistas) refer to a kind of poetics. How do you describe the creative makings (the poetics) of community, of the Kingdom of God, or of the revolution?
We want poems, long-form journalism, personal stories of transformation, short bursts of feelings, nuggets of insight, and inspiration. Pick your aspect of the topic and expand with personal experience, researched wisdom, or spiritual insight.
Submit up to three poems for consideration. If formatting does not translate well in a text file, send a PDF, too.
We're looking for creative non-fiction essays, investigative articles or research-based pieces on the topic above. While a reference to your personal experience is welcome, readers need wisdom from other sources as well (interviews, books, articles, theologians, social-justice activists, academics, and moms and dads). Pitches should be one page, touch on the wider context of your topic and name at least one other source you will be consulting. Here's a request: Please think of yourself as a Geez contributing editor. Ask yourself: what would readers who are social-justice oriented and at the fringes of faith want to read on this topic? If your pitch is accepted you'll usually have two to four weeks to complete the piece.
Consider the topic above and send original photographs (i.e., you took the photo) or illustrations that provoke or pacify, animate or incite. Or, if you know of a photographer or illustrator who can deliver an awesome photo essay or series of drawings and is willing to get big play in a premium little magazine for a modest honorarium, please pass this pitch along.
The Geez project is a discussion among people often at the fringes of faith seeking social justice. Our readers and writers express this through art, activism (a creative critique of those in power and the structures that keep them there, and the promotion of alternative practices that subvert such powers), contemplation, and a more-grounded, interconnected approach to living.
Before pitching, please read our guidelines for writers. Ideally we would like to respond personally to every piece of correspondence we receive. But given the number of submissions we receive – and having tried to respond to all – we realize it is just not possible. If you do not hear back from us within six weeks, assume that we were unable to use your submission.


A vital element in any Terran Fleet, the Aegis strengthens the already formidable energy shields of the vessels around it. Distributed through the heart of the Terran Fleet, every vessel sheltering in their umbrella becomes a sturdy redoubt, and the largest ships all but impervious to long range fire. With these invaluable vessels interspersing the formation, the Terran advance becomes an inexorable march to victory.
Contains 3 Aegis Class Shield Cruisers.


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Boeing to supply components for USAF AC-130U gunships
1 March 2012	(Last Updated March 1st, 2012 04:30)
Boeing has been awarded a contract for the supply of components to complete the installation of 25mm Ammunition Storage and Handling Systems (ASHS) on the US Air Force's (USAF) four AC-130U gunships.
Under the first $4.6m delivery order of the original $11.4m indefinite delivery, indefinite quantity (IDIQ) contract, the company will also provide spare parts to support sustainment of the ASHS.
The 25mm ASHS system consists of five main components that include the magazine assembly, conveyor assembly, loader/downloader, magazine transfer unit (MTU), and the hydraulic drive subassembly (HDS).
With a capacity of 3,000 rounds, the ASHS is compatible with GAU-12/U Gatling guns and can also feed ammunition to the 25mm cannons located inside the aircraft's cargo hold.
Boeing has been the prime maintainer of the ASHS since 1994 and also provided five major subassemblies for building six ammunition storage systems under the $11.1m contract, initially awarded in October 2011.
A heavy-armed ground-attack aircraft variant of the C-130 Hercules aircraft, the Lockheed AC-130 gunship is used by the USAF to conduct close air support, air interdiction and force protection missions including defending air bases and other military facilities.
The AC-130U is equipped with the AN/APQ-180 synthetic aperture radar for long-range target detection and identification and also features Inertial Navigation Systems (INS) and a Global Positioning System (GPS) for accurate target tracking.
The gunships have been used by the USAF during Operation Enduring Freedom in Afghanistan and Operation Iraqi Freedom in Iraq and also by US Special Operations forces during attacks on suspected al-Qaeda militants in Somalia in 2007.
Manufacturing of the parts will commence immediately at Boeing Fort Walton Beach facility and the components will be delivered to Robins Air Force Base, in Georgia, US in December 2012.
Image: The USAF AC-130U gunship can perform close air support and armed reconnaissance missions. Photo: Senior Airman Andy M. Kin, USAF.
Avionic Night Vision Lighting Systems
Helicopter and Aircraft Weapon Systems
Airbus completes preliminary design review of Skynet 6A project


The Call for Entries titled The Great Indoors Award 2015: Forever Now is now open!
CALL FOR ENTRIES - This Interior Design Competition is being held for the fifth time and this year is themed "Forever Now", focusing on the question of our relationship to the now(and the new).
Only non-residential interiors or interior installations will be considered for the Award. Only projects realized between September 2013 and August 2015 are eligible for participation.
Winners will be announced at an award ceremony in Maastricht, the Netherlands on 30-31 October 2015. One representative from each nominated firm will be invited to attend the award ceremony at the organization's expense.
The Great Indoors Award competition is hosted by Frame Publishers (Amsterdam), Stichting Bureau Europa (Maastricht) and Stichting Marres, House for Contemporary Culture (Maastricht).
For each project you sign up with, you are required to pay a fee of 145 Euro.
Four prizes of 5,000 Euro will be awarded for the most innovative interior designs for public and commercial spaces in each of the competition categories.


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Q: Unable to check checkboxes inside SysTreeView32 I am trying to change Internet Explorer settings using AutoIt, but I can't check items in a SysTreeView32 using ControlTreeView(). My code :
var controlTreeView = "[CLASS:SysTreeView32; INSTANCE:1]";
var treeView = AutoItX.ControlGetHandle(hwnd, controlTreeView);
AutoItX.ControlTreeView(hwnd, treeView, "Check", "#6|#6, "");

Code with command "Select" works fine, but it just selects the item.
var controlTreeView = "[CLASS:SysTreeView32; INSTANCE:1]";
var treeView = AutoItX.ControlGetHandle(hwnd, controlTreeView);
AutoItX.ControlTreeView(hwnd, treeView, "Select", "#6|#6, "");

How can I (un-) check them instead of just selecting?

A: As help file for ControlTreeView() describes, the Check command only works "if the item supports it". Because of mentioned security-sensitive dialog, this command will not work.
A solution:
Use the Select command, focus the window/dialog and send a space keystroke:
Global $sWindow   = 'Internet Options'
Global $sControl  = '[CLASS:SysTreeView32; INSTANCE:1]'
Global $sCheckbox = '#6|#6'

Func _getTreeViewHandle()
    WinWaitActive( $sWindow, '', 2 )
    Return ControlGetHandle( $sWindow, '', $sControl )
EndFunc

Func _checkTreeViewCheckbox( $hControl )
    ControlTreeView( $sWindow, '', $hControl, 'Select', $sCheckbox )
    WinActivate( $sWindow )
    Send( '{SPACE}' )
EndFunc

Global $hTreeView = _getTreeViewHandle()
_checkTreeViewCheckbox( $hTreeView )

Tested in IE v11.
Second approach:
Adjust the registry with RegWrite(). Example:
Global $iReturnCode = RegWrite('HKEY_CURRENT_USER\Software\Microsoft\Internet Explorer\Main', 'UseSWRender', 'REG_DWORD', '0x00000000')

You have to adjust the key for your OS version.

redpajama_set_name: RedPajamaStackExchange


I'm not a fan of Candlestick Park. It makes me angry to think about paying $90 for an experience that is basically like watching a game in a giant urinal. But I'm getting melancholy for the impending loss of the fan experience when the team moves in 2014.
He never addressed what he did, and his fans wanted to be like him. Sickest of all, despite public outcry, Wayne Pacelle and the Humane Society of the United States put their seal of approval on Vick getting another dog. This was supposed to be the dog he was "getting for his children".
wholesale jerseys from china "A very long history" Soon after, police head to Hillsborough. This couple have been married for 20 years and a nasty fight has erupted over a T shirt. He put it on the table. I knew it was going to pay off for John today. John got a bingo. B m is a Vietnamese stuffed baguette, an ode to the influence introduced by the French during its colonial period. The baguette is stuffed with all kinds of Vietnamese goodies mostly pork, chilies, herbs and other yummy things. In Hanoi, we loved the little street food cart Bahn Mi 25, selling the best tasting ones wholesale jerseys from china.


