<s>[INST] How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$? How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$?
I got answer $\frac{1}{\sqrt{2}}$, but the real answer is $\frac{\sqrt{2}}{2}$. Anyway, calculator for both answers return same numbers.
 [/INST] The number $\frac{1}{\sqrt{2}}$ is defined to be the number such that, when you multiply it by $\sqrt{2}$, you get $1$. Symbolically, it is the solution to $x\cdot \sqrt{2}=1$. That's how division works - it's the inverse of multiplication. What is $\frac{\sqrt{2}}2\cdot \sqrt{2}$? Why, it's $\frac{2}2=1$ - so it must be $\frac{1}{\sqrt{2}}$, as it satisfies the definition of division for $\frac{1}{\sqrt{2}}$.
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