Mathematics is a subject that so many students struggle with that online math tutoring services have exploded in popularity over the last decade. Young math students in middle school and high school sometimes feel like they will never understand the topics they learn about day-in day-out at school. Turning to professional tutors is definitely a step in the right direction for anyone struggling with complex topics such as geometry.
Any math students that are currently studying geometry will be familiar with the topic of this article, the Father of Geometry himself: Euclid. This Ancient Greek mathematician devised a groundbreaking algorithm that has stood the test of time and continues to be a vital tool in modern mathematics.
Let’s take a closer look at Euclid’s Algorithm, see what exactly it’s about, and try to understand this algorithm’s longevity which has made it one of the most famous and commonly-used geometric algorithms studied in middle school nowadays.
What is Euclid's Algorithm?
Put simply, Euclid’s Algorithm is a straightforward but highly powerful method used in mathematics to find the Greatest Common Divisor (GCD) of two positive integers. The GCD of two numbers is the biggest positive integer that divides both of these numbers individually without leaving a remainder. Euclid’s Algorithm has a fascinating history. The algorithm’s roots are deep and can be traced back to Euclid’s incredible piece of work entitled, “Elements,” which dates back to about 300 BCE. It is considered one of the oldest, most elegant, and easy-to-understand algorithms in mathematics.
Euclid's Algorithm Explained
Now that we know what Euclid’s Algorithm is used for, let’s dive into the operations involved. The algorithm follows a recursive process – this is what makes it so easy to understand and apply to problems in math class. If we want to find the GCD of two numbers, let’s call them a and b, where a is greater than or equal to b, the steps we must follow are:
- Divide the bigger number, a, by the smaller number, b.
- If the remainder, let’s call it r, is zero, then the GCD of a and b is b.
- If the remainder is not equal to zero, replace a with b and b with r, and repeat the process from step 1.
- The algorithm ends when the remainder becomes zero, and the current value of b represents the GCD of the original numbers a and b.
To make sure we’re all on the same page with the process, let’s lay out an example using real numbers, as opposed to labeling them. We want to find the GCD of 56 and 84 using Euclid’s Algorithm.
- We divide the bigger number by the smaller number: 84 ÷ 56 = 1 with a remainder of 28
- Next, we replace the bigger number with our original smaller number, and the smaller number with our remainder: 56 ÷ 28 = 2 with a remainder of 0
- Now that the remainder is zero, we have the GCD of 56 and 84.
In this case, our answer, GCD of 56 and 84, is 28.
Why Euclid's Algorithm Matters
Euclid’s Algorithm has several essential qualities that make it a significant algorithm in the history of mathematics, and it has a host of practical applications in today’s day and age. The qualities that make Euclid’s Algorithm stand out are its simplicity, its time efficiency, and the fact that it’s a basis for more advanced algorithms.
This algorithm is very easy to understand, as demonstrated by the fact that students learn about it in middle school. This makes it accessible to students of varying levels and ages, regardless of any gaps in their mathematical education and knowledge.
Euclid’s Algorithm is extremely time efficient, even when we are dealing with finding the GCD of very large numbers, and Its recursive nature minimizes that the number of steps required to find the GCD.
Finally, as mentioned, Euclid’s Algorithm lays the foundation for a host of more advanced algorithms in various fields, such as number theory, cryptography, and computer science. Understanding this foundational algorithm allows for further exploration of complex mathematical concepts.
Euclid’s Algorithm represents a mathematical concept that is truly timeless. By giving us a simple, straightforward, repeatable algorithm to use to find the GDC of two positive integers. Its simplicity and efficiency will help students grasp not just this and related topics, but also much more complex mathematical concepts and algorithm as they progress in their mathematics education.