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News > Action required: Government announcement on Tax Free shopping
Action required: Government announcement on Tax Free shopping
On Friday 11 September the Government announced that it plans to stop offering tax free shopping to all international visitors after 31 December 2020. This will mean losing a £3.5 billion sector and undermining Britain's global appeal and competitiveness. We know from talking to our business community that this is yet another blow, at a time when many are still struggling due to the ongoing COVID-19 pandemic. The decision is not just something that will impact retailers, it will have a ripple effect across many sectors including hospitality and culture. Similarly, it is not only a London problem – the reality is that the tax free incentive does encourage visitors to the UK and this gives people a clear reason to go elsewhere. We are urging the Government to rethink its decision.
We urgently need your help to express the devastating impact this decision will have on a wide range of sectors across the whole of the UK.
Please see more information below about what action can be taken over the next few weeks.
Write directly to the Chancellor: Rt Hon Rishi Sunak MP, Chancellor of the Exchequer, 1 Horse Guards Road, London, SW1A 2HQ / email: public.enquiries@hmtreasury.gov.uk
Contact your local MP – you can find your MP and contact details here:
https://members.parliament.uk/FindYourMP
View related news stories
A New Partnership
Over the next five years the Cheapside Business Alliance will invest £2.2m
The Cheapside Business Alliance would like to announce "Working towards a Superfast City".
Cheapside Business Alliance March News Letter
Cheapside Business Alliance February Enewsletter


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Mancos Trails Group breathes new life into Colorado Trail spur
In 1973, it was part of the Colorado Trail dream: the Mancos Spur.
Since then, the dream has fallen by the trailside with little recollection.
But now, the Mancos Trails Group aspires to rebrand the original vision and add the spur to the Colorado Trail system, including updated trail signs.
"This has the potential to put the Mancos Spur on the map. Literally." Robert Meyer, board chairman of the Mancos Trails Group, said in a news release. The plan doesn't require building new trails because the current national forest trail system supports the entire spur.
"We couldn't ask for anything better," Meyer said.
The spur utilizes 23½ miles of local trails that include Chicken Creek, West Mancos and Sharkstooth as it travels from Mancos State Park and Jackson Lake and connects to the Colorado Trail near Taylor Lake. The Mancos State Park trailhead is set in a 7,800-foot-high ponderosa forest and travels through several climatic zones and ecosystems, including an abundance of alpine tundra. Eventually it passes over two passes at 12,000 feet before connecting with the Colorado Trail.
Mancos State Park Manager Scot Elder said that the park "is very excited to partner with the U.S. Forest Service and Mancos Trails Group to improve this beautiful and important section of trail."
In addition, Christopher Bouton, a trails, wilderness, and dispersed recreation specialist with the San Juan National Forest, said, "The general public sometimes doesn't realize that there are currently many miles of underutilized trails in the local area. It is exciting to be able to partner with the Mancos Trails Group and the Mancos State Park to help maintain this trail connection to the Colorado Trail by improving the trail condition and performing more regular maintenance."
For more information, see Mancos Trails Group website at http://www.mancostrails.org.


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What is the EU customs union and does it prevent trade deals?
Once largely only mentioned by policy enthusiasts and campaigners, this agreement is now the subject of rowdy debate.
The customs union is the subject of fierce debate (Victoria Jones/PA)
April 4 2019 13:35 PM
In an effort to dislodge the parliamentary "logjam", Theresa May has entered into Brexit talks with Jeremy Corbyn.
https://www.belfasttelegraph.co.uk/news/uk/what-is-the-eu-customs-union-and-does-it-prevent-trade-deals-37983436.html
https://www.belfasttelegraph.co.uk/news/uk/17ab1/37983432.ece/AUTOCROP/h342/bpanews_22cac498-3b42-4bfe-a945-d33706b5c0e6_1
But the Labour leader will be aiming to convince the Prime Minister to rub out one of her red lines and renew membership of the EU customs union.
Here the Press Association answers your burning questions on the subject:
– First things first, what exactly is a customs union?
You're in safe hands doing business in the EU.
The #CustomsUnion turns 50 this year. Learn more about what Customs Union does for your business → https://t.co/YnmaX6YGGF #50CU pic.twitter.com/0wCIuX7VJr
— European Commission 🇪🇺 (@EU_Commission) August 12, 2018
To make trade more seamless, countries within the union agree not to impose customs checks on each other's goods or to charge tariffs – taxes on imports and exports.
The EU's customs union includes all member states, plus Monaco. Goods within these countries can travel around unimpeded.
But under the EU's rules, members of the customs union can only negotiate as a whole with the rest of the world, and cannot choose to negotiate tariffs with outsiders.
– Could the UK strike trade deals while in the customs union?
Yes. But…
The Prime Minister has pledged to leave the customs union, arguing it would hinder the UK's ability to strike deals.
Because the UK would not be able to negotiate tariffs, it could only make deals on other areas, such as regulation.
Stefan Enchelmaier, a professor of European law at Oxford University, agreed that this leaves Britain with little bargaining power.
But he added: "The UK would be free to allow in chlorinated chicken and hormone beef."
These are two infamous examples of goods that could be imported from the US if the UK relaxes regulations post-Brexit. They are currently banned in the EU.
Prof Enchelmaier also pointed to other difficulties in negotiating trade deals, saying the EU has faced difficulties with India because India wanted certain levels of freedom of movement along with trade.
"These concerns are not going to go away when the UK faces India," he said.
– Does the customs union solve the hard border issue in Ireland?
No. That highly contentious area would remain because checks would still need to be carried out on safety standards when goods crossed the border.
– Who said what before the referendum?
The customs union was not at the forefront of the conversation by any means in the run-up to the vote.
As London mayor, Boris Johnson had been an advocate even of the single market, which would mean continued freedom of movement.
"Myself, I think it would be better to stay in the single market, to have the benefits of a customs union, the benefits of completely untrammelled free trade, rights of establishment, all those kinds of things, those are very important for British business," the Tory told LBC in May 2013.
But now the former foreign secretary describes such an arrangement very differently while criticising Mrs May's deal.
"Not only are we prevented from doing proper free trade deals, but we must endure the utter humiliation of watching the EU commission negotiate access to UK markets – with no UK say on the deals," he wrote in The Telegraph last month.
– What's next?
Businesses, are you preparing for #Brexit?
All you need to know about customs duties, import and export licences, customs authorisations, VAT and excise rules: https://t.co/2ntBJ4U0PE#Prepare4Brexit pic.twitter.com/5rI393ERn7
— EU Tax & Customs 🇪🇺 (@EU_Taxud) April 4, 2019
Mrs May has sought the Labour leader's assistance to "break the logjam" in the House of Commons in order to get a deal through. With three failed attempts on her current plan, a compromise is necessary to win Mr Corbyn's support.
Labour backs a customs union, as well as a second referendum on any deal – two of the Prime Minister's red lines that she did not want to cross.
But a Number 10 source said: "Unless there is compromise on both sides, it is unlikely we can find a way forward."
However, any new deal would need support from within her ranks, too.
So far, Mrs May has seen two ministerial resignations since announcing the talks and has received terse warnings from hardline Brexiteers.
Harry Dunn case: Police chief 'considering' CCTV and stop and search at RAF base
By Josh Payne, PA A police chief is considering the use of CCTV and stop and search exercises at an RAF base after footage emerged of a driver on the wrong side of the road near where Harry Dunn was killed, the...
By Aoife Moore, PA Irish Premier Leo Varadkar has said he believes big companies should pay their taxes as he welcomed Apple chief executive Tim Cook to Dublin.
Key questions around the HS2 rail project
By Neil Lancefield, PA Transport Correspondent A leaked report warns HS2 could cost up to £106 billion.


