Multiplication of the Complex numbers

A process of multiplying two or more complex numbers to find their product is called the multiplication of the complex numbers.

Introduction

Let $a+ib$ and $c+id$ represent two complex numbers in mathematical form, where $a,$ $b,$ $c$ and $d$ are real numbers algebraically. The complex numbers often participate in multiplication. So, it is essential to learn how to multiply two or more complex numbers in mathematics to find their product. There is a procedure to find the product and it is called the multiplication of the complex numbers.

The two complex numbers are simply denoted by $z_1$ and $z_2$ respectively in mathematics.

$(1).\,\,\,$ $z_1 \,=\, a+bi$

$(2).\,\,\,$ $z_2 \,=\, c+di$

The multiplication of the complex numbers is expressed as $z_1 \times z_2$ and their product is written as $z_1 z_2$ mathematically.

$z_1 \times z_2$ $\,=\,$ $z_1 z_2$

$\implies$ $z_1 z_2$ $\,=\,$ $z_1 \times z_2$

Now, let’s learn how to multiply the complex numbers in mathematics.

$z_1 \times z_2$ $\,=\,$ $(a+ib) \times (c+id)$

Methods

There are two methods to multiply the complex numbers in mathematics.

Basic method

Learn how to multiply a complex number by another complex number fundamentally to find their product.

Shortcut method

Learn how to multiply the complex numbers in shortcut method to calculate the product of them.

Examples

List of problems to learn how to multiply complex numbers for getting product of them in mathematics.

$(1).\,\,$ Find $(1+2i)(3+4i)$

$(2).\,\,$ Evaluate $(5-4i)(7+5i)$

$(3).\,\,$ Calculate $(9-i)(2-7i)$