Introduction
While learning about complex numbers, you might run into this question:
"Why does multiplying by i suddenly become a 90° rotation? Is this just something to memorize, or is there a real reason behind it?"
Here's the short answer — there is a reason. If you split a complex number into a magnitude and an angle, the product of two complex numbers multiplies the magnitudes and adds the angles. Since the imaginary unit i happens to have an angle of 90°, multiplying by i increases the angle by 90° — that is, it rotates by 90°.
A complex number is a point on a plane
A complex number is a number made of a real part and an imaginary part.
$$z = a + bi$$Here \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit, defined so that \(i^2 = -1\).
We can represent this complex number as a point on a plane. Put \(a\) on the horizontal axis (the real axis) and \(b\) on the vertical axis (the imaginary axis); then \(z = a + bi\) becomes the point at coordinates \((a,\, b)\). This plane is called the complex plane.
For example:
- \(z = 2\) → the point (2, 0) — on the real axis
- \(z = i\) → the point (0, 1) — on the imaginary axis
- \(z = 1 + i\) → the point (1, 1) — along the diagonal of the first quadrant
The modulus and argument of a complex number
A point on the plane can be described by two pieces of information.
The modulus (absolute value) \(r\) — the distance from the origin to the point. We find it with the Pythagorean theorem.
$$r = |z| = \sqrt{a^2 + b^2}$$The argument \(\theta\) — the angle measured from the positive real axis (to the right) around to the point. Counterclockwise is taken as positive.
Using these two values, we can write a complex number in polar form.
$$z = r(\cos\theta + i\sin\theta)$$"The point at distance \(r\) from the origin in the direction \(\theta\)" is exactly the complex number \(z\).
For instance, \(z = i\) is the point (0, 1), so:
- Modulus: \(|i| = \sqrt{0^2 + 1^2} = 1\)
- Argument: from the real axis (right) up to the imaginary axis (up) = 90°
Multiplication = multiply the moduli, add the arguments
What happens when we multiply two complex numbers \(z_1 = r_1(\cos\theta_1 + i\sin\theta_1)\) and \(z_2 = r_2(\cos\theta_2 + i\sin\theta_2)\)?
Expanding algebraically:
$$z_1 \cdot z_2 = r_1 r_2 \bigl[(\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2) + i(\sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2)\bigr]$$Applying the angle addition formulas — \(\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta\) and \(\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta\) — we see that these are exactly \(\cos(\theta_1+\theta_2)\) and \(\sin(\theta_1+\theta_2)\).
Therefore:
$$z_1 \cdot z_2 = r_1 r_2 \bigl(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\bigr)$$In one line, the key conclusion:
$$|z_1 \cdot z_2| = r_1 \cdot r_2 \qquad \arg(z_1 \cdot z_2) = \theta_1 + \theta_2$$The modulus (absolute value) is the product of the two moduli, and the argument is the sum of the two arguments.
Why multiplying by i is a 90° rotation
Now apply this directly to multiplying by i.
The modulus and argument of the imaginary unit i:
- Modulus: \(|i| = 1\)
- Argument: \(\arg(i) = 90°\)
So multiplying any complex number \(z\) by \(i\) gives:
- Modulus: \(|z| \cdot 1 = |z|\) — unchanged
- Argument: \(\arg(z) + 90°\) — exactly a 90° counterclockwise rotation
Because the modulus is 1, nothing grows or shrinks; because the argument is 90°, it turns by precisely that much. The rotation is not a coincidence — it is the inevitable result of this addition of angles.
A concrete example
Starting from the point 1 on the real axis, multiplying by i four times takes you all the way around and back.