It is a social choice, in MMT, above all for societies with currency-issuing governments. A currency-issuing government faces no revenue constraint. If it voluntarily requires itself (a self-imposed constraint) to match its net spending with bond sales, it does so on its own terms. It cannot be cajoled by 'bond vigilantes' into paying high rates of interest on public debt against its will. Any interest it does pay is at its own discretion. This is in contrast to the weaker position of financially constrained currency-using governments, such as local, state and eurozone governments, who not only can be forced to pay higher rates of interest on their debt but must do so in a currency that they themselves do not issue. The absence of a revenue constraint in the case of a currency-issuing government means that it can initiate or encourage production along not-for-profit lines to the extent that this is feasible given resource constraints and political realities. In principle, sufficient grassroots pressure from the left could compel government to work toward socialism or be voted out of office, just as overwhelming political pressure from the right could compel a government to adopt capitalist-friendly measures or suffer the consequences at the next election.
The capacity of a currency-issuing government to initiate and foster activity that is not for profit opens the way for shaping the economy in whatever manner happens to reflect the collective will, at least as this is effectively expressed through unionism, activism, protest, voting behavior, lobbying and other means. Decisions over – and mechanisms for determining – what to produce, how to produce it, the mode of production, the kinds of enterprises in operation (public sector, corporate, small businesses, cooperatives, communes), the conditions under which production takes place (autocratic or democratic) and how to distribute the results of production are open to a democratic determination. MMT in itself is neutral on most such questions.
Even on questions where MMT does take a particular position – most notably, its emphasis on the job guarantee as a way of maintaining full employment alongside price stability – there is scope for widely diverging views on the appropriate design of the program. It is possible to see the job guarantee in a minimalist way, essentially as a means of keeping marginal workers job ready for capitalist employers when they are temporarily discarded from other employment. Someone of this view might prefer the job-guarantee wage to be set below the current minimum wage so as not to disrupt the existing wage structure. Others, in contrast, will want the job guarantee to provide a living wage plus defined benefits as part of a broader social wage so as to strengthen the position of workers relative to capitalists and force private-sector employers to improve pay and conditions or perish. And some also see a job guarantee as a mechanism for broadening, over time, the kinds of activities that are regarded as socially useful.
Taxes drive the currency. Government ensures a level of demand for currency by imposing taxes and other obligations (such as fees and fines) in it and requiring that final settlement only be conducted in the currency. The imposition and enforcement of taxes and other obligations denominated in the currency ensure that government can spend its currency into existence in exchange for goods and services.
The value of the currency is what must be done to obtain it. The reciprocal of the economy-wide average wage indicates the amount of labor-power that the average worker must supply to obtain a unit of the currency. Conversely, it indicates how much labor-power of average complexity capitalists can command with a unit of the currency. To the extent that government sets the wages for some kinds of labor (a minimum wage as well as a public sector pay scale) it influences other wages and therefore influences the value of the currency. It can exert more or less influence on currency value depending on how much authority it exercises over pay and conditions.
Government spending precedes tax payments. A currency-issuing government, whether capitalist or socialist, needs to issue the currency before non-government can use it to pay taxes or for other purposes. If the government issues bonds to match deficits, it must first provide the funds with which the bonds are to be purchased. Current practice somewhat obscures this, but the point is still clear on close inspection. Currently, the funds used to purchase newly issued bonds are created through central bank advances of reserves to primary dealers in so-called reverse repo operations. In exchange for the reserves, primary dealers must supply collateral in the form of already existing bonds, and these bonds required for collateral can only exist because of past government spending.
Government deficit equals non-government surplus. Basic accounting relationships apply irrespective of a society's politics. Funds created through government spending that have not been taxed back are held as financial assets by non-government. If some publicly provided goods and services are positively priced, purchases of them will function as tax payments. In a purely socialist economy, prices of publicly produced consumer goods would be set high enough that supply was not exhausted by the workers engaged in consumption-goods production to ensure that workers in the investment-goods sector as well as people outside the labor force could also purchase them. As under capitalism, government spending in excess of taxes would result in a financial surplus for non-government. The same accounting relationship would also apply in a socialist society that replaced currency with (non-transferable) labor certificates. Any unspent labor certificates would remain in the non-government sector until used to purchase publicly supplied output, at which point they would essentially be taxed out of the economy. If desired, the labor certificates could be time limited, in which case the expiration of a certificate would function like a tax payment or purchase.
Vertical and horizontal transactions. In MMT, transactions between government and non-government are considered vertical. These transactions directly alter the net financial assets of non-government. For instance, when government buys labor services with its currency (or labor certificates), members of non-government receive a financial asset (currency or labor certificates) that are not offset by a corresponding liability within non-government. Rather, the offsetting liability is in the government sector (currency or labor certificates on issue are a liability of government). In contrast, transactions between members of non-government are considered horizontal. They do not directly alter net financial assets because any financial asset created within the non-government sector will be offset by a liability within the same sector. A private bank loan, for example, is an asset of the bank and liability of the borrower, while the deposit created through lending is a liability of the bank and asset of the borrower. Overall, the financial impact on non-government nets to zero.
Monetary circuits. Under capitalism, the endogeneity of both net financial assets and private money means that the private monetary circuit M – C … P … C' – M' as conceived by Marx and later circuit theorists is fully consistent with MMT. A discussion of this point is provided in an earlier post. There will be two-way feedback between the private monetary circuit and public monetary circuit because private bank lending is influenced by the level of activity, which depends in part on exogenous fiscal policy settings, but also because the rate of reflux in the public monetary circuit (through payment of taxes) will vary endogenously as a result of private lending and spending decisions. Under socialism, with non-transferable labor certificates and no profit seeking, the situation would be modified. The government and non-government financial balances would still vary endogenously. But the monetary circuit would be much simplified, essentially beginning with government spending and perhaps public bank loans for social purposes, depending on institutional particularities, and ending with payment for publicly supplied goods and services, tax payments and, if applicable, expiry of time-limited labor certificates.
Job guarantee. As noted earlier, MMT indicates that full employment alongside price stability can be achieved through a job guarantee. In principle, it would be possible in either a capitalist or socialist society for government to maintain price stability through the use of direct wage and price controls (incomes policy), but even here full employment would most effectively be maintained through a job guarantee rather than reliance on trickle-down effects or hidden unemployment. For reasons of flexibility and efficiency, it might make sense for production levels in a socialist economy to be responsive to demand in much the same way as under capitalism. Unforeseeable or seasonal changes in the composition of demand would result in unforeseeable or seasonal changes in the need for various types of workers. Underemployed workers could transition temporarily to the job-guarantee sector. In a socialist system, this would not need to entail a pay cut.


May buy 1 for yourself and 1 additional as a gift. May use 1 per person. Online registration required. Must start classes before voucher expiration date. Expires 4 months after purchase. See the terms that apply to all deals.
For this course, no prior experience with the piano is required. Through easy-to-understand lessons, you can advance from a complete novice to the level of an early intermediate pianist. The course includes time-stamped video lessons, extensive lesson descriptions and detailed notes to help you along the way.
SkillSuccess provides quality online courses to help you improve your life. Whether you want to learn an instrument or a language, learn how to create your own website or how to salsa dance, there are dozens of online courses for you to choose from.


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Find Me Local
So, You Have MS. Now What
Grace Augustine | Hiawatha, Iowa
A concise in depth look at the autoimmune disease, Multiple Sclerosis. In this poignant account, Ms. Augustine chronicles her own journey with this disease and gives clinical explanation, as well as practical ways to deal with the symptoms and losses brought on by this disorder.
About Grace Augustine (Iowa City, Iowa Author)
Editor and award winning novelist, Grace Augustine, grew up in Montana. Her work career has taken her from title and abstracting to administrative assistant church work to co-owning a small town weekly newspaper to her current profession as a successful editor and author.
Several books in her Acorn Hills series are Indie Publishing award winners. This series is romance and life issues for the over 50 group. Grace currently has books published in Contemporary Romance, Metaphysical Romance, Prose/poetry, self help, and Christian Romance Suspense genres.
She was diagnosed with multiple sclerosis in 2003 and is a public speaker and advocate for autoimmune diseases, especially multiple sclerosis.
Mystic Capers
Sultry Sensations
Holiday Hide & Seek
Moonlight & Music
Richard's Relics


