If the square root of two is irrational, why can it be created by dividing two numbers? I'm doing pre-calculus and got a bit caught on this question&hellip; I looked online and it said that a number is rational if it can be the quotient of two integers. So I did this:



I know that it is irrational because if this was actually rational, mathematicians would have figured out. So my question is, why is the square root of two irrational?
Squaring the number you posted:

$$1.41421356237 \cdot 1.41421356237 = 1.9999999999912458800169$$

so the result is not $2$, so the number is not $\sqrt{2}$ (but rather a rational approximation thereof).
"Does that mean all irrational numbers are infinite?"

The irrational numbers are exactly those numbers that require infinitely many digits right of the radix point, no matter which base you are using. (Thanks Beanluc: Normally we say "decimal point" but that would obviously wrong in bases other than base 10, so the generic term is "radix point". )

A number like 1/3 needs infinitely many digits in base 2, 10, 16 but not in base 3, 6, 9, 12. $2^{1/2}$ needs infinitely many digits in every base.