# What are complex numbers?Thanx.

Feb 9, 2016

Complex numbers are numbers of the form $a + b i$ where $a$ and $b$ are real numbers and $i$ is defined as $i = \sqrt{- 1}$.

(The above is a basic definition of complex numbers. Read on for a little more about them.)

Much like how we denote the set of real numbers as $\mathbb{R}$, we denote the set of complex numbers as $\mathbb{C}$. Note that all real numbers are also complex numbers, as any real number $x$ may be written as $x + 0 i$.

Given a complex number $z = a + b i$, we say that $a$ is the real part of the complex number (denoted $\text{Re} \left(z\right)$) and $b$ is the imaginary part of the complex number (denoted $\text{Im} \left(z\right)$).

Performing operations with complex numbers is similar to performing operations on binomials. Given two complex numbers ${z}_{1} = {a}_{1} + {b}_{1} i$ and ${z}_{2} = {a}_{2} + {b}_{2} i$

${z}_{1} + {z}_{2} = {a}_{1} + {b}_{1} i + {a}_{2} + {b}_{2} i = \left({a}_{1} + {a}_{2}\right) + \left({b}_{1} + {b}_{2}\right) i$

${z}_{1} - {z}_{2} = {a}_{1} + {b}_{1} i - \left({a}_{2} + {b}_{2} i\right) = \left({a}_{1} - {a}_{2}\right) + \left({b}_{1} - {b}_{2}\right) i$

${z}_{1} \times {z}_{2} = \left({a}_{1} + {b}_{1} i\right) \left({a}_{2} + {b}_{2} i\right)$
$= {a}_{1} {a}_{2} + {a}_{1} {b}_{2} i + {a}_{2} {b}_{1} i + {b}_{1} {b}_{2} {i}^{2}$
$= {a}_{1} {a}_{2} + {a}_{1} {b}_{2} i + {a}_{2} {b}_{1} i - {b}_{1} {b}_{2}$ (remember $i = \sqrt{- 1}$)
$= \left({a}_{1} {a}_{2} - {b}_{1} {b}_{2}\right) + \left({a}_{1} {b}_{2} + {a}_{2} {b}_{1}\right) i$

${z}_{1} \div {z}_{2} = \frac{{a}_{1} + {b}_{1} i}{{a}_{2} + {b}_{2} i}$
$= \frac{\left({a}_{1} + {b}_{1} i\right) \left({a}_{2} - {b}_{2} i\right)}{\left({a}_{2} + {b}_{2} i\right) \left({a}_{2} - {b}_{2} i\right)}$
$= \frac{\left({a}_{1} {a}_{2} + {b}_{1} {b}_{2}\right) + \left({a}_{2} {b}_{1} - {a}_{1} {b}_{2}\right) i}{{a}_{2}^{2} + {b}_{2}^{2}}$
$= \frac{{a}_{1} {a}_{2} + {b}_{1} {b}_{2}}{{a}_{2}^{2} + {b}_{2}^{2}} + \frac{{a}_{2} {b}_{1} - {a}_{1} {b}_{2}}{{a}_{2}^{2} + {b}_{2}^{2}} i$

For division, we used the fact that $\left(a + b i\right) \left(a - b i\right) = {a}^{2} + {b}^{2}$. Given a complex number $z = a + b i$ we call $a - b i$ the complex conjugate of $z$ and denote it $\overline{z}$ It is a useful property (as seen above) that $z \overline{z}$ is always a real number.

The complex numbers have many useful applications and attributes, but one which is often encountered early is their use in factoring polynomials. If we limit ourselves to only real numbers, a polynomial such as ${x}^{2} + 1$ cannot be factored further, however if we allow for complex numbers, then we have ${x}^{2} + 1 = \left(x + i\right) \left(x - i\right)$.

In fact, if we allow for complex numbers, then any single-variable polynomial of degree $n$ may be written as the product of $n$ linear factors (possibly with some being the same). This result is known as the fundamental theorem of algebra, and, as the name indicates, is very important to algebra and has broad application.