# Multiplying Complex Numbers

The multiplication of complex numbers is realized by applying the distributive property and taking into account that **i**** ^{2} = −1**.

**(a + b i) · (c + di) = (ac − bd) + (ad + bc)i **

(5 + 2 *i*) · (2 − 3 *i*) =

= 10 − 15*i* + 4*i* − 6 *i*^{2} = 10 − 11*i* + 6 = 16 − 11*i*

### Polar Form

The** multiplication** of two** complex numbers** is another** complex number** such that:

The modulus is the product of the modules.

Its argument is the sum of the arguments.

6_{45°} · 3_{15° } = 18_{60°}

# Multiplication by a Complex Number of Module 1

**The multiplication of a complex number z = r _{α} for 1_{β} is rotated at angle β around the origin.**

r_{α} · 1_{β} = r_{α + β}

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