What is an irrational number and where did it come from?
Lets start with the basics from a Ted-Ed video on "Making sense of irrational numbers:
The video teaches us that the irrational numbers were discovered because the Greek Hippasus was trying to measure the diagonal length of a square. When he tried to figure out the square root of 2, he realized that this could not expressed as a rational number or ratio of just 2 integers like Pythagoras was suggesting.
Rational means that the number can be expressed in ratios. Because this equation could not be expressed in ratios, then they needed to figure out how to label these types of numbers that don't fit the definition. Irrational numbers are those numbers which cannot be proven to be rational through division.
He proved that the square root of 2 could not be expressed by just 2 integers through this equation when solved using division. He kept getting a long number that was not whole when he was trying to solve the diagonal circumference of a square.
Lets dig deeper into this definition and define what is an integer.
An Integer is a whole number that is not a fraction. It is complete within itself.
So when trying to figure out the square root of 2, we always get a long number that is not a whole number hence the creation of irrational number.