Facebook's Android and iPhone apps aren't great. The iPhone app has had bugs that cause it to drain battery in the background, and it could be using up to 20% of your battery on Android. What's more, Facebook once even reportedly made its Android app crash on purpose once.
Rather than put up with the awful app, you can use Facebook's fairly full-featured mobile site instead. Add the website to your home screen and you can launch it in one tap, just like the app. On Android, you can even get push notifications from Facebook via Google Chrome.
On the iPhone, you can uninstall the Facebook app by locating the Facebook app icon on your home screen, long-pressing it, and tapping the "x" to uninstall it.
Once you've done that, open the Safari web browser and head to facebook.com. Sign in with your Facebook account. Next, tap the "Share" button on the bar at the bottom of the screen—it's the icon that looks like a square with an up arrow coming out of it—and tap the "Add to Home Screen" icon in the bottom row of action icons.
Facebook will get its own icon on your iPhone's home screen, and you can quickly tap that icon to quickly open the Facebook website. As Facebook isn't installed as an app, it won't be able to run in the background and access your location, which is good—those things drain your battery. You can rearrange the Facebook icon and other app icons as normal. Just long-press it and drag it around to move it to other screens or change its position on the current screen. You could also place it it inside an app folder.
Sadly, the mobile site won't be able to display notifications on the iPhone. If you really want Facebook notifications, you can always enable email notifications on the Facebook website. You'll get your Facebook notifications emailed to you, and they'll appear in Mail or your email app of choice.
On Android, you can generally locate the Facebook app icon in your app drawer, long-press it, and drag it to a trash icon or something similar to uninstall it. This may work differently on different phones, depending on the customizations your manufacturer made to your phone's version of Android. If this doesn't work, open the Settings page, tap the "Apps" category, tap the "Facebook" app, and tap "Uninstall".
If there is no "Uninstall" button, it's likely because your manufacturer preinstalled it on your phone, and you aren't allowed to uninstall it. A "Disable" button should appear here instead, though; tap that to disable the app instead.
Now it's time to replace it with the mobile site. Open your preferred web browser. For simplicity's sake, we'll provide instructions for Google Chrome. This process will also work with other web browsers like Mozilla Firefox, but the option may be in a different place. Other browsers likely won't support push notifications like Google Chrome does, either.
Head to facebook.com in your web browser app and sign in. When you first visit the website, you'll be informed that Facebook wants to send you notifications. Tap "Allow" and you'll get Facebook notifications via Google Chrome.
Next, you'll want to add the icon to your home screen. In Chrome, tap the menu button and then tap "Add to Home screen." Other browsers will have similar options in their menus.
A Facebook icon will appear on your home screen alongside your app shortcut icons and widgets, allowing you to get to Facebook with a single tap. You can long-press the icon and drag it around to place it somewhere more convenient.
The only downside: The Facebook mobile site, much like the app, doesn't let you read your messages. Instead, it'll ask you to download Facebook Messenger app. You can, however, read your messages at the very basic mbasic.facebook.com mobile site if you really need to.
If you're looking for something with even more features than the mobile site, Metal for Facebook (and Twitter) is the way to go. It's basically a wrapper for the mobile site—this means it loads the mobile site within its own "shell" of sorts—but also provides and expands upon the features already found on Facebook Mobile.
If you've been using the Facebook Mobile site (or at least checked it out), then Metal is going to look very familiar. All the existing functionality of the mobile site is still in place, but Metal somehow makes it feel much more like a native app. For example, if you navigate to the notification tab and refresh the page, Metal will simply reload your notifications, where the mobile site will actually refresh the entire page and take you back to your feed. It's the little things, man.
Moreover, Metal adds one of Android's most useful features to Facebook: a side menu. This provides quick access to a slew of different pages with just a tap: search, news feed, your profile, friend requests, groups, pages, and a lot more. This in itself makes Metal so much nicer than just using Facebook's mobile site.
Along with enhancements to Facebook's mobile site itself, Metal also has its own settings. For example, Metal can generate Facebook notifications if you're into that sort of thing. It also has a nice security feature called "Metal Lock" that enables password verification before you can launch the app—it's worth noting that this is a separate password from Facebook. It also works with Nexus Imprint, Android's fingerprint system. It's pretty brilliant.
Beyond that, Metal offers themes to keep things looking clean, as well as a feature that I've personally grown to love: the Metal Bar. Basically, this is a persistent notification that gives quick access to your feed, friend requests, messages and notifications—best of all, it does so in a floating window! This means it leaves whatever app you're currently in running in the foreground and just display the Metal window on top of it. This is by far my favorite feature of Metal.
Finally, there are a handful of other Facebook-specific features found in Metal. The option to enable Facebook check-ins is there, as well as some choices for link control (open links in the app or in the browser). You can also block images for faster loading and lower data usage, as well as define which version of the site loads: mobile or desktop. The are options for "Facebook Basic" and "Facebook Zero" as well, though the latter is only available on specific carriers.
Almost all of Metal's features are completely free, but there is also a paid version of the app called Metal Pro that allows you to support the developer. It also adds the option for a Material Design theme, which is a nice bonus—and worth the $1.99 asking price in itself, honestly.
Sure, this is an old tip, but it's amazing how many people are still struggling with Facebook's app when there's such a better option available. The mobile site looks almost exactly like the app, and with modern mobile web browsers faster than ever, the Facebook website is pretty snappy. Switch to Metal on Android, and the experience gets even better. Add in the ability to send messages and (again, on Android) push notifications, and this is a killer solution.
This also works for other social media services that provide half-decent mobile websites, of course. You could use this same process for Twitter, for example—which is also supported by Metal on Android. Just sayin'.


Real Estate Multiple Listings and mortgage rate info for LU VERNE, IA Companies.
If we haven't found a LU VERNE, IA company's page yet, their phone number may be included.
Links in the end columns are links to companies working in LU VERNE who'd like your business.


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Jan 30, '2330 January, 2023 Jan 31, '2331 January, 2023 Feb 1, '231 February, 2023 Feb 2, '232 February, 2023 Feb 3, '233 February, 2023 Feb 4, '234 February, 2023 Feb 5, '235 February, 2023
3030 August, 2021 3131 August, 2021 11 September, 2021 22 September, 2021 33 September, 2021 44 September, 2021 55 September, 2021
66 September, 2021 77 September, 2021 88 September, 2021 99 September, 2021 1010 September, 2021 1111 September, 2021 1212 September, 2021
1313 September, 2021 1414 September, 2021 1515 September, 2021 1616 September, 2021 1717 September, 2021 1818 September, 2021 1919 September, 2021
2727 September, 2021 2828 September, 2021 2929 September, 2021 3030 September, 2021 11 October, 2021 22 October, 2021 33 October, 2021


The Oregon Lakes Association Board of Directors are elected from the General Membership on two-year, staggered terms. Elections are held at the annual conference, which is generally held in September through an open nomination process. The Board manages the financial affairs of the association as well as is responsible for putting on the conference, publishing the regular newsletter, and conducting outreach such as the scholarship program, the newly revised www.OregonLakes.org, as well as OLA's social media presence.
As the Board itself generally only meets in person once a year, the logistics of this work is largely handled via email and through teleconferences making a Board position readily accessible to members from all corners of the state.


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President Biden celebrates two years in office
@nathanking
Published January 21, 2023 at 3:04 PM
It's two years since U.S. President Biden was sworn into office, and he used the opportunity Friday to highlight what he sees as the many accomplishments he and his team have achieved over the last 700-plus days in the White House.
In reality, however, it has been a mixed picture of successes and setbacks, both domestically and abroad.
CGTN's White House correspondent Nathan King reports.
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Bar Hill Community Primary School (CB23 8DY) pupils and paying 12.8 p/kWh estimated .
Downham Feoffees Primary Academy (CB6 2ST) 210 pupils and paying 12.8 p/kWh estimated .
Ely College (CB6 2SH) 1298 pupils and paying 12.8 p/kWh estimated .
Lantern Community Primary School (CB6 2WL) pupils and paying 12.8 p/kWh estimated .
Nene Park Academy (PE2 7EA) 1129 pupils and paying 12.8 p/kWh estimated .
North Cambridge Academy (CB4 2JF) 750 pupils and paying 12.8 p/kWh estimated .
Sawtry Junior Academy (PE28 5SH) 260 pupils and paying 12.8 p/kWh estimated .
Sawtry Village Academy (PE28 5TQ) 1200 pupils and paying 12.8 p/kWh estimated .
Swavesey Village College (CB24 4RS) 1315 pupils and paying 12.8 p/kWh estimated .
West Town Primary Academy (PE3 6DD) 352 pupils and paying 12.8 p/kWh estimated .


