In daily life, we tend to think of numbers as perfectly rounded and workable quantities, but do you know that some numbers are not of this kind? Let’s explore what rational and irrational numbers are in this article.
In the last paragraph, we mentioned numbers that are perfectly expressible. They are called rational numbers and can be expressed as fractions of two integers. For example, all integers on the real number line are rational numbers, as any integer n is n / 1. If you go down the decimal places, all numbers you’ll find are rational numbers. That’s because they are still expressible as a specific integer divided by a certain power of 10. For example, the floating point number 3.1415926535 is rational because one can express it as 31415926535 / 10.
The world of rational numbers doesn’t only include integers and decimals with a finite number of places. In fact, some rational numbers have an infinite number of decimal places. Take 1/3 for an example. The answer is 0.333333……, and this series of numbers continue forever. Even though you cannot express it exactly in the base of decimal, you can always turn to fractions and mention that it’s 1/3. Therefore, it is a rational number as well. As a result, numbers with repeating decimal places, no matter where the cycle starts, are always rational numbers.
Rational numbers are all that you, your computer, a supercomputer, or anything in the observable universe can imagine. However, the vast majority of numbers are not in this category. They are irrational numbers, which are those that cannot be expressed as fractions of integers. This may seem weird, as you may think you can find a fraction with the precise value of the number in question as you approach that number. But the truth is, some numbers are proven to be inexpressible with the ratios of integers. Therefore, any way to store them in computer memory will result in approximations instead of exact values.
In fact, there is no way to precisely define the value of irrational numbers unless they are the result of an algebraic expression or geometric definition. For example, π is the circumference of a circle divided by its diameter, and √ 2 is the side length of a square of area 2 and also the answer to the equation x2 = 2. In fact, most well-defined irrational numbers are mathematical constants or the result of multiplying them by some rational number. That’s because most of the numbers between those constants are not useful, and thus are not worth defining (and may not have an exact definition).
In the previous paragraphs, we mentioned that it’s proven that some numbers are irrational. Well, here is one of the famous ones, proving that √ 2 is irrational. Suppose that √ 2 can be expressed as p / q in which p and q are coprime (i.e., whose GCD is 1).
Since q is an integer, 2p2 is divisible by 2. Thus, we can substitute p as 2x, so p2 = 2x2. Then, we can work down the fact and prove that q is divisible by 2 as well:
Thus, we can infer that q is divisible by 2 as well. Since p and q now have a common factor of 2, it cannot exist in its simplified form. Every fraction (x / y) can be simplified into one where x and y are coprime. Therefore, those two numbers that can make √ 2 rational does not exist, and √ 2 is therefore, irrational.
In this article, we’ve introduced rational and irrational numbers, how to define them, and how to know if a certain number is rational. Remember that, although the concept of irrational numbers may seem a bit complicated, they exist as provable facts. If we’ve missed anything we should have mentioned, please leave those points in the comments below.