Complex Numbers: One Mark Questions, Book Back Answers and Notes

About Complex Numbers

Medium ~180 min study

The equation $x^2 + 1 = 0$ has no real solution, because the square of a real number is never negative. To get around this we introduce the imaginary unit $i$ with $i^2 = -1$. Numbers of the form $x + iy$, with $x$ and $y$ real, are complex numbers. With them, every polynomial equation of degree $n$ has exactly $n$ roots, counting repeated roots.

The chapter keeps moving between algebra and geometry. The complex number $x + iy$ is the point $(x, y)$ in the Argand plane, or the vector from the origin to that point. Adding two complex numbers is the parallelogram law. Multiplying by $i$ turns the point through $90^\circ$ anticlockwise about the origin. The conjugate is a reflection in the real axis, and the modulus is the distance from the origin. Keeping this picture in mind makes locus questions much easier.

For the exam, be quick with powers of $i$, the properties of modulus and conjugate, finding square roots of a complex number, and principal arguments. De Moivre's theorem is used for powers such as $(1 + i)^{10}$ and for the $n$-th roots of unity. Locus questions, such as finding the curve traced by $z$ when $|z - 2| = 3$, come up regularly in the book back exercises.

What you'll learn

• Add, subtract, multiply and divide complex numbers in rectangular form.
• Plot complex numbers as points and vectors in the Argand plane.
• Convert between rectangular, polar and Euler forms and find the principal argument.
• Use the properties of conjugate and modulus to simplify expressions.
• Find square roots of a complex number and the n-th roots of a complex number.
• Evaluate powers of complex numbers using De Moivre's theorem.
• Find and describe the locus of $z$ from conditions such as $|z - z_0| = r$.

Before you start

• The real number system and standard algebraic identities.
• Solving quadratic equations and using the discriminant.
• Basic trigonometric ratios, radian measure and coordinate geometry.

Complex Numbers explained

Algebra and Geometry of Complex Numbers

The Imaginary Unit and Rectangular Form

We define $i$ by $i^2 = -1$. Its powers repeat in a cycle of four: $i, -1, -i, 1$. So to find $i^{n}$, divide $n$ by 4 and use the remainder; for example $i^{103} = i^{3} = -i$. A complex number in rectangular form is $z = x + iy$, where $x = \text{Re}(z)$ is the real part and $y = \text{Im}(z)$ is the imaginary part. Two complex numbers are equal only when their real parts are equal and their imaginary parts are equal.

Algebraic Operations and Their Properties

To add or subtract, combine real parts and imaginary parts separately. To multiply, expand as usual and replace $i^2$ by $-1$. Addition and multiplication are commutative and associative, and multiplication distributes over addition, just as for real numbers. One thing does change: complex numbers cannot be ordered, so a statement like $2 + i > 1 + 3i$ has no meaning.

Conjugate and Modulus

The conjugate of $z = x + iy$ is $\bar{z} = x - iy$. Since $z\bar{z} = x^2 + y^2$ is real, multiplying the numerator and denominator by the conjugate of the denominator is how we divide. The modulus is $|z| = \sqrt{x^2 + y^2}$, the distance of $z$ from the origin. Useful results are $|z_1 z_2| = |z_1||z_2|$, $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$, and the triangle inequality $|z_1 + z_2| \leq |z_1| + |z_2|$. The distance between $z_1$ and $z_2$ is $|z_1 - z_2|$, which is what locus questions use.

The Argand Plane

Each complex number $x + iy$ matches exactly one point $(x, y)$. The horizontal axis is the real axis and the vertical axis is the imaginary axis. Addition follows the parallelogram law, and multiplying by $i$ rotates a point through $90^\circ$ anticlockwise about the origin. A circle with centre $z_0$ and radius $r$ is simply $|z - z_0| = r$.

Polar Form and Euler Form

If $r = |z|$ and $\theta$ is an argument of $z$, then $z = r(\cos \theta + i \sin \theta)$. Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$ lets us write this compactly as $z = re^{i\theta}$. The principal argument, $\text{Arg}\, z$, lies in $(-\pi, \pi]$; find the acute angle $\alpha = \tan^{-1}\left|\frac{y}{x}\right|$ and adjust for the quadrant. Polar form suits multiplication and division: moduli multiply or divide, and arguments add or subtract.

De Moivre's Theorem and Roots

De Moivre's theorem states $(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta$ for every integer $n$. For a rational power it gives one of the values. It is the tool for finding the $n$-th roots of a complex number. The $n$-th roots of unity are $e^{i\,2k\pi/n}$ for $k = 0, 1, \ldots, n-1$. They lie on the unit circle as the vertices of a regular $n$-sided polygon, and their sum is 0 for $n \geq 2$.

Common mistakes to avoid

• Taking $\tan^{-1}\frac{y}{x}$ as the argument without checking the quadrant. For $z = -1 - i$ the principal argument is $-\frac{3\pi}{4}$, not $\frac{\pi}{4}$.
• Writing $\sqrt{-4}\sqrt{-9} = \sqrt{36} = 6$. The rule $\sqrt{a}\sqrt{b} = \sqrt{ab}$ fails when both are negative; the correct value is $(2i)(3i) = -6$.
• Finding the multiplicative inverse of $z$ by only changing the sign of the imaginary part. The inverse is $\frac{\bar{z}}{|z|^2}$, so you must also divide by $x^2 + y^2$.
• Applying De Moivre's theorem to a number not in the form $\cos \theta + i \sin \theta$. For example $\sin \theta + i \cos \theta$ must first be rewritten as $\cos\left(\frac{\pi}{2} - \theta\right) + i \sin\left(\frac{\pi}{2} - \theta\right)$.

Test yourself on these with the practice test, then check the worked reasoning in the solved MCQs.

Frequently asked questions

What is the difference between an imaginary number and a complex number?

A purely imaginary number has the form $iy$, where $y$ is a non-zero real number. A complex number is $x + iy$ with both parts allowed. Every real number is a complex number with $y = 0$, and every purely imaginary number is a complex number with $x = 0$.

How do I find the principal argument of a complex number?

First find the acute angle $\alpha = \tan^{-1}\left|\frac{y}{x}\right|$. Then use the quadrant: first quadrant gives $\alpha$, second gives $\pi - \alpha$, third gives $-\pi + \alpha$ and fourth gives $-\alpha$. The answer must lie in $(-\pi, \pi]$.

Why do we multiply by the conjugate when dividing?

Because $(x + iy)(x - iy) = x^2 + y^2$ is a real number. Multiplying the numerator and denominator by the conjugate of the denominator makes the denominator real, so the answer can be split into a real part and an imaginary part.

What does the triangle inequality mean geometrically?

In the Argand plane, $0$, $z_1$ and $z_1 + z_2$ form a triangle whose sides have lengths $|z_1|$, $|z_2|$ and $|z_1 + z_2|$. No side of a triangle is longer than the sum of the other two, so $|z_1 + z_2| \leq |z_1| + |z_2|$.

When should I use polar form instead of rectangular form?

Use rectangular form for addition and subtraction, where you just combine like parts. Use polar form for multiplication, division, powers and roots, because moduli simply multiply and arguments simply add. Finding $(1 + i)^{8}$ by expanding is slow; in polar form it is $(\sqrt{2})^8(\cos 2\pi + i \sin 2\pi) = 16$.

What are the roots of unity used for?

The $n$-th roots of unity help solve equations like $z^n = 1$ and give the $n$-th roots of any complex number, since those are one root multiplied by each root of unity. They lie at the vertices of a regular $n$-gon on the unit circle. For the cube roots $1, \omega, \omega^2$, remember $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$.