im just wondering since irrational numbers supposedly dont end and dont repeat either, why is it not a possibility that after a huge bunch of numbers they all start over again or are only a single repeating digit.
In: Mathematics
ViHart on YouTube has a… Good explanation. I think it would be great if they just slowed down, but if you watch it on half speed, it’s better. The idea is to first assume that it is rational. If it is, it can be written as a/b where a and b are both whole numbers. In fact, all rational numbers have a simplest form. In the simplest form, they share no factors. Example 6/8 is not simplest form because both 6 and 8 are divisible by 2. Therefore 3/4 is the simplest form.
Basically, you can prove that a and b are even (if you start with the assumption that it’s rational). That’s a problem because a and b are the generalized form. Meaning you essentially proven that there is no simplest form. If you were to write √2 as a/b, a and b would have to always be even, forever, no matter how many times you divide by 2. It’s a paradox. When you encounter a paradox, you look at your assumptions. Since we only had one assumption, the work is quite easy from here. Our assumption must have been wrong.
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