Imaginary Number

Imaginary Number

Definition. A number of the form $bi$, where $b\in\mathbb{R}$, $b\neq0$, and $i=\sqrt{-1}$, is called an imaginary number.

For example: $\sqrt{-25}=5i$, $2i$, $\dfrac{2}{7}i$, $-\sqrt{11}\,i$, and so on — in each case, the number is a real multiple of $i$.

Standing properties of $i$, used throughout the chapter:

1. $i\times0=0$ (multiplying by zero always gives zero, even for $i$).
2. If $a\in\mathbb{R}$, then $\sqrt{-a^2}=a\,i$ — the square root of a negative perfect square converts directly to a real multiple of $i$. (More generally, $\sqrt{-a}=\sqrt a\,i$ for any positive real $a$; this is the single most-used conversion in the chapter, turning any "square root of a negative number" into an ordinary real surd times $i$.)
3. If $a,b\in\mathbb{R}$ and $ai=bi$, then $a=b$ — imaginary numbers can be compared and cancelled just like their real coefficients, provided both sides are genuinely written as (real number)$\times i$. …