"Every man hears only what he understands" (Goethe). Nowhere else, this is so true than in the Great Lakes region. For almost two decades, the Eastern part of DRC has been the epicentre of violence and seemingly never ending conflicts. While a lot of initiatives have been tried, numerous conferences took place for no avail either. People seem talking to but not understanding each other. A justified question is asking why peace seems so elusive that part of the country. The quick response will be that the conflict is complex.
In fact, there is no single conflict, no clear-cut front between good and evil. The reality is that there are multiple layers of contradictory interests that collude in a context of abject poverty and lawlessness. Several actors, willingly or not, help fuel the conflict while those who pay the brunt of this scramble for influence and money are left powerless. Who are be the main characters in this war of shadows?
The main actors are the different states of the region. Besides the government of DRC, which has interest to control his territory, all neighbouring countries are more or less, directly or not, implicated in the state of instability in the region. Access to resources has always been the principal cause of conflicts between states. Recent discoveries in the region just made this matter worse.
Apart from states, individuals or groups also benefit from the chaos. The porous nature of DRC's borders and the inexistence of the state made it possible for smuggling networks, fuelled by greed, to have a firm footing in Eastern DRC. Those networks naturally finance different armed groups to protect their trade. Militias become therefore organised gangs and almost the only alternative for youth employment.
Another category to consider is what we shall call "ignorant sponsors". That will include all those businesses or NGOs who pay 'taxes' to different militias in order to go ahead with their normal dealing. They are inadvertently financing the perpetuation of violence.
Last, but not least, are all those conflicts related to land, ethnicity, or simple neighbour quarrels that go out of proportion. That is where, in my view, the basis of all conflicts lays, the element that justifies the unthinkable. The conflict in the East, or more accurately, the conflicts, is seen as an ethnic struggle: saving the land from invaders or defending the oppressed minority from a homicidal majority. The reality though is far from such black and white view: Ethnic animosity is a product both of top-down manipulation, and of bottom-up tensions, fed by traumatic memories, resentments and embedded stereotypes (Stearns, 2013). So conflicts are fuelled by psychological and cultural factors, not necessary tangible facts. It's most likely 'perception'. How can we therefore help bring peace in the region?
First, it cannot be done through exclusively involving heads of states. Countries have interests not friends. Most of the time, those interests just serve an elite. Thus, peacebuilding should be done at the grassroot level.
Building peace in today's conflicts calls for long-term commitment to establish an infrastructure across the levels of a society, an infrastructure that empowers the resources for reconciliation from within that society and maximizes the contribution from outside. In short, constructing the house of peace relies on a foundation of multiple actors and activities aimed at achieving and sustaining reconciliation (Lederach, 1997).
In our case, there is a need to rewire our perception of things. We need to redefine what our values are, to re-educate the mass about what is right and what is wrong. There cannot be any reconciliation without justice. There cannot be justice without reparation to the victim. Reparative justice is better than repressive justice. The former ills, the latter just focus on punishment or vengeance.
Another thing to consider is that the destruction of the social fabric that in many places creates this situation of unbalance, where everything is tolerated. It is important to re-establish those structures that were important to maintain order and justice in our traditional societies. While rituals play a role here; the system of 'bami' is an important aspect as well. Whatever it pertains, it is important to create or reinvent structures that will allow people to vent their grievances in a peacefully matter and be answer in a just way. Festivals, tournaments, fairs, etc. are ways our ancestors devised to maintain peace and create relationships.
Peacebuilding is a difficult exercise. It requires a lot of responsibility and a great dose of accountability. For the moment, a lot of our 'elders' and 'bami' seem to be corrupted and don't fulfil their ancestral duty: protecting their people. I do not know if there is a structure in place to fill this vacuum currently. I heard there are a lot of local initiatives that yield good results but are not supported. Maybe it is time for those organisations to coordinate their actions in order to make a powerful impact. There are a lot of ideas that have been tossed around (cf. Borello, 2013) but so far not a lot of people are taking initiatives to implement a general plan that will bring people together and solve the big problem of youth unemployment.
It is important for local organisations to be relevant on the ground. As stated before, there is a need for coordination in order to maximise the impact in the region. Creating jobs for people will help reduce the attraction to enrol in militias. All this requires a coordinate effort of civil society in partnership with local businesses and maybe international organisations. There are a lot of options to explore. There are plenty of initiatives that need to be developing in a larger scale.
My father once told me that, "if you cannot see the sky anymore, you have to become the sky yourself". Peace in the East of DRC is achievable but we need to figure out how to neutralise those networks that have been built to exploit the existing chaos. It is up to the locals to find common grounds in order to live in peace. Peacebuilding is a process. It takes everybody's efforts and willingness for compromise to reach lasting harmony. Only by establishing structures that allow people to express their frustrations we will be able to live together. Only by creating opportunities for people to live a decent live we will break the cycle of violence.
Mbuku Ngomanseke is a logistician and computer information specialist living in the diaspora. He tweets at www.twitter.com/blaisengo.


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Home 2020 MAF plan to host Malaysian Open next month
MAF plan to host Malaysian Open next month
The Malaysian Athletics Federation (MAF) plan to host the Malaysian Open Athletics Championship in early January as a platform for national athletes' preparation ahead of several sporting events next year.
MAF President, Datuk SM Muthu said, the federation would continue to monitor the current situation of the COVID-19 pandemic before giving the "green light" to hold the championship.
He also disclosed that should the situation allows the tournament to be held, the body plans to invite several Asian countries to join the event.
However, he added, if the situation does not allow, the championship will be held internally only involving national athletes, reducing the risk of COVID-19 infection among athletes and officials.
MAF had previously organised an indoor tournament last month to assess the current performance of the national athletics squad following the absence of the 2020 tournament due to the pandemic.
It was previously reported that MAF planned to hold the Malaysian Open from December 11-13 as well as National Under Distance Trials on November 7, but both tournaments had to be cancelled following the spread of Covid-19 in the country.
The national athletics squad has not participated in any international tournaments since last March following the enforcement of the Movement Control Order (MCO).
Meanwhile, Muthu confirmed that the athletics body will get the doping test result of national marathon runner, Tan Huong Leong first before deciding whether the 2 hours 23 minutes and 13 seconds recorded by the Johor-born runner at the Taipei City Marathon Championship is considered a new national record.
Currently, they are still waiting for the result from the organiser which is expected to take about two weeks.
Tan reportedly set a new national record after finishing 6th overall in the full marathon event (42km) at the tournament recently, surpassing his own record of 2 hours 24 minutes and 04 seconds. – Oleh Siti Zafirah Kamal
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Third time the charm for Van Quyet


3 min read ✭ In this post, you will learn how #hashtags have the ability to broaden your reach and expand your audience! We're diving into the importance of using hashtags and how to find great ones to market your nonprofit. Check it out!
Hashtags are supported by most, if not all, social platforms. They have the ability to show your organization's posts to a wider audience, as whoever clicks a certain hashtag is then shown all content relating to it. For example, if your nonprofit wants to promote a fundraiser, you might take to Instagram to spread the word. Whatever hashtag you decide to include in the caption of your photo or video will allow your organization's content to be seen by not only people who are following you directly, but also people who click on a hashtag – many of which may not be following you at all. Using hashtags may even generate more interest in your organization! People who were previously unfamiliar with your nonprofit are now potentially interested because the hashtag they searched is relevant to your content. This is why it's important to not only use hashtags, but to use ones that pertain to your organization and its mission.
While hashtagging your posts might seem like a mindless addition to your content, there should be some strategy involved within your process. Additionally, you will definitely benefit from adhering to some best practices and making sure that you are nor overdoing nor under-doing it when it comes to using hashtags.
If users are spelling your hashtag wrong or are misremembering what your actual hashtag is, then people won't find your post. While your followers will still see most of what you post regardless, making hashtags difficult to remember may decrease your audience reach as less and less new people are able to see your posts. From a branding perspective, it's important to use words or phrases that are catchy and can easily be picked up.
A platform like Instagram yields better to using many hashtags versus a platform like Twitter where one or two strong hashtags are sufficient. Understanding the status quo of how people use whichever platform you are on is vital to your social media strategy. Knowing when to do more and when to cut back is essential, because you definitely don't want to come across spamming your followers with tons of hashtags. This also ties into the importance of using relevant hashtags on your organization's social media and content posts. The better the hashtags you use, the less you'll actually have to use. However, we'll dive into that in the next section.
If you create a hashtag unique to your nonprofit organization, you can't expect people to already be using the same hashtags you are using. Make sure to clearly communicate your hashtag effectively and share why someone would want to include it on their post. Again, this is more from a branding perspective, but it's definitely an important practice to note. The more eyes that see it, the more your organization is on top-of-mind.
If you want to optimize your posts' reach and increase its visibility, then it's important to look into what hashtags are already being used. That is honestly a must. Similar to the reason why you want to communicate your hashtag effectively, you can't just make something up and expect other people to happen to be using the same thing. Being intentional with your hashtag strategy is how you can maximize its effects. For starters, take a look into other causes or organizations who are rocking their social. What hashtags are they rocking? Take a note and go off that.
First, you should find trending hashtags in your niche. It's pretty vital to your social media strategy to identify which topics are trending, both in general and in your nonprofit's space. You can use the trends to your advantage. If tons of people are already searching or clicking those hashtags, that's a really great way for you broaden your reach. However, making sure the trending hashtag is somewhat relevant to your voice and your organization is just as important. Definitely don't just tag whatever is trending the most, but do look to see if you can use any to your advantage.
A great tool to help you identify relevant hashtags in the nonprofit space is hastagify.me. When you search specific terms, the search engine then identifies several related hashtags that are popular in relation to your search. For example, the most used hashtags relating to #Nonprofit are #philanthropy, #socialmedia, #nptech, #marketing, #hiring, #jobs, #fundraising, and #charity!
Another really cool resource is called Tagboard, which helps you to find relevant cross-platform hashtags. While some hashtags may only be pertinent to Twitter and others more popular on Instagram, Tagboard compiles conversations happening cross-platform. This is a fantastic way to find hashtags that are relevant across all platforms and not just one.
Additionally, finding relevant hashtags manually is really great too! As previously mentioned, doing your research is one of the best ways you will actually understand what conversation is happening throughout your nonprofit space. You can search a hashtag that you think might be relevant to your brand, and click on relevant and top-performing posts to see what other ones the different accounts are using. Looking into competitors and other similar causes in your space is a wonderful strategy as well.
Just test some different things out and you will see how much more viewership your posts will gain. Strategic hashtags are truly amazing and are something we implement in our social strategy every day!
What hashtags are you using? Do some perform better than others? Give us some feedback on how you think we're tackling social strategy. We'd love to see what you think! Tweet us at @CommunityBoost, follow us on Facebook, or leave a comment below. Don't forget to subscribe to our page and #follow us on social media to keep updated with daily content. We're really excited about what is coming and want you to see it all!


