Virtual laboratory/Mathematics/Algebra/Complex numbers on the plane

Complex numbers on the plane

An extension of the topic of complex numbers. We picture the number $a+bi$ as a point in the plane (a real axis and an imaginary axis); $i^2=-1$, multiplication by $i$ is a 90° rotation, and multiplication by any number is a rotation combined with scaling. This geometric interpretation was given by Jean-Robert Argand (1806, the "Argand plane") and popularized by Carl Friedrich Gauss (1831) — it made the once "imaginary" numbers tangible.

Complex numbers on the plane
Mathematics · Algebra · Complex numbers on the plane (1806)
An operation on z=a+biz = a + bi — see its geometric meaning:
Multiplying a complex number by ii corresponds on the plane to…
number z
2 + i
result
3 + i
modulus |result|
3.162
argument
18.4°
🎯 Challenge: by how much does the multiplication rotate?
streak: 0

I will give you a multiplier w — guess by how many degrees it rotates the vector. When multiplying, angles add: the rotation angle = arg(w).

Complex numbers become tangible

Numbers of the form z=a+biz = a + bi, where i2=−1i^2 = -1, were long considered “imaginary” and suspect — until a geometric interpretation made them tangible. Jean-Robert Argand in 1806 (and earlier, unnoticed, Caspar Wessel in 1799) showed that a complex number can be represented as a point in the plane — hence the name “Argand plane”. Carl Friedrich Gauss popularized this interpretation (1831).

A point in the complex plane:

z=a+bi,i2=−1z = a + bi, \qquad i^2 = -1

The horizontal axis is the real part aa, the vertical axis the imaginary part bb. We treat zz as a vector from the origin to the point (a,b)(a, b).

The operations have simple geometry:

Addition is a translation of vectors. Multiplication is a rotation combined with scaling — and multiplying by ii is a pure 90° rotation, because i⋅(a+bi)=ai+bi2=−b+aii\cdot(a+bi) = ai + bi^2 = -b + ai, i.e. the point (a,b)(a,b) maps to (−b,a)(-b, a).

The invariant of multiplication:

∣z1z2∣=∣z1∣ ∣z2∣,arg⁡(z1z2)=arg⁡(z1)+arg⁡(z2)|z_1 z_2| = |z_1|\,|z_2|, \qquad \arg(z_1 z_2) = \arg(z_1) + \arg(z_2)

— when multiplying, the moduli multiply, and the arguments (angles) add. That is why multiplying by ii (modulus 1, argument 90°) rotates without changing length, while multiplying by w=1+iw = 1 + i (modulus 2\sqrt{2}, argument 45°) rotates by 45° and stretches the vector by 2\sqrt{2} times.

Why this matters for exams and beyond:

Thanks to this interpretation, complex numbers became a powerful tool in geometry, physics (alternating current, quantum mechanics), engineering and signal processing. They are an extension of the real numbers in which every operation has a geometric meaning.

Remember: in the plane, addition is a translation and multiplication is a rotation + scaling — multiplying by ii is a 90° rotation.

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Theory & formulas

A complex number as a point

We represent the number z=a+biz=a+bi as the point (a,b)(a,b) in the plane: the horizontal axis is the real part and the vertical axis the imaginary part. The imaginary unit satisfies i2=−1i^2=-1.

Modulus and argument

  • modulus (distance from 00): ∣z∣=a2+b2\lvert z\rvert=\sqrt{a^2+b^2}
  • the argument φ\varphi is the angle the ray 0z0z makes with the real axis
  • trigonometric form: z=∣z∣(cos⁡φ+isin⁡φ)z=\lvert z\rvert(\cos\varphi+i\sin\varphi)

Multiplication = rotation and scaling

Under multiplication the moduli multiply and the arguments add: ∣z1z2∣=∣z1∣ ∣z2∣,arg⁡(z1z2)=arg⁡z1+arg⁡z2\lvert z_1 z_2\rvert=\lvert z_1\rvert\,\lvert z_2\rvert,\qquad \arg(z_1 z_2)=\arg z_1+\arg z_2 Multiplication by ii (modulus 11, argument 90∘90^\circ) is a pure 90∘90^\circ rotation.

Notation: z=a+biz=a+bi a complex number, aa the real part, bb the imaginary part, ii the imaginary unit (i2=−1i^2=-1), ∣z∣\lvert z\rvert the modulus, φ\varphi the argument.

Frequently asked questions

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What does multiplying a complex number by i correspond to geometrically?

A 90° rotation of the point about the origin of the coordinate system.

›

How do we represent a complex number in the plane?

As the point (a,b): the horizontal axis is the real part a and the vertical axis the imaginary part b.

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What happens when complex numbers are multiplied?

Their moduli multiply and their arguments (angles) add: |z₁z₂|=|z₁||z₂|.

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