text: Associated Topics || Dr. Math Home || Search Dr. Math
  
  Decimal Square Roots
  
  Date: 10/19/98 at 19:44:43
  From: Ron Flitcraft
  Subject: Decimal answers in square roots
  
  Dr. Math,
  
  Recently my Algebra 2 teacher stated in front of the class that
  decimal points cannot be the answers to square roots. He said that any
  decimal point that is the answer for a square root will be an
  approximation. I doubted it and put a few numbers into my calculator.
  When my friend and I found a couple of numbers that didn't fill the
  number of the calculator space completely, our Algebra teacher admitted
  he was wrong.
  
  A couple of days later he put those same numbers in and multiplied them
  by themselves. They came close to but were not the exact number -
  omething like 46.999999999. I have devoted myself to finding a square
  root that comes out to a decimal point for an answer and checks when it
  is squared. Is there such a number? I have tried for quite some time
  
  Date: 10/19/98 at 20:01:25
  From: Doctor Sam
  Subject: Re: Decimal answers in square roots
  
  Ron,
  
  A very interesting question, but I'm not exactly sure what you mean.
  I am guessing that by "a decimal point" you mean a terminating decimal
  like .6 rather than a decimal with infinitely many digits. (Do you see
  my confusion? All decimals have "decimal points" but some decimals
  have a fixed number of digits to the right of the decimal point while
  others do not.)
  
  If you mean "a fixed number of decimal digits" then there are many.
  Here is an easy way to create such a number: pick a decimal and square
  it. So for example, (1.5)^2 = 2.25 and the square root of 2.25 is
  exactly 1.5
  
  But maybe you mean that you want to take the square root of a whole
  number, not a number with decimal digits to the right of its decimal
  point. If that is what you mean, then your teacher is correct: the only
  whole numbers that have a perfect and exact square root are numbers
  that are perfect squares, like 4, 9, 16, 25, 49. Any whole number that
  is not a perfect square will have infinitely many decimal digits in its
  square root. Your calculator can only display 9 or 10 digits and so
  rounds the answer. This gives a very good approximation of the square
  root, but it is only an approximation.
  
  For example, my calculator gives the square root of 2 as 1.414213562.
  Now if I enter (square root of 2)^2 the calculator gives 2. But if I
  take the displayed answer and square it I get:
  
  (1.414213562)^2 = 1.99999999999
  
  Whole numbers either have whole number square roots or their square
  roots are irrational. An irrational number has infinitely many digits
  in its decimal expansion, so no matter how accurate your calculator or
  computer is it will never be able to compute all those digits and will
  give you only an approximation.
  
  I hope that helps.
  
  - Doctor Sam, The Math Forum
  http://mathforum.org/dr.math/
  
  Associated Topics:
  Middle School Square Roots
  
  Search the Dr. Math Library:
  
   Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words
  
  Submit your own question to Dr. Math
  Math Forum Home || Math Library || Quick Reference || Math Forum Search