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What is a number?
When I considered what people generally want in calculating, I found that it always is a number.
Mohammed ben Musa al-Khowarizmi.
From The Treasury of Mathematics, p. 420
H. O. Midonick
Philosophical Library, 1965
The individual is what he is and has the significance that he has not so much in virtue of his individuality, but rather as a member of a great human community ....
From Ideas and Opinions, p. 15
Wings Books,
To paraphrase Albert Einstein, a number in and by itself has no significance and only deserves the designation of number by virtue of its being a member of a group of objects with some shared characteristics. The most common characteristic of numbers is that they can be added and multiplied to produce other numbers in their group. However, not all objects that can be added or multiplied are designated as numbers.
As a matter of fact, there are many different kinds of numbers.
rational and irrational
real and complex
algebraic and transcendental
hyperreal numbers
square and triangular numbers
Let's talk a little about each of these in turn.
Rational and Irrational numbers
A number r is rational if it can be written as a fraction r = p/q where both p and q are integers. In reality every number can be written in many different ways. To be rational a number ought to have at least one fractional representation. For example, the number
((√5 + 1)/2)2 + ((√5 - 1)/2)2
may not at first look rational but it simplifies to 3 which is 3 = 3/1 a rational fraction. On the other hand, the number √5 by itself is not rational and is called irrational. This is by no means a definition of irrational numbers. In Mathematics, it's not quite true that what is not rational is irrational. Irrationality is a term reserved for a very special kind of numbers. However, there are numbers which are neither rational or irrational (for example, infinitesimal numbers are neither rational nor irrational). Much of the scope of the theory of rational numbers is covered by Arithmetic. A major part belongs to Algebra. The theory of irrational numbers belongs to Calculus.
Using only arithmetic methods it's easy to prove that the number √5 is not rational. Just to remind, √5 stands for the number whose square equals 5. Thus Arithmetic can show that, when squared, no rational number gives 5. It's impossible to derive in Arithmetic that such a number actually exists.
Proof that for no rational number r = p/q, (p/q)2 = 5
Assume that a rational number r exists such that r2 = 5. In the representation r = p/q assume p and q are mutually prime, i.e. have no common divisors. The fraction p/q in this case is called irreducible. It would be a very legitimate question to ask whether every rational number has an irreducible representation. See if you know enough to answer this question.
Thus 5q2 = p2 so that p2 is divisible by 5. 5 is a prime number (it has no other divisors but itself and 1). Therefore I can use a theorem by Euclid (Elements, VII.30) (see also Ref [2]) to claim that since 5 is a factor of p2 it also divides p, i.e. p = 5s. Substituting this into 5q2 = p2 and dividing by 5 gives q2 = 5s2. By the same token, q is then divisible by 5 which makes the fraction p/q reducible. Contradiction.
(In fact for any integer n, which is not a square of another integer, √n is irrational. The above proof is commonly used to prove the irrationality of √2. This is because √2, being the length of the diagonal in a unit square, was the first number proved not to be rational. There is in fact great many ways to establish this result with various degrees of intuitive appeal.)
Intuitively the number √5 should exist. Indeed, apply the Pythagorean Theorem to the right-angled triangle with sides 1 and 2. Then (hypotenuse)2 = 12 + 22 = 5. Thus there exists a line segment whose length should be equal √5. For, otherwise, what else might it be equal to? As I already mentioned, it takes methods beyond Arithmetic to introduce irrational numbers. It is interesting to observe the characteristic unity of Mathematics as reflected in this example. In Arithmetic we pose and successfully solve a variety of problems until we unwittingly ask to solve a very innocently looking equation x2 = 5. However, Arithmetic proves powerless to solve the equation. Geometry suggests that a solution should exists. Calculus actually solves the problem.
As often the case in Mathematics, there are many ways to define irrational numbers. We'll use the one suggested by Georg Cantor (1845-1918), the father of modern Set Theory. Incidentally, Cantor's doctoral thesis was titled "In mathematics the art of asking questions is more valuable than solving problems." (see Ref [4])
Without giving a rigorous definition let me say that Cantor defined irrational numbers as limits of convergent sequences of rational numbers (see Ref [1]). The idea is quite intuitive. For simplicity, I'll handle only numbers between 0 and 1. Start with writing rational numbers as decimal fractions 1/5 = 0.2, 3/8 = 0.375, 1/3 = 0.333..., 1/7 = 0.142857. As you see, some rational numbers are written as finite decimal fractions (1/5 and 3/8). To write others will take an infinite sequence of digits. That's why we write ellipses in the decimal expansion of 1/3. The more 3s you write the closer the decimal fraction comes to 1/3 but no finite expansion will ever equal 1/3 exactly. In case of 1/7 brackets designate a period. In other words, the decimal expansion of 1/7 = 0.142857142857142857142857... - the period 142857 is repeated indefinitely. The expansion of 1/7 also requires an infinite sequence of digits. In the math parlance 1/7 is the limit of a convergent sequence
a1 = 0.1, a2 = 0.14, a3 = 0.142, ...
The sequence converges because |1/7 - an| < 10-n which only states an obvious fact that any fraction 0.... that starts with n 0s ought to be less than the fraction that has 1 in the nth position. But this is true for an arbitrary sequence of digits, not necessarily periodic. Thus Cantor studied such decimal expansions and observed that periodic expansions correspond to rational numbers whereas the non-periodic ones correspond (by his definition) to the irrational numbers. Please note that every sequence of digits corresponds to a convergent sequence of rational numbers and, therefore, represents a number, rational or irrational. (We'll talk of positive numbers. Any sequence of digits is represented by a nondecreasing sequence of rational number. The decimal portions of such sequences are bounded by one of the terms in the sequence .9, .99, .999, ... Therefore, the decimal portions have a limit bounded by .999... = 1.)
Lemma 1
Between any two different rational numbers a and b there is at least one other rational number.
Between any two different irrational numbers a and b there is at least one other rational number.
Between any two different rational numbers a and b there is at least one other irrational number.
Between any two different irrational numbers a and b there is at least one other irrational number.
In all four statements assertion of existence of a single number can be replaced with the assertion of existence of infinitely many numbers.
Proof of Lemma 1
a < (a + b)/2 < b and (a + b)/2 is rational.
Let decimal expansions of two irrational numbers a and b (a < b) first differ in the nth digit. Consider the number bn obtained from b by cutting all the digits after the nth. Then a < bn < b. Of course, being a finite decimal expansion, bn is a rational number.
Let the decimal expansions of two rational numbers a and b (a < b) first differ in the nth digit. Then, as before, consider bn: a < bn < b. Assume b has non-zero digits after the nth. Think of how to amend the proof if it does not. Let bm be the first non-zero digit of b after bn. Append to bn m - n zeros and then any random sequence of digits. Every time you append a digit you obtain a new rational number between a and b. The sequence is convergent to a limit between a and b. If you make an effort to avoid periodic sequences the limit will be irrational.
Will Rose suggested a different way of growing bn into an irrational number. Pick any number r known to be irrational. It could be the square root of 2, e, π or any other irrational number. Since the product of two rational numbers is rational, r·10M is bound to be irrational for any integer M. Take N so large as to make r·10-N smaller than bm10-m. Then use the digits of this number in lieu of the random selections as above.
(An Aside: Is number 1234567891011121314151617181920212223... periodic?).
The proof is a variation on the above.
