Question

What is the product of two complex numbers?

Answer 1

Complex numbers:

complex numbers are written in the form, $\mathrm{x}+\mathrm{iy}$ where $\mathrm{x}$ and$\mathrm{y}$ belongs to real numbers and $\mathrm{i}=\sqrt{-1}$ .

The product of complex numbers is also complex, meaning there will be real parts as well as imaginary parts.

Example.

Let, $\mathrm{a}=2+3\mathrm{i}$ and $\mathrm{b}=4+5\mathrm{i}$

Their product will be $\mathrm{a}\times \mathrm{b}$.

$\mathrm{a}\times \mathrm{b}=\left(2+3\mathrm{i}\right)\left(4+5\mathrm{i}\right)$

$\Rightarrow \mathrm{a}\times \mathrm{b}=2\left(4\right)+2\left(5\mathrm{i}\right)+3\mathrm{i}\left(4\right)+3\mathrm{i}\left(5\mathrm{i}\right)$

$\Rightarrow \mathrm{a}\times \mathrm{b}=8+10\mathrm{i}+12\mathrm{i}+15{\mathrm{i}}^2$

$\Rightarrow \mathrm{a}\times \mathrm{b}=15\left(-1\right)+22\mathrm{i}+8\left[i^2=-1\right]$

$\Rightarrow \mathrm{a}\times \mathrm{b}=-15+22\mathrm{i}+8$

$\Rightarrow \mathrm{a}\times \mathrm{b}=22\mathrm{i}-7$

Let, $\mathrm{a}=2-3\mathrm{i}$ and $\mathrm{b}=2+3\mathrm{i}$

Their product will be $\mathrm{a}\times \mathrm{b}$.

$\mathrm{a}\times \mathrm{b}=\left(2-3\mathrm{i}\right)\left(2+3\mathrm{i}\right)$

$\Rightarrow \mathrm{a}\times \mathrm{b}=2^2-{\left(3\mathrm{i}\right)}^2\left[\because a^2-b^2=\left(a+b\right)\left(a-b\right)\right]$

$\Rightarrow \mathrm{a}\times \mathrm{b}=4-9i^2$

$\Rightarrow \mathrm{a}\times \mathrm{b}=4+9\left[\because i^2=-1\right]$

$\Rightarrow \mathrm{a}\times \mathrm{b}=13$

Hence, the product of two complex numbers is sometimes complex and sometimes purely real.