The square root of 2, or the (1/2)th power of 2, written in mathematics as 👁 {\displaystyle {\sqrt {2}}}
or 👁 {\displaystyle 2^{1/2}}
, is the positive irrational number that, when multiplied by itself, equals the number 2.^[1] To be more correct, it is called the principal square root of 2, to tell it apart from the negative square root of 2 (👁 {\displaystyle -{\sqrt {2}}}
), which would also equal 2 if multiplied by itself.

Geometrically, the square root of 2 is the length of a diagonal across a square with sides with a length of one; this can be found with the Pythagorean theorem. Because of that, it is also called the Pythagoras's constant.^[2]

The number 👁 {\displaystyle {\sqrt {2}}}
is not rational. Here is the proof.^[3]

1. Assume that 👁 {\displaystyle {\sqrt {2}}}
   is rational. So there are some numbers 👁 {\displaystyle a,b}
   such that 👁 {\displaystyle a/b={\sqrt {2}}}
   .
2. We can choose a and b so that either a or b is odd. If a and b were both even, then the fraction could be simplified (for example, instead of writing 👁 {\displaystyle {\tfrac {2}{4}}}
   , we could write 👁 {\displaystyle {\tfrac {1}{2}}}
   instead).
3. If both sides of the equation are squared, then we get a² / b² = 2 and a² = 2 b².
4. The right side is 👁 {\displaystyle 2b^{2}}
   . This number is even. So the left side must be even too, which means that 👁 {\displaystyle a^{2}}
   is even. If an odd number is squared, then an odd number will be the result. And if an even number is squared, an even number would be the result too. So 👁 {\displaystyle a}
   is even.
5. Because a is even, it can be written as: 👁 {\displaystyle a=2k}
   .
6. The equation from the step 3 is used. We get 2b² = (2k)²
7. An exponentiation rule can be used (see the article) – the result is 👁 {\displaystyle 2b^{2}=4k^{2}}
   .
8. Both sides are divided by 2. So 👁 {\displaystyle b^{2}=2k^{2}}
   . This means that 👁 {\displaystyle b}
   is even.
9. In step 2, we said that a is odd or b is odd. But in step 4, it was said that a is even, and in step 7, it was said that b is even. If the assumption we made in step 1 is true, then all these other things have to be true, but since they disagree with each other they can not all be true; that means that our assumption is not true.

Therefore, it is not true that 👁 {\displaystyle {\sqrt {2}}}
is a rational number. So 👁 {\displaystyle {\sqrt {2}}}
must be irrational.

• Imaginary unit
• Mathematical constant
• Methods of computing square roots
• nth root
• Square number

1. ↑ "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-28.
2. ↑ Weisstein, Eric W. "Pythagoras's Constant". mathworld.wolfram.com. Retrieved 2020-08-28.
3. ↑ "Irrationality of the square root of 2". www.math.utah.edu. Archived from the original on 2023-06-05. Retrieved 2020-08-28.