COMPLEX NUMBERS

Breaking free from the 1D number line. Unlocking the 2D plane through the geometry of rotations and the square root of negative one.

01 // The Missing Unit

For centuries, mathematicians believed $\sqrt{-1}$ was an impossible paradox. Then they realized it wasn't a quantity at all—it was an operation. Multiplying by $i$ represents a perfect 90° rotation in 2D space.

i = \sqrt{-1}

i^1

= i

i^2

= -1

i^3

= -i

i^4

= 1

Real

Imaginary

i

1

-i

-1

02 // The Argand Diagram

2D Coordinates

A complex number $z = a + bi$ is not a point on a line. It is a coordinate in 2D space. The horizontal axis measures the Real component ( $a$ ), and the vertical axis measures the Imaginary component ( $b$ ).

Argand Visualizer

Complex Coordinate (z)

4+3i

Real Part (a)4

Imaginary Part (b)3

Modulus |z|

\approx 5.0

Angle θ

\approx 37^\circ

Real Axis (x)

Imaginary Axis (y)

03 // The Algebra of 2D Space

Addition

Treat $i$ just like any other variable ( $x$ ). Combine your Real terms together, and your Imaginary terms together.

(3+2i) + (1+4i) = 4+6i

Multiply

Expand using FOIL. The only trick is remembering that $i^2$ collapses back into a Real number ( $-1$ ).

(2i)(3i) = 6i^2

6(-1) = -6

Conjugate

Flip the sign of the imaginary part. Multiplying a complex number by its conjugate mathematically destroys $i$ , yielding a pure Real number.

z = a + bi

\bar{z} = a - bi

Elementary Algebra Completed

You have mastered the foundational systems of modern mathematics.

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