1.12 Complex numbers in Cartesian form

The real numbers are not enough to solve every polynomial equation: no real number squared gives $-1$, so $x^2 + 1 = 0$ has no real solution. Complex numbers extend the real numbers with one new number, $\mathrm{i}$, whose square is $-1$. With this single addition, every quadratic equation has solutions, and (as later results show) so does every polynomial equation. Complex numbers are also a working tool outside pure mathematics: electrical engineers use them to describe alternating current, and they appear throughout signal processing and quantum physics.

The imaginary unit

For every real number $x$, $x^2 \ge 0$. A negative number therefore has no real square root, and an equation such as $x^2 = -9$ has no real solution. Complex numbers are built by adding one number to the real numbers that removes this obstacle.

Once $\mathrm{i}$ exists, every negative number has square roots. For $a > 0$, the number $\sqrt{-a}$ is defined as $\mathrm{i}\sqrt{a}$

Powers of i

Multiplying by $\mathrm{i}$ repeatedly produces a cycle of four values:

After $\mathrm{i}^4 = 1$, the cycle starts again: $\mathrm{i}^5 = \mathrm{i}^4 \times \mathrm{i} = \mathrm{i}$

The cycle has a geometric meaning. On an Argand diagram the four values $1, \mathrm{i}, -1, -\mathrm{i}$ sit on the axes, each a quarter turn anticlockwise from the one before, so multiplying by $\mathrm{i}$ acts as a rotation through $90°$ about the origin. This rotation idea is developed fully in 1.13.

Negative powers follow the same pattern, since $\mathrm{i}^{-1} = \frac{1}{\mathrm{i}} = \frac{\mathrm{i}}{\mathrm{i}^2} = -\mathrm{i}$

The four values in the cycle add up to zero: $\mathrm{i} + (-1) + (-\mathrm{i}) + 1 = 0$. Any four consecutive powers of $\mathrm{i}$ therefore also add up to zero, which makes long sums of powers quick to evaluate.

Complex numbers in Cartesian form

Combining real numbers with real multiples of $\mathrm{i}$ gives a new, larger set of numbers.

For $z = 4 - 7\mathrm{i}$, $\mathrm{Re}(z) = 4$ and $\mathrm{Im}(z) = -7$

Two special cases have their own names. When $b = 0$, the number $z = a$ is real, so every real number is also a complex number: $\mathbb{R}$ is a subset of $\mathbb{C}$, and the chain of number sets becomes $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$

Equality of complex numbers

Two complex numbers are equal when, and only when, their real parts are equal and their imaginary parts are equal:

The reason is that if $a + b\mathrm{i} = c + d\mathrm{i}$ then $a - c = (d - b)\mathrm{i}$. The left side is real. If

This fact turns one equation in complex numbers into two equations in real numbers. It is the basis of many methods for solving complex equations, and it is used repeatedly in the rest of the topic.

The Argand diagram

A real number can be shown as a point on a number line. A complex number $a + b\mathrm{i}$ needs two real numbers to describe it, so it is shown as a point in a plane.

Real numbers lie on the real axis, and purely imaginary numbers lie on the imaginary axis. Representing $z$ as a vector $\begin{pmatrix} a \\ b \end{pmatrix}$ from the origin is often more useful than representing it as a point, because addition of complex numbers then matches addition of vectors.

Modulus

The distance of a complex number from the origin on the Argand diagram measures its size. By Pythagoras' theorem, the point $(a, b)$ is at distance $\sqrt{a^2 + b^2}$

For a real number $z = a$, the modulus is $\sqrt{a^2} = |a|$

Argument

The position of $z$ on the Argand diagram is also fixed by its distance from the origin together with its direction. The direction is measured by an angle.

Some arguments can be read straight from an Argand diagram: a positive real number has argument $0$, $\arg \mathrm{i} = \frac{\pi}{2}$, $\arg(-1) = \pi$

Adding and subtracting complex numbers

Complex numbers are added and subtracted by combining real parts and imaginary parts separately, in the same way as collecting like terms in algebra:

For example, $(3 - 2\mathrm{i}) + (-5 + 6\mathrm{i}) = -2 + 4\mathrm{i}$ and $(3 - 2\mathrm{i}) - (-5 + 6\mathrm{i}) = 8 - 8\mathrm{i}$

Addition and subtraction as vectors

On an Argand diagram, adding complex numbers is the same as adding their vectors. With $z_1 = 1 + 3\mathrm{i}$ and $z_2 = 4 + \mathrm{i}$

Subtraction works in the same way. Since $z_2 + (z_1 - z_2) = z_1$

A useful consequence follows. The length of the vector from $z_2$ to $z_1$ is $|z_1 - z_2|$

Multiplying complex numbers

To multiply two complex numbers, expand the brackets as in ordinary algebra and then replace $\mathrm{i}^2$ by $-1$:

There is no need to memorise this formula; expanding each time is just as quick and less error-prone. Multiplying by a real number $k$ multiplies both parts by $k$, so $k(a + b\mathrm{i}) = ka + kb\mathrm{i}$, which on the Argand diagram is an enlargement of the vector by scale factor $k$

Part 3 of this example shows the rotation property again: $3 + 2\mathrm{i}$ is the point $(3, 2)$, and $\mathrm{i}(3 + 2\mathrm{i}) = -2 + 3\mathrm{i}$

The complex conjugate

Changing the sign of the imaginary part of a complex number gives a closely related number with very useful properties.

