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Complex Numbers

See imaginary numbers become a real geometric plane.

Follow the mechanism1 / 3

A number has two components. The real part gives a horizontal coordinate and the imaginary part gives a vertical coordinate.

complex plane|z| = 3.61 · arg 56°
Focus the idea
A complex number is a point and vector on the complex planeThe complex number z equals 2 + 3i has real part 2, imaginary part 3, modulus 3.61, and argument 56 degrees. The real part is horizontal and the imaginary part is vertical.realimaginaryz = 2 + 3ii·z = −3 + 2i+90°z = a + bi
real part a2
imaginary part b3
modulus |z|3.61
i·z−3 + 2i

The real part moves z horizontally and the imaginary part moves it vertically. Together they locate one point: 2 + 3i.

The complex plane is a representation of numbers of the form a + bi. It makes i² = −1 visible as a quarter-turn applied twice.