Complex Numbers and the Number Line

Why the one-dimensional number line is not big enough - adding a vertical imaginary axis to make the complex plane, reading a number position from its two parts, and writing it in polar form.

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The number line has one dimension: positives to the right, zero in the middle, negatives to the left. Every real number has a place on it, and there is no room for anything else. Complex numbers need more room — so we add a second axis and turn the line into a plane.

Concept From a line to a plane
1 + 2i real imaginary r
The horizontal axis carries the real numbers, exactly as the old number line did. The vertical axis carries the imaginary numbers. Together they make the complex plane.
Every complex number is written in Cartesian form:
where is the real part, is the imaginary part, and .

A real number is not excluded by this — it simply has and sits on the horizontal axis. The plane contains the old number line rather than replacing it.

Concept Reading a number's position

The two parts act like coordinates. The real part gives the horizontal step, the imaginary part gives the vertical step, and together they place the number in one of the four quadrants:

— first quadrant
a < 0,\; b > 0 — second quadrant
a < 0,\; b < 0 — third quadrant
a > 0,\; b < 0 — fourth quadrant

This is the same reading you already do with the point — the complex plane simply gives those two coordinates a new meaning.

Example Locating 1 + 2i

Identify the parts of and find its distance from the origin.

Step 1 — the real part:
Step 2 — the imaginary part:
Step 3 — both positive, so the number sits in the first quadrant, at the point
Step 4 — the distance:
⟹ first quadrant,

That distance is found exactly as it was for any point in the plane — Pythagoras on the two parts.

Theorem The polar form

Because a complex number is a point in a plane, it can also be described by a distance and an angle instead of two steps:

Here is the distance from the origin, and is the angle measured anticlockwise from the positive real axis.
Multiplying out recovers the two parts:

Anticlockwise is the positive direction for angles, exactly as in polar coordinates.

Example Polar back to Cartesian

A complex number has and . Write it as .

Step 1 — the real part:
Step 2 — the imaginary part:
⟹
Note The four special directions
Positive real — angle , for example
Positive imaginary — angle , for example
Negative real — angle , for example
Negative imaginary — angle , for example

Seen this way, multiplying by is a quarter turn anticlockwise. This is why the polar form makes multiplication, division and powers so much easier — the topic of the next lesson.

Summary
  1. The number line is one-dimensional; the complex plane adds a vertical imaginary axis.
  2. Cartesian form is z = a + bi, where a is the real part and b the imaginary part.
  3. The signs of a and b place the number in one of the four quadrants.
  4. The distance from the origin is r = √(a² + b²); for 1 + 2i it is √5 ≈ 2.24.
  5. Polar form is z = r(cos θ + i sin θ), with a = r cos θ and b = r sin θ.
  6. Angles are measured anticlockwise from the positive real axis: 0°, 90°, 180°, −90°.
283. Relations and Functions
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