The Imaginary Unit and the Complex Number System

This Maharashtra Std XI (Commerce) Mathematics and Statistics chapter extends the number system beyond the real numbers so that every quadratic equation has a solution. The chapter draws on the same standard, well-established treatment of complex numbers used in mathematics curricula nationally.

Why we need a new number

A simple equation such as $x^{2}+1=0$ has no real solution, because $x^{2}\ge 0$ for every real number $x$, so $x^{2}+1$ can never be $0$. To close this gap we define a new number whose square is $-1$.

The imaginary unit $i$

The imaginary unit is the number $i$ defined by

$i=\sqrt{-1},\qquad\text{equivalently}\qquad i^{2}=-1.$

With $i$ available, $x^{2}+1=0$ gives $x^{2}=-1$, so $x=\pm i$.

What a complex number is

Complex number

A complex number is any number of the form

$z=a+bi,\qquad a,b\in\mathbb{R},$

where $a$ is the real part, written $\operatorname{Re}(z)=a$, and $b$ is the imaginary part, written $\operatorname{Im}(z)=b$. The set of all complex numbers is denoted $\mathbb{C}$.

Note that the imaginary part $b$ is itself a real number — it is the coefficient of $i$, not $bi$. Every real number $a$ is also complex, since $a=a+0i$; and a number of the form $bi$ with $b\ne 0$ (real part $0$) is called purely imaginary.

Equality of complex numbers

Square root of a negative real number

Using $i=\sqrt{-1}$, for any positive real $k$ we write $\sqrt{-k}=\sqrt{k}\,\cdot i$. For example $\sqrt{-9}=3i$ and $\sqrt{-5}=\sqrt5\,i$.