# Complex number multiplication

Calculator nad formula for multiplying a complex number

## Multiplying a complex number

Complex number multiplication

 Input Multiplier + i Multiplicand + i Decimal places 0 1 2 3 4 6 8 10 Result Product

## Formula for multiplication of complex numbers

This paragraph describes how to multiply two complex numbers. As an example we use the two numbers $$3 + i$$ and $$1 - 2i$$. So it should be calculated

$$(3+i)·(1-2i)$$

According to the permanence principle, the calculation rules of real numbers should continue to apply. Therefore, we will first multiply the parenthesis as normal. So we write

$$(3+i)·(1-2i)=$$
$$(3·1)+(3·(-2i))+i+(i·(-2i))=$$
$$3-6i+i-2i^2$$

Besides expressions with $$i$$ the formula also contains $$i^2$$. We can easily replace this $$i^2$$. By the definition of $$i$$ we have $$i^2 = -1$$. So we replace $$i^2$$ by the number $$-1$$ and continue to calculate with the result from above as usual.

$$3-6i+i-2i^2=$$
$$3-6i+i-2·(-1)=$$
$$3-5i+2=5-5i$$

The result of the calculation is $$5 - 5i$$.

This article describes the multiplication of complex numbers in normal form. Easier to calculate is the multiplication of complex numbers in polar form.