# How do complex numbers apply in real life?

Apr 20, 2017

A few thoughts...

#### Explanation:

When you travel in a straight line, it makes sense to measure the distance you travel in the form of a single real number.

When considering rotations, then we often measure the angle as a single real number too, but what if you make one or more complete revolutions? You may not be interested in the total angle turned through, but just the resulting difference between the starting and finishing orientations. If that is so, then it may make more sense to think of the angle as a complex number of modulus $1$. Then rather than add angles, you simply multiply complex numbers.

More generally, any affine transformation in $2$ dimensions corresponds to a combination of complex number operations:

• Scaling: Multiplication by a real quantity.

• Rotation: Multiplication by a value on the unit circle.

• Reflection: Complex conjugate.

So many everyday circumstances where basic $2$ dimensional geometry is involved can be naturally expressed in the language of complex numbers.