Extending the real number system

There is no real-number solution to $x^2=-1$. To extend the number system, we define the imaginary unit as the dimensionless number $i$ that satisfies

where $i$ is the imaginary unit (no SI unit). Its powers follow a repeating cycle: $i^0=1$, $i^1=i$, $i^2=-1$, $i^3=-i$ and $i^4=1$. This pattern helps when simplifying higher powers of $i$.

A complex number can be written in Cartesian form as

where $z$ is a complex number (no SI unit), $a$ is its real component (no SI unit) and $b$ is the coefficient of its imaginary component (no SI unit). Its real part is the real number $\operatorname{Re}(z)=a$. The imaginary part is the real number $\operatorname{Im}(z)=b$—that is, $b$, not $bi$.

Two complex numbers are equal exactly when their real parts match and their imaginary parts match. In particular, $a+bi=0$ implies $a=0$ and $b=0$.

Conjugate, modulus and argument

To find the complex conjugate of $z=a+bi$, reverse the sign of its imaginary component:

where $\overline z$ is the conjugate of $z$ (no SI unit). The product of a complex number and its conjugate is a non-negative real number:

The modulus of a complex number is its non-negative distance from the origin in the complex plane:

where $|z|$ is the modulus of $z$ (no SI unit). It follows that $z\overline z=|z|^2$.

An argument of a non-zero complex number is the angle measured anticlockwise from the positive real axis to the line segment joining the origin to that number. We write $\arg z=\theta$, where $\theta$ is an argument of $z$ (radians). Adding any whole number of turns leaves the direction unchanged, so arguments differ by multiples of $2\pi$. Calculators usually return a principal argument in a specified interval, commonly $-\pi<\theta\le\pi$.

Before accepting an inverse-tangent result, use the signs of $a$ and $b$ to identify the quadrant. On its own, $\tan^{-1}(b/a)$ may indicate the wrong quadrant. It’s also undefined as written when $a=0$.

Arithmetic in Cartesian form

Let $w=c+di$, where $w$ is a second complex number (no SI unit), $c$ is its real component (no SI unit) and $d$ is the coefficient of its imaginary component (no SI unit). Add and subtract component by component:

For multiplication, expand normally, then replace $i^2$ by $-1$:

To write a quotient in Cartesian form, multiply its numerator and denominator by the conjugate of the denominator:

provided $w\ne0$. This works because $(c+di)(c-di)=c^2+d^2$, which is real.

You should be able to handle sums, differences, products and quotients both by hand and with technology. On a calculator, select complex-number mode and enter brackets carefully. Check whether the displayed answer is in Cartesian or polar form. Technology can calculate integer powers directly in Cartesian form. For a small power done by hand, repeated multiplication and the cycle of powers of $i$ are enough. In $z^n$, $n$ is an integer exponent (no SI unit).

The complex plane and Argand diagrams

The complex plane is a coordinate plane whose horizontal axis represents the real part and whose vertical axis represents the imaginary part. An Argand diagram plots $z=a+bi$ at the point $(a,b)$. Equivalently, it shows the position vector from the origin to $(a,b)$.

The modulus $|z|$ is therefore a distance, while the argument $\theta$ is a directed angle. Reflecting $z$ in the real axis gives its conjugate $\overline z$.

When drawing an Argand diagram, label the axes $\operatorname{Re}$ and $\operatorname{Im}$ rather than $x$ and $y$. Use equal scales wherever the geometric interpretation matters, and show an argument using an anticlockwise arc from the positive real axis.

Quadratic equations with non-real solutions

Consider the quadratic equation

where $A$, $B$ and $C$ are real coefficients (no SI units), $x$ is the unknown number (no SI unit) and $A\ne0$. Its solutions are

The discriminant is the real number

where $D$ is the discriminant (no SI unit). If $D<0$, write $D=-q$ with $q>0$, where $q$ is the positive magnitude of the negative discriminant (no SI unit). Since $\sqrt{-q}=i\sqrt q$, the two solutions become

With real coefficients, non-real solutions come as a conjugate pair. On the real graph

where $f(x)$ is the real output of the quadratic function (no SI unit), $D<0$ tells us that the parabola has no real $x$-intercepts. The equation still has solutions; its two solutions simply lie outside the real number system.

Language and mathematical knowledge

The labels “imaginary” and “complex” come from historical vocabulary. They don’t describe numbers that are less legitimate or necessarily more complicated. The number $i$ has a precise algebraic definition, just as negative numbers and irrational numbers are characterized within extensions of earlier number systems. Complex numbers therefore provide a useful TOK example of how language may initially obstruct understanding, even when the underlying definitions are exact.

Modulus–argument form

Cartesian form uses perpendicular components to describe a complex number. Modulus–argument form, also called polar form, describes a non-zero complex number by its distance from the origin and its direction:

The notation $\operatorname{cis}\theta$ abbreviates $\cos\theta+i\sin\theta$, and $r=|z|>0$. When $z=0$, the modulus is zero, but its argument is undefined.

The geometry leads straight to the conversion formulas:

The tangent equation alone cannot identify the quadrant. Check the signs of $a$ and $b$, or use a technology command that finds the angle from both coordinates.

An argument is not unique:

where $k$ is any integer (no SI unit). If the question asks for a principal argument, use the convention specified in the question or by the technology being used.

Exponential form

Exponential form, sometimes called Euler form, writes the number as

Euler's relation

shows that exponential and polar forms represent exactly the same modulus and argument.

You should be able to convert in either direction, by hand or with technology:

• Cartesian to polar or exponential: calculate the modulus, then determine the argument in the correct quadrant.
• Polar or exponential to Cartesian: use $a=r\cos\theta$ and $b=r\sin\theta$.
• Polar to exponential: replace $\operatorname{cis}\theta$ by $e^{i\theta}$, or reverse that replacement.

Watch the calculator settings. Check whether it is interpreting angles in radians or degrees, and keep exact values such as $\pi/3$ when the answer should be exact.

Products, quotients and integer powers

Let

To multiply the numbers, multiply their moduli and add their arguments:

For division, divide the moduli and subtract the arguments:

In exponential form, the same rules come directly from the index laws:

For an integer power, de Moivre's result gives

So the power raises the modulus to that power and multiplies the argument by the exponent. Finding roots of complex numbers is not required here.

Geometric interpretation

On an Argand diagram, complex numbers act like two-dimensional vectors when they are added. The sum of $z_1$ and $z_2$ is the diagonal of the parallelogram formed by their corresponding vectors. Subtraction works by adding the opposite vector.

Multiplication has another geometric interpretation. Multiplying by $z_2=r_2\operatorname{cis}\theta_2$ scales every distance from the origin by the factor $r_2$, then rotates every direction anticlockwise through $\theta_2$. When $0<r_2<1$, the “stretch” is a contraction. If $r_2=1$, the transformation is a pure rotation.

The polar multiplication rule now has a clear geometric meaning: lengths change multiplicatively, while directions change additively.

Adding sinusoids with a common frequency

Complex numbers offer a quick way to combine sinusoidal quantities with the same angular frequency but different amplitudes and phase shifts. Consider

Represent each source by the phasor

The phasors are added in Cartesian form:

Now convert the result back to polar form:

The combined voltage is

This method works because every term has the same $\omega$. If the sinusoids have different frequencies, phasor addition cannot collapse them into one sinusoid.

Euler's identity, elegance and extension

Substituting $\theta=\pi$ into Euler's relation gives

This identity is often called beautiful because a single short statement connects the additive identities $0$ and $1$ with the imaginary unit $i$, the circle constant $\pi$ and the exponential constant $e$. It