Numbers are basics to our life. We need numbers to count and measure things. For example, I am sure you would have seen a tailor measure and a cut piece of cloth when you go to stitch your dress.
So, let’s take a line segment given below:
Mathematically, we say the line segment above is measurable if we can express the length in some units say m ⁄ n inches, where both m and n are positive integers.
For example, if the length is say 14 and half inches, then we can say it is 29 ⁄ 2 inches in fraction where m is 29 and n is 2.
Now, any number that we can express in the form of m ⁄ n, where both m and n are integers, is a Rational Number. To be more specific, the m and n should not have a common factor (except number 1) in their simplest form.
Ok. I hear you. You want to know what a Common Factor is.
- I will give an example:
Take the numbers 5 ⁄ 10, and 25 ⁄ 50. You can reduce both of them to their simplest form as ½. Now if you take 1 and 2, they do not have a common factor.
That is 1 and 2 are not divisible by a same number other than 1. So they don’t have a common factor.
In the example 5 ⁄ 10, both 5 and 10 are divisible by 5. So they have a common factor: 5 (other than 1). But in their reduced form, ½, there is no common factor for 1 and 2 other than 1.
Okay, now that you have understood what a Rational Number is, we will go back to our question: Are all things measurable? Or to be precise, are all line segments measurable the way we explained above with Rational Numbers?
Before we check that, a short digression but something you need to know before we answer that question:
Just to add some interesting history to the numbers, the Mathematics of Pythagoreans had only Rational Numbers.
I am sure you would not ask me who the Pythagoreans are. Hope you still remember the famous Pythagorean Theorem that you studied in your school days
For those who scratch their head here it is:
In any right angle triangle the equation a2+ b2 = c2 relates to the sides of the triangle where c is the length of the hypotenuse and a and b are lengths of the other two sides.
Pythagoras who gave us the above famous theorem lived during the 6th Century B.C. He and his followers believed in the philosophical principle that Mathematics is the basis of everything in the Universe and everything, from music to movements of planets, can be explained with numbers.
Okay, I hear you. You want to know what has Pythegorean Theorem got to do with Real and Rational Numbers. Read our next Part to understand that.
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