A complex number z = a + bi carries two real numbers,
a and b. Two numbers — that is exactly
what a point in the plane needs. So picture z as the point with
coordinates (a, b): the real part runs along the horizontal axis,
the imaginary part up the vertical one. This picture is the
complex plane, or Argand diagram.
Suddenly complex numbers are geometry. The reals are the horizontal axis,
i sits one unit up at (0, 1), and
every complex
number is an arrow from the origin to its point.
How far out? The modulus
The most natural measurement of a point is its distance from the origin. For
z = a + bi that distance is the modulus
|z|, and the
Pythagorean theorem
hands it to us directly.
Step 1 — drop the right triangle. From the point
(a, b), the horizontal leg has length a
and the vertical leg has length b; the hypotenuse is the arrow to
z.
Step 2 — apply Pythagoras. The squared hypotenuse is the sum of the squared
legs:
|z|^2 = a^2 + b^2.
Step 3 — take the root (the modulus is a length, so non-negative):
|z| = \sqrt{a^2 + b^2}.
Notice this is exactly the z\bar{z} = a^2 + b^2 from the
conjugate — so |z| = \sqrt{z\bar{z}}. For example
z = 3 + 4i has |z| = \sqrt{9 + 16} = \sqrt{25} = 5.
Which direction? The argument
Distance alone doesn't pin down a point — you also need a direction. The
argument \arg z is the angle
\theta the arrow makes with the positive real axis, measured
anticlockwise. Together, the pair (r, \theta) — modulus and
argument — locates z just as well as (a, b)
does.
Step 1 — read the legs from the angle. In the right triangle, the adjacent
leg is r\cos\theta and the opposite leg is
r\sin\theta, where r = |z|. That is just
trigonometry on the triangle:
a = r\cos\theta, \qquad b = r\sin\theta.
Step 2 — rebuild z from (r, \theta).
Substitute into z = a + bi:
z = r\cos\theta + (r\sin\theta)\,i = r(\cos\theta + i\sin\theta).
This is the polar form. Going the other way — from
(a, b) back to (r, \theta) — use
r = \sqrt{a^2 + b^2}, \qquad \theta = \operatorname{atan2}(b, a).
(The \operatorname{atan2}(b, a) function is just
\arctan(b/a) done carefully, so it returns the correct angle in
all four quadrants rather than getting confused by signs.)
Represent z = a + bi as the point
(a, b) in the plane. Then:
-
z corresponds to the point
(a, b) (real part across, imaginary part up);
-
its modulus (distance from the origin) is
|z| = \sqrt{a^2 + b^2}, with
|z| = \sqrt{z\bar{z}};
-
its argument is \theta = \operatorname{atan2}(b, a),
and the polar form is
z = r(\cos\theta + i\sin\theta) with
r = |z|.
If z = a + bi is the point (a, b),
then a complex number is essentially a two-dimensional
vector
— and addition agrees perfectly. Adding componentwise,
(a + bi) + (c + di) = (a + c) + (b + d)i,
is exactly the tip-to-tail
vector
addition of (a, b) and
(c, d). The modulus |z| is the
vector's length, and the argument is its direction. So everything you know about adding
arrows transfers straight across.
What complex numbers add beyond vectors is a genuine multiplication
— and as the next page shows, that multiplication has a beautiful geometric meaning that
plain vectors lack.