Start: \(z = 1\), argument 0°
$$i \cdot 1 = i \quad\Rightarrow\quad 0° + 90° = 90°,\; \text{point }(0,1)$$$$i \cdot i = i^2 = -1 \quad\Rightarrow\quad 90° + 90° = 180°,\; \text{point }(-1,0)$$$$i \cdot (-1) = -i \quad\Rightarrow\quad 180° + 90° = 270°,\; \text{point }(0,-1)$$$$i \cdot (-i) = -i^2 = 1 \quad\Rightarrow\quad 270° + 90° = 360° = 0°,\; \text{point }(1,0)$$Four multiplications make one full turn (360°) back to the start. This is exactly why \(i^4 = 1\) holds.
The geometric meaning of \(i^2 = -1\)
We can now explain why \(i^2 = -1\) holds, geometrically.
Multiplying by i once rotates by 90°. Multiplying by i twice gives \(i^2\), which is 90° + 90° = a 180° rotation.
A 180° rotation sends a point to its reflection through the origin — every coordinate flips sign. Rotating the real number 1 by 180° sends it to −1. Therefore:
$$i^2 = -1$$This is not an arbitrary axiom. It is the algebraic expression of a geometric fact: "90° twice = 180° = sign reversal."
Try it yourself
The blue arrow is the complex number \(z\). Use the sliders to adjust its argument and modulus, and the buttons to choose the multiplier \(w\); the orange arrow (\(z \cdot w\)) updates in real time.
×i preset: confirm a 90° counterclockwise rotation with no change in size. ×(1+i) preset: watch a 45° rotation together with the modulus growing by a factor of \(\sqrt{2}\). ×2 preset: see the size double with no rotation. ×(−1) preset: confirm a 180° rotation = reflection through the origin.
The general rule for multiplying by any complex number
Not just i — multiplying by any complex number \(w = c + di\) works by the same logic.
Expanding algebraically:
$$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$$Writing this in matrix form:
$$\begin{pmatrix}c & -d \\ d & c\end{pmatrix} \begin{pmatrix}a \\ b\end{pmatrix} = \begin{pmatrix}ac - bd \\ ad + bc\end{pmatrix}$$The matrix \(\begin{pmatrix}c & -d \\ d & c\end{pmatrix}\) represents a rotation-and-scaling transformation.
- Angle of rotation: the argument of \(w\)
- Scale factor: \(|w| = \sqrt{c^2 + d^2}\)
Geometrically, complex multiplication is exactly the same as a linear transformation that "rotates by the argument of \(w\) and scales by the modulus of \(w\)."
Key takeaways
| Operation | Change in size | Change in angle |
|---|---|---|
| ×i | none (×1) | +90° counterclockwise |
| ×(1+i) | ×√2 | +45° counterclockwise |
| ×2 | ×2 | none |
| ×(−1) | none (×1) | +180° (reflection through origin) |
| ×(−i) | none (×1) | −90° clockwise |
Studying this with AI
The link between complex numbers and rotation deepens in many directions. Some questions worth asking an AI:
- "Explain why Euler's formula \(e^{i\theta} = \cos\theta + i\sin\theta\) holds, connecting it to the polar form."
- "Show me, side by side, that complex multiplication has the same structure as a rotation matrix."
- "Why does signal processing use complex exponentials of the form \(e^{i\omega t}\)? How are waves and rotation connected?"
Wrapping up
i is not a "nonexistent number." On the complex plane, i is "the unit arrow pointing 90° above the real axis." Multiplying by i becomes a 90° rotation for just two reasons — complex multiplication adds arguments, and the argument of i is 90°.
\(i^2 = -1\) is the natural consequence of this logic, and multiplying by any complex number reduces to a simple principle: "rotate by the argument, scale by the modulus." The reason electrical engineering's AC signals, signal processing's Fourier transform, and physics' waves and quantum mechanics all rely on complex numbers as a core tool is precisely this: the structure of "rotation and scaling" naturally expresses oscillation and wave phenomena.
Related reading — Read first: What is trigonometry actually for? · Why does matrix multiplication multiply so strangely? → Up next: Why does the dot product give cos?