Consider this puzzle: What is the square root of $-1$? We know that $1 \times 1 = 1$. We also know that $(-1) \times (-1) = 1$. There is no real number that, when multiplied by itself, results in a negative number. Thus, $\sqrt{-1}$ cannot exist on the number line. [Page No. 64]

There is no real number whose square is $-1$, so $\sqrt{(-1)}$ does not sit anywhere on the real number line.

The reason is the sign rule from earlier in this chapter: a positive times a positive is positive, and a negative times a negative is also positive. So, any real number multiplied by itself gives a result that is zero or positive never negative. Hence, no real number squares to $-1$.

To handle this, mathematicians created a brand-new kind of number, written $i$, defined so that $i^2 = -1$. These are called imaginary numbers, and they open up a whole new dimension beyond the real line (the complex numbers). Despite the name, they are very real in use electrical engineering, signal processing and quantum mechanics all rely on them. For Grade 9, the key idea is simply that $\sqrt{(-1)}$ cannot live on the real line and needs this new number $i$.