A-Level Further Mathematics: Complex Numbers

Complex numbers elevate mathematics by providing a complete system where every polynomial equation has a solution. Moving beyond the real numbers, they unlock powerful techniques in engineering, physics, and advanced mathematics itself. For your A-Level studies, mastering them is not just about performing calculations but developing a deep, geometric intuition for a new dimension of numbers.

From Real to Complex: Foundations and Arithmetic

A complex number is an expression of the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined by the property $i^2=-1$. The real part, $a$, and the imaginary part, $b$, are denoted as $\mathrm{Re}(z)$ and $\mathrm{Im}(z)$ respectively. This elegant definition resolves the historical impossibility of taking square roots of negative numbers.

Arithmetic with complex numbers follows algebraic rules, treating $i$ as a variable but remembering $i^2=-1$.

• Addition/Subtraction: Combine real and imaginary parts separately: $(a+bi)\pm (c+di)=(a\pm c)+(b\pm d)i$.
• Multiplication: Expand brackets as usual: $(a+bi)(c+di)=ac+adi+bci+bd{i}^2=(ac-bd)+(ad+bc)i$.
• Division: Requires the concept of the complex conjugate. For $z=a+bi$, its conjugate is $\bar{z}=a-bi$. To divide, multiply the numerator and denominator by the conjugate of the denominator:

$$\frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{(c+di)(c-di)}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}.$$ This creates a real denominator, allowing separation into standard $a+bi$ form.

Geometric Interpretation: The Argand Diagram

The Argand diagram provides the crucial geometric insight. It is a coordinate plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part. The complex number $z=a+bi$ corresponds to the point with coordinates $(a,b)$.

This representation allows us to describe a complex number using polar coordinates rather than Cartesian ones. The distance from the origin $(0,0)$ to the point $(a,b)$ is called the modulus of $z$, denoted $|z|$. By Pythagoras' theorem, $|z|=\sqrt{a^2+b^2}$. The angle measured anticlockwise from the positive real axis to the line segment connecting the origin to $(a,b)$ is called the argument of $z$, denoted $\arg (z)$, usually given in radians within the range $-\pi <\theta \le \pi$ (the principal argument).

This leads to the modulus-argument (polar) form of a complex number: $z=r(\cos \theta +i\sin \theta )$, where $r=|z|$ and $\theta =\arg (z)$.

Loci in the Complex Plane

The power of the Argand diagram shines when describing sets of complex numbers that satisfy certain conditions, known as loci. These often produce familiar geometric shapes.

• $|z-(a+bi)|=r$ describes a circle with centre at $(a,b)$ and radius $r$.
• $|z-u|=|z-v|$ describes the perpendicular bisector of the line segment joining the points representing complex numbers $u$ and $v$.
• $\arg (z-(a+bi))=\theta$ describes a half-line (ray) starting from, but not including, the point $(a,b)$, making an angle $\theta$ with the positive real direction.

Understanding and sketching these loci is a key skill, often tested by combining conditions, such as finding complex numbers that satisfy both a modulus and an argument condition (e.g., the intersection of a circle and a half-line).

de Moivre's Theorem: Powers and Roots

de Moivre's theorem states that for any integer $n$ and real number $\theta$, $$[r(\cos \theta +i\sin \theta ){]}^n=r^n(\cos n\theta +i\sin n\theta ).$$ This provides an exceptionally efficient method for raising a complex number in modulus-argument form to a power, and, more importantly, for finding its roots.

To find the $n$th roots of a complex number $z=r(\cos \theta +i\sin \theta )$, we seek numbers $w$ such that $w^n=z$. de Moivre's theorem guides us to the solution: $$w_k=r^{1/n}\left(\cos \left(\frac{\theta +2k\pi }{n}\right)+i\sin \left(\frac{\theta +2k\pi }{n}\right)\right),$$ for $k=0,1,2,...,n-1$. Crucially, there are exactly $n$ distinct $n$th roots. On an Argand diagram, these roots are equally spaced around a circle of radius $r^{1/n}$, a beautiful geometric consequence of the theorem. This theorem is also instrumental in proving trigonometric identities by equating real and imaginary parts of $(\cos \theta +i\sin \theta {)}^n$.

Euler's Formula and Exponential Form

The connection between complex numbers, trigonometry, and exponential functions is crystallized in Euler's formula: $$e^{i\theta }=\cos \theta +i\sin \theta .$$ This allows us to write any complex number in a third, extremely compact form: the exponential form $z=r{e}^{i\theta }$, where $r=|z|$ and $\theta =\arg (z)$.

This form makes multiplication, division, and application of de Moivre's theorem almost trivial, as it leverages the standard laws of indices:

• Multiplication: $(r_1{e}^{i{\theta }_1})(r_2{e}^{i{\theta }_2})=(r_1{r}_2)e^{i({\theta }_1+{\theta }_2)}$.
• Division: $\frac{r_1{e}^{i{\theta }_1}}{r_2{e}^{i{\theta }_2}}=\frac{r_1}{r_2}{e}^{i({\theta }_1-{\theta }_2)}$.
• de Moivre's: $(r{e}^{i\theta }{)}^n=r^n{e}^{in\theta }$.

Euler's formula is not just a notational convenience; it is a profound result that underpins much of higher mathematics, physics, and signal processing.

Common Pitfalls

1. Incorrect Argument Calculation: When finding $\arg (a+bi)$, blindly using $\arctan (b/a)$ can give the wrong quadrant. Always sketch the point on an Argand diagram. For example, $\arg (-1-i)=-\frac{3\pi }{4}$, not $\arctan (1)=\frac{\pi }{4}$.
2. Misapplying de Moivre's Theorem for Roots: Forgetting that there are n distinct $n$th roots, each found by using different values of $k$ ($0$ to $n-1$), is a common error. Writing only the root for $k=0$ is incomplete.
3. Confusing Loci Definitions: Misinterpreting $|z-a|=|z-b|$ as the midpoint rather than the perpendicular bisector, or treating $\arg (z-a)=\theta$ as a full line instead of a half-line starting at $a$. Always test your interpretation with a simple point.
4. Algebraic Errors with Conjugates: When simplifying expressions, incorrectly assuming $(z{)}^2=|z{|}^2$ (it doesn't) or mishandling the conjugate in multi-step problems. Remember $z\bar{z}=|z{|}^2$ is a key identity.

Summary

• Complex numbers $a+bi$ extend the real number system, with $i^2=-1$. Arithmetic uses standard rules alongside the complex conjugate for division.
• The Argand diagram provides a geometric plane for visualization, leading to the modulus $|z|$ (distance) and argument $\arg (z)$ (angle), and the modulus-argument form $r(\cos \theta +i\sin \theta )$.
• Loci equations describe geometric paths in the complex plane, such as circles, lines, and half-lines.
• de Moivre's theorem, $[r(\cos \theta +i\sin \theta ){]}^n=r^n(\cos n\theta +i\sin n\theta )$, efficiently calculates powers and, vitally, reveals all $n$ distinct $n$th roots of a complex number.
• Euler's formula, $e^{i\theta }=\cos \theta +i\sin \theta$, unifies concepts, giving the compact exponential form $r{e}^{i\theta }$ which simplifies operations and deepens the theoretical framework.