The Number Whose Square Is Minus One, and Why Maths Cannot Do Without It

The equation $x^2+1=0$ asks for a number whose square is $-1$. No real number works, because every real square is zero or positive.

So we define a new number $i$ with

$$i^2=-1$$

and build complex numbers from it. With them, every quadratic equation has solutions. This chapter covers the form $a+ib$, arithmetic and powers of $i$, modulus and conjugate, and the Argand plane.

**A complex number is written $z=a+ib$, where $a$ and $b$ are real, $a$ is the real part and $b$ is the imaginary part; $i$ is needed so that quadratics with negative discriminants have roots.

Examples.**

- $z=3-4i$: $\mathrm{Re}(z)=3$, $\mathrm{Im}(z)=-4$
- $z=-2i$: $\mathrm{Re}(z)=0$, $\mathrm{Im}(z)=-2$
- $z=7$: $\mathrm{Re}(z)=7$, $\mathrm{Im}(z)=0$
- $\sqrt{-16}=\sqrt{16}i=4i$

Worked example. Solve $x^2+4x+13=0$.

$$x=\frac{-4\pm \sqrt{16-52}}{2}=\frac{-4\pm \sqrt{-36}}{2}=\frac{-4\pm 6i}{2}=-2\pm 3i$$

Check $x=-2+3i$: $(-2+3i{)}^2+4(-2+3i)+13=(4-12i-9)-8+12i+13=0$.

An everyday example. Electrical engineers designing power supply circuits use complex numbers to handle alternating currents, where voltage and current rise and fall out of step.

The substance. **The imaginary part of $3-4i$ is $-4$, not $-4i$**, and every real number is a complex number with imaginary part $0$.

**Add and subtract real and imaginary parts separately, multiply by expanding and using $i^2=-1$, divide by multiplying top and bottom by the conjugate of the denominator, and reduce powers of $i$ using $i^4=1$.**

Let $z_1=2+3i$ and $z_2=4-5i$.

$$z_1+z_2=6-2i{z}_1-z_2=-2+8i$$
$$z_1{z}_2=8-10i+12i-15i^2=23+2i$$
$$\frac{z_1}{z_2}=\frac{(2+3i)(4+5i)}{(4-5i)(4+5i)}=\frac{-7+22i}{41}$$

**Powers of $i$** repeat every four: $i,-1,-i,1$.

$$i^{35}=i^{32}\cdot i^3=-i{i}^{-39}=\frac{1}{i^{39}}=\frac{1}{-i}=i$$

Square roots of negatives.

$$\sqrt{-9}\times \sqrt{-16}=3i\times 4i=12i^2=-12$$

An everyday example. Turning a point on a map through a right angle about the origin is the same as multiplying the complex number for that point by $i$.

The trap. **$\sqrt{a}\sqrt{b}=\sqrt{ab}$ fails when both are negative**: $\sqrt{(-9)(-16)}=\sqrt{144}=12$, but the correct product is $-12$.

**For $z=a+ib$, the modulus is $|z|=\sqrt{a^2+b^2}$ and the conjugate is $\bar{z}=a-ib$; key properties are $z\bar{z}=|z{|}^2$ and $|z_1{z}_2|=|z_1||z_2|$.

Worked example 1.** $z=3+4i$.

$$|z|=\sqrt{9+16}=5\bar{z}=3-4iz\bar{z}=9+16=25=|z{|}^2$$

Inverse. Since $z\bar{z}=|z{|}^2$,

$$z^{-1}=\frac{\bar{z}}{|z{|}^2}=\frac{3-4i}{25}$$

Worked example 2 — product rule. $z_1=3+4i$, $z_2=1-i$.

$$z_1{z}_2=3-3i+4i-4i^2=7+i|z_1{z}_2|=\sqrt{50}=5\sqrt{2}$$
$$|z_1||z_2|=5\times \sqrt{2}=5\sqrt{2}$$

More properties: $z+\bar{z}=2\mathrm{Re}(z)$, $z-\bar{z}=2i\mathrm{Im}(z)$ and $\overline{z_1{z}_2}=\overline{z_1}\overline{z_2}$.

An everyday example. The strength of an alternating signal is given by the modulus of its complex representation, however the signal's timing is shifted.

The substance. **$|z_1+z_2|$ is usually not $|z_1|+|z_2|$** — modulus multiplies nicely but does not add.

**Represent $z=a+ib$ by the point $(a,b)$, with real parts on the horizontal real axis and imaginary parts on the vertical imaginary axis; the modulus $|z|$ is the distance of that point from the origin.

Worked example 1.** $z=-3+4i$ is the point $(-3,4)$ in the second quadrant, at distance $|z|=\sqrt{9+16}=5$ from the origin.

Worked example 2 — conjugate. $\bar{z}=-3-4i$ is $(-3,-4)$: **the reflection of $z$ in the real axis.

Worked example 3 — distance.** The distance between $1+2i$ and $4+6i$ is

$$|(4+6i)-(1+2i)|=|3+4i|=5$$

Worked example 4 — a locus. All $z$ with $|z|=2$ lie at distance $2$ from the origin: **a circle of radius $2$.

An everyday example. A delivery app showing a shop at $3$ km east and $4$ km north of you** is effectively plotting $3+4i$, and the straight-line distance is its modulus, $5$ km.

The substance. Complex numbers cannot be ordered: a statement such as $3+4i>2+i$ has no meaning, even though their moduli can be compared.