For example, $(3 + 2\mathrm{i})^* = 3 - 2\mathrm{i}$, $(-4 - \mathrm{i})^* = -4 + \mathrm{i}$

Properties of the conjugate

For $z = a + b\mathrm{i}$:

The last result is the most useful, because it shows that the product of a complex number and its conjugate is always a non-negative real number. The conjugate also behaves well with arithmetic: $(z_1 + z_2)^* = z_1^* + z_2^*$

The identities $z + z^* = 2\,\mathrm{Re}(z)$ and $zz^* = |z|^2$

Dividing complex numbers

Division uses the fact that $zz^*$ is real. To divide by $c + d\mathrm{i}$, multiply the numerator and the denominator by the conjugate of the denominator, $c - d\mathrm{i}$. The denominator becomes the real number $c^2 + d^2$

The method is the same idea as rationalising a denominator such as $\frac{1}{2 + \sqrt{3}}$ by multiplying by $2 - \sqrt{3}$

The result of part 2 gives a general formula for the reciprocal: $\frac{1}{z} = \frac{z^*}{|z|^2}$

When $|z| = 1$, the formula $\frac{1}{z} = \frac{z^*}{|z|^2}$

An application: impedance in AC circuits

In an alternating current (AC) circuit, each component resists the current by an amount called its impedance, which has two parts: a resistance $R$ that dissipates energy, and a reactance $X$ caused by coils and capacitors, which shifts the timing of the current relative to the voltage. Engineers combine the two into one complex number, $Z = R + X\mathrm{i}$ (they write $j$ for $\mathrm{i}$

Solving equations by equating parts

Many equations in a complex unknown $z$ can be solved by writing $z = x + y\mathrm{i}$ with $x, y \in \mathbb{R}$, simplifying each side into the form real part $+$

Equations involving z and its conjugate

When an equation contains both $z$ and $z^*$, it usually cannot be solved by rearranging for $z$, because $z$ and $z^*$

Square roots of a complex number

The equation $z^2 = w$ asks for the square roots of the complex number $w$. Writing $z = x + y\mathrm{i}$ gives $z^2 = (x^2 - y^2) + 2xy\mathrm{i}$

The two square roots are negatives of each other, just as $\pm 3$ are the square roots of $9$. A shortcut avoids the quartic equation. Taking the modulus of both sides of $z^2 = w$ gives $|z|^2 = |w|$

Simultaneous equations

Linear simultaneous equations with complex coefficients are solved by substitution or elimination, exactly as with real coefficients. Divisions by complex numbers are carried out with the conjugate.

Quadratic equations with complex roots

For the quadratic equation $az^2 + bz + c = 0$ with real coefficients, the discriminant $\Delta = b^2 - 4ac$

The roots have the same real part and opposite imaginary parts, so they are a conjugate pair.

The graph of $y = x^2 - 4x + 13$ shows why there are no real roots: its minimum point is $(2, 9)$, so the curve never reaches the $x$-axis. The real part of the complex roots,

On an Argand diagram the two roots $2 \pm 3\mathrm{i}$ are reflections of each other in the real axis.

Some equations of higher degree are quadratics in disguise. An equation such as $z^4 + pz^2 + q = 0$ is a quadratic in $z^2$

Forming a quadratic from a complex root

For roots $p \pm q\mathrm{i}$, the sum is $2p$ and the product is $(p + q\mathrm{i})(p - q\mathrm{i}) = p^2 + q^2$

with real coefficients. Conversely, if a quadratic with real coefficients has one non-real root, the quadratic formula shows that the other root is its conjugate.

Exam-style questions on complex numbers in Cartesian form

Questions on this content appear on both calculator and non-calculator papers. On a non-calculator paper every step of complex arithmetic must be shown. Where a graphic display calculator (GDC) is allowed, its complex mode can check divisions and powers, but questions that say "show that" or ask for exact values still need full working.

The next question is a typical parameter question: a condition such as "purely imaginary" or "real" is a statement about one of the two parts, which gives an equation for the unknown.

Self-check

• To do:

  I can simplify $\mathrm{i}^n$ for any integer $n$ and write square roots of negative numbers in terms of $\mathrm{i}$.

• To do:

  I can state the real part, imaginary part, conjugate and modulus of $z = a + b\mathrm{i}$