Complex numbers

Complex numbers extend the real number system so equations like \(x^2+1=0\) can be solved. In high school, you usually write a complex number in Cartesian form: \[ z = a + bi, \] where \(a,b\in\mathbb{R}\) and \(i^2=-1\).

The imaginary unit \(i\)

Useful cycle: \[ i^1=i,\quad i^2=-1,\quad i^3=-i,\quad i^4=1, \] and then the pattern repeats every 4 powers.

Real and imaginary parts

For \(z=a+bi\):

• \(\Re(z)=a\) (real part)
• \(\Im(z)=b\) (imaginary coefficient)

Example. If \(z=3-4i\), then \(\Re(z)=3\), \(\Im(z)=-4\).

Basic operations

Add/subtract by combining like terms:

\[ (a+bi)\pm(c+di)=(a\pm c)+(b\pm d)i. \]

Multiply using expansion and \(i^2=-1\):

\[ (a+bi)(c+di)=ac+adi+bci+bd\,i^2=(ac-bd)+(ad+bc)i. \]

Example. \((2+3i)(1-4i)=2-8i+3i-12i^2=14-5i.\)

Conjugate and modulus

The conjugate of \(z=a+bi\) is \[ \overline{z}=a-bi. \] Multiplying by conjugates removes imaginary parts: \[ z\overline{z}=(a+bi)(a-bi)=a^2+b^2. \]

The modulus (distance from origin in Argand plane) is: \[ |z|=\sqrt{a^2+b^2}. \]

Example. For \(z=3-4i\), \(\overline{z}=3+4i\), \(|z|=5\).

Division in Cartesian form

To divide by \(c+di\), multiply top and bottom by \(c-di\):

\[ \frac{a+bi}{c+di} =\frac{(a+bi)(c-di)}{(c+di)(c-di)} =\frac{(a+bi)(c-di)}{c^2+d^2}. \]

Example. \(\dfrac{1+2i}{3-i}\):

\[ \frac{1+2i}{3-i}\cdot\frac{3+i}{3+i} =\frac{(1+2i)(3+i)}{10} =\frac{1+7i}{10} =\frac{1}{10}+\frac{7}{10}i. \]

Argand plane (graph view)

Plot \(z=a+bi\) as the point \((a,b)\): horizontal axis is real part, vertical axis is imaginary part. This lets you interpret \(|z|\) as geometric distance.

Why this matters: solving quadratics

If \(x^2+4=0\), then \(x^2=-4\), so \[ x=\pm 2i. \] Complex numbers ensure every quadratic has two roots (counting multiplicity), and later this extends to all polynomial equations.

History

Complex numbers first appeared when mathematicians tried to solve cubic equations in the 16th century. Algebraic formulas sometimes produced square roots of negative numbers, even when final answers were real.

Mathematicians like Cardano, Bombelli, and later Euler developed rules for these numbers. What began as a formal trick became a major mathematical system used in engineering, physics, and signal processing.

Derivation

Why the conjugate removes \(i\) in denominators

For \(c+di\): \[ (c+di)(c-di)=c^2-d^2i^2=c^2+d^2, \] which is real and positive (unless \(c=d=0\)). This is why multiplying by conjugates converts complex division into standard real-number division.

Why \(|z|^2=z\overline{z}\)

For \(z=a+bi\): \[ z\overline{z}=(a+bi)(a-bi)=a^2+b^2=|z|^2. \] So modulus is directly linked to conjugation and the Pythagorean distance in the Argand plane.

Checkpoints

Pause and reason before continuing. Discuss with a classmate or write your reasoning in a notebook.

Applications

AC circuits and impedance

Electrical engineers model alternating-current circuits with complex numbers because amplitude and phase are naturally represented together as one complex quantity.

Waves and signals

Sound, radio, and image-processing systems use complex numbers to describe oscillations and frequency content efficiently (e.g., Fourier methods).

Geometry and rotations

Multiplying by certain complex numbers corresponds to rotating and scaling points in the plane, making complex numbers useful in geometry and computer graphics.

References

• Bombelli, R. (1572). L'Algebra (early systematic rules for imaginary numbers).
• Euler, L. (18th century works) connecting complex numbers with trigonometry and exponentials.
• Standard high-school extension algebra curricula on Cartesian form, conjugates, and division.