Collectively, rational and irrational numbers are called real. Thus a real number is either rational or irrational. (In passing, Ian Stewart questions the wisdom of calling irrational numbers real: How can things be real if you can't even write them down fully?) From Lemma 1 it may appear that rational and irrational numbers play symmetric roles in the set of real numbers. One of the many Cantor's contributions was in showing that there are various kinds of infinities. While it is true that there are infinitely many rational numbers and infinitely many irrational numbers, in a well defined sense, the are more irrational numbers than rational.
Cantor began with the notion that counting or enumeration of objects brings finite sets into a 1-1 correspondence with finite segments of the set of integers. When you say you have two hands you mean that it's possible to assign the number 1 to one hand and the number 2 to another and no hands will be left uncounted. Right? Is it what you mean? Cantor applied the notion of the 1-1 correspondence to infinite sets such as the sets of integers, rational, irrational or real numbers.
With the notion of the 1-1 correspondence came quite a few surprises. (Actually the fact that handling infinities is wrought with surprises was already known to medieval thinkers, see Ref [3]. Check also Hilbert's paradox of the Grand Hotel.) For example, the sets of odd numbers, even numbers, complete squares or cubes, the set of integers greater than 1996 all can be brought into a 1-1 correspondence with the set of all integers (of which they are subsets.) Indeed, the following table summarizes how it's done.
Odd numbers
Even numbers
Greater than 1996
n 2n-1 2n n2 n3 n+1996
1 1 2 1 1 1997
3 5 6 9 27 1999
4 7 8 16 64 2000
... ... ... ... ... ...
Let N, E, and O denote the sets of all counting numbers, all even and all odd numbers, respectively. Then, on the one hand, N~E and N~O (where tilde signifies existence of a 1-1 correspondence). On the other hand, N~N = E+O. To many it may look like 1 = 2. But we can also adopt a more positive attitude.
I'll call two sets A and B equivalent if A~B. What we discovered was that infinite sets may be (and in fact are) equivalent to their own subsets and that the union of two equivalent (infinite) subsets is equivalent to either of them. Actually we may have gained an insight into the fundamental differences between finite and infinite sets. No finite set is equivalent to a proper subset of its own whereas every infinite is. This is known as the Dedekind's Theorem, but is often taken as a definition of infinite sets.
A set A is called countable (or enumerable, or denumerable, and, sometimes, even numerable) if A~N, i.e. if it's equivalent to the set of all integers. Our second discovery can be formulated as
Union of two countable sets is countable.
Let us denote the two sets A and B. We are given that A ~ N and B ~ N. Since N ~ E and N ~ O we get A ~ E and B ~ O. Therefore A + B ~ E + O = N.
It follows from Lemma 2 but also can be proven directly that the union of a finite number of countable sets is countable. A more exciting fact is given by
Union of a countable number of countable sets is countable.
For a proof please refer to a separate page.
A Cartesian product of two countable sets is countable. (Cartesian product of two sets A and B consists of pairs (a, b) where a ∈ A (a is element of A) and b ∈ B.)
Cantor's Set Theory is a fascinating topic in its own right but what was the purpose of swaying away from the discussion of various sets of numbers? Besides being useful in the forthcoming discussion on algebraic and transcendental numbers, there was a more immediate goal.
The set Q of all rational numbers is equivalent to the set N of all integers.
It's a direct consequence of Lemma 3. To get an enumeration of Q, write in the diagram above 1/1 instead of 11, 1/2 instead of 12 and so on. Put another way, the function F(m/n) = (m + n - 1)(m + n - 2)/2 + m enumerates all possible representations of positive rational numbers. But every rational number has infinitely many fractional representations. Therefore, Q is at most countable. It's clearly infinite. Proving that it's indeed countable takes a little more perseverance and I prefer to refer to Ref [5].
Another appealing proof can be found on a separate page. And a third one is based on enumeration of fractions on the Stern-Brocot tree.
The set Q of all rational numbers is equivalent to the set N of all integers. The set I of all irrational numbers is not countable.
The proof is a variant of what is known as the diagonal argument. Again I shall only consider numbers between 0 and 1. Irrational numbers have been defined as decimal (non-periodic) fractions. Assume it is possible to enumerate all such decimals. Let choose an enumeration and list the decimals in the corresponding order:
a1 = 0.a11a12a13a14...
where amn stands for the nth digit of the mth decimal. The approach I employ is Cantor's famous diagonal process. To remind, we made an assumption that all the decimals between 0 and 1 have been listed in the sequence above. I'll get a contradiction by showing that at least one decimal is missing from the list. The decimal b = 0.b1b2... is constructed a digit by digit. Select b1 to be any digit but a11. Select b2 to be any digit but a22. And in general, select bn to be any digit but ann. Then b can't equal any decimal an, n = 1, 2, 3, ... because b differs from a1 in the first digit; it differs from a2 in the second digit and so on.
(An absolutely delightful argument of a recent vintage the does not refer to the diagonal process is well worth looking into.)
From Lemma 1 it appeared that the distributions of rational and irrational numbers have much in common. Between any two numbers of one kind there always existed numbers of the same and of the other kind. However, rational numbers form a countable set whereas irrational form a set which is not countable.
To conclude the discussion of rational and irrational numbers:
There are infinite sets for which no 1-1 correspondence exists. There are various kinds of infinite sets. This topic belongs to the realm of the Set Theory and, more particularly, to the Theory of Transfinite Numbers (Cardinals).
The property expressed in Lemma 1 is known as density of both rational and irrational numbers on the straight line and, as such, belongs to the General Topology. Sets could be dense without being equivalent.
Expectations based on experience with finite sets should not be extrapolated to the sets infinite.
Algebraic and Transcendental numbers
Now let us consider polynomial equations with integer coefficients:
Pn(x) = anxn + an-1xn-1 + ... + a1x + a0 = 0
Real roots of such equations are said to be algebraic. In other words, a number a is called algebraic if it satisfies an algebraic equation: Pn(a)=0 for some polynomial Pn(x) with integer coefficients. Rational numbers and integers are all algebraic. Indeed, for a given rational number r = p/q take n = 1 and P1(x) = qx - p. Obviously P1(r) = 0. √5 is also algebraic for it solves the equation P2(x) = x2 - 5 = 0.
If an ≠ 0 then the polynomial is said to be of order n.
Polynomials of order 1 have two integer coefficients each, a1 and a0. It follows from Lemma 3 that the set of all polynomials of order 1 is countable. Directly or by induction, one can prove a general assertion. For every n the set of polynomials of order n is countable. The Fundamental Theorem of Algebra claims that a polynomial of order n has exactly n (perhaps complex) roots. Thus the set of all numbers that satisfy some equation of order n is countable. The subset of this set that consists of real numbers is as well countable. Applying once more Lemma 3 we get
The set of all algebraic numbers is countable.
Real numbers that are not algebraic are called transcendental. Thus the set R of all real numbers is the union of two sets - algebraic and transcendental. The set R is not countable while there are countably many algebraic numbers. Therefore the set of all transcendental numbers is not countable either. Given the difficulty of establishing whether a given number is algebraic or not, this was one of Cantor's early surprising results.
It's known that π and e are transcendental. e was proven to be transcendental by Hermite in 1873, and π by Lindemann in 1882. eπ is transcendental from the Gelfond's Theorem. At least one of πe and π + e is transcendental. It is not known if ee, ππ or πe are transcendental.
Note that although most of the real numbers are transcendental (since there is only countably many algebraic numbers), transcendentality is proven a number at a time.
Real and Complex numbers
The union of rational and irrational numbers forms the set of real numbers. Complex numbers are pairs c = (x, y) of two real numbers. Since the situations of having to handle two numbers instead of one is more complex than handling a single number, the terminology is well justified. An impetus for developing the theory of complex numbers came originally from unsolvability of a simple equation x2 + 1 = 0 in the set of real numbers. In the set of complex numbers the equation has two solutions ±i. The rest of the complex numbers could also be defined by adding this new number i to the set of reals and postulating that usual arithmetic operations (addition, subtraction, multiplication) apply to the expanded set and all the laws known to hold for these operations hold for the new set as well. Such an action is known as the closure with respect to the given algebraic operations. This is very satisfactory that without breaking existing laws and by adding a single new number we actually get a set of numbers in which every polynomial equation with real coefficients has a solution. This is known as the Fundamental Theorem of Algebra. Moreover, the theorem also holds for every polynomial with complex coefficients.
Complex numbers have unusual properties especially when it comes to differentiation. With real numbers differentiation is a starting point of Calculus and Real Analysis. With complex numbers it leads to the Analytic Function Theory.
As was just defined, a complex number c is a pair (x, y) of two real numbers. x in the pair is called the real part and y is the imaginary part of the complex number c. This notion of imaginary part is rather unfortunate for the following reason.
Complex number (x, y) is associated with a point (x, y) in the standard coordinate system in the plane. Points (x, 0) lie on a very real X-axis whereas points (0, y) lie on a no less real Y-axis. When it's desirable to emphasize the algebraic nature of complex numbers it's customary to write (x, y) = x + iy thus making distinction between the real part x and the imaginary part y more palpable. However, there are situations where the word "imaginary" furnishes the most suitable Description of the circumstances.
In the plane the distance between two points c1 = (x1, y1) and c2 = (x2, y2) is defined as dist(c1, c2)2 = (x1 - x2)2 + (y1 - y2)2. Which is quite natural due to the Pythagorean Theorem. Then the circle with radius r and center at c0 = (x0, y0) consists of all points c = (x, y) such that dist(c, c0) = r. In other words,
(1) (x - x0)2 + (y - y0)2 = r2.
This is an equation of a real (i.e. one that could be drawn with a compass) circle in a real (although consisting of complex numbers) plane. Now, mathematicians did not stop at a simple equation x2 = 1 but were curious enough to take matters further by changing the sign on the right-hand side. The new equation x2 = -1 turned out to be the source of much of the discoveries in the last two hundred fifty years. Continuing in their usual manner, mathematician did not stop at changing signs at the equation (1) either. Since, i2 = -1, they got
(2) (x - x0)2 + (y - y0)2 = (ir)2.
which is really an imaginary circle with a real center c0 but an imaginary radius ir. Getting this far, it must be said that the polynomial equation (2) has no real solutions (x, y) or, which is the same, no complex solutions c. However, if, for simplicity's sake, we consider a circle centered at the origin
(3) x2 + y2 = (ir)2
then it'll become obvious that equations (2) and (3) do have solutions. For example, the pair (0, ir) solves (3). This should probably be called a really imaginary number (see Ref [7]).
Perfect numbers
Every number n is divisible by at least 1 and n. The sum of all divisors of n is called s(n). If n = paqb..., where p, q ... are all primes, then s(n) = (1+ p + ... + pa)(1 + q + ... qb)... . n is perfect if it equals the sum of its divisors including 1 but excluding n. In other words, n is perfect if s(n) = 2n.
The numbers for which s(n) < 2n are known as deficient. For example, s(8) = 1 + 2 + 4 + 8 = 15 and s(15) = 1 + 3 + 5 + 15 = 24, so that 8 and 15 are deficient. The numbers for which s(n) > 2n are called abundant. 12 is such a number, because s(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28 > 2·12.
Euclid in his Elements, IX.36 proved that if, for a prime p, p+1 = 2k, then 2k-1p is perfect. Leonhard Euler (1707-1783) in a paper published posthumously, showed that every even perfect number has Euclidean form. Euler was absolutely blind for the last 18 years of his life. So it's likely this result was among those obtained by already blind Euler.
The first perfect number is 6 (= 1 + 2 + 3). 28 (= 1 + 2 + 4 + 7 + 14) is the second. Then come 496 and 8128. The fifth perfect number 33550336 has been calculated by Hudalrichus Regius (1536), not 500 hundred years ago.
Surreal numbers
Surreal numbers have been invented by John Conway who, among many other things, created the famous Game of Life. Donald Knuth's brochure Surreal Numbers gives a very popular but rigorous introduction into the development of surreal numbers successively starting with literally nothing. This is the only aspect of the theory that I intend to mention here.
A surreal number (see a short introduction or in relations to games) is a pair of sets {XL, XR} where indices indicate the relative position (left and right) of the sets in the pair. The first number to be constructed is 0 = {,} for which both left and right sets are empty. The rest of surreal numbers (included are the numbers we discussed so far and myriads of numbers some of which I have a difficulty imagining) are formed starting with 0 and applying just two simple rules. Truly, it's a creation of something from nothing.
Martin Gardner (Ref [9]) describes application of surreal numbers to the theory of games also developed by John Conway.
Every one knows how to compute a square of a given number especially if it's not too large. 22 = 4, 32 = 9, 42 = 16 and so on. Why this operation of multiplying a number by itself is called squaring? The reason is best grasped from the following picture.
A square with a side of length n can be visualized as comprising a grid of n×n smaller squares of size 1×1.
Triangular numbers owe their name to a similar construction, now of triangles. Actually we already used triangular numbers in the proof of Lemma 3 above. Triangular numbers are in the form 1, 3, 6, 10, 15, ... The general formula is that for every n the number N = n(n+1)/2 is triangular and describes the number of points arranged in a triangular shape with n points on a side as shown below.
Of course it needs to be proven that the total number of crosses arranged as in the diagram is given by the formula N = n(n+1)/2. However, we can observe that the number of crosses in every triangle equals the sum
1 + 2 + ... + n,
where n is the number of crosses on the side of the triangle. Thus, along the way, we are going to prove that
1 + 2 + ... + n = n(n+1)/2.
Another diagram will make this statement obvious
Indeed, two triangles, each with n crosses on the side, together form an n×(n+1) rectangle. You may question whether pointing to the diagram proves anything. No, of course by itself the diagram does not prove anything. But it does help to visualize and perhaps discern the proof. Indeed, we seek a formula for the sum of integers from 1 through n. This sum represents the number of crosses in a triangle. Two triangles of which one is rotated 180° form a rectangle, i.e. a shape in which all rows have the same number of crosses. Rows in the first triangle count crosses 1,2,3... in the increasing order whereas in the second we have rows with n, n-1, ... crosses in the decreasing order. Let us mimic this procedure in an algebraic manner.
First take two sums (one in increasing, another in decreasing order) (1 + ... + n) + (n + ... + 1). Now, regroup the terms by combining first terms in both sums, then second terms and so on:
[1+n] + [2+(n-1)] + [3+(n-2)] + ... + [(n-1)+2] + [n+1] = n(n+1)
Quite rigorously. Q.E.D.
It is obvious that all Euclid's perfect numbers are triangular.
J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag, NY, 1996.
R. Courant and H. Robbins, What is Mathematics?, Oxford University Press, 1966.
H. Davenport, The Higher Arithmetic, Harper&Brothers, NY.
W. Dunham, The Mathematical Universe, John Wiley & Sons, NY, 1994.
M. Gardner, Penrose Tiles To Trapdoor Ciphers, W.H.Freeman and Co., NY, 1989.
D. Knuth, Surreal Numbers, Addison-Wesley Publishing Co., 1974.
K. Kuratowski and A. Mostowski, Set Theory, North-Holand, Amsterdam, 1967.
M. Gardner, Time Travel and Other Mathematical Bewilderments, W.H.Freeman and Co., NY, 1988.
R. Rucker, Infinity and the Mind, Princeton University Press, Princeton, NJ, 1955
H. Schwerdtfeger, Geometry of Complex Numbers, Dover Publications, NY, 1979.
I.Stewart, Nature's Numbers, BasicBooks, 1995.
Abstract Algebra OnLine
Eric's Treasure Trove of Mathematics
Triangular numbers (an online and iPod video by Julio de la Yncera)
What's a number?
Irrational number to an irrational power may be rational
Bisector of an imaginary angle may be real
Distances from the square root of 2
Almost every integer has a digit 3 in it
Pi is a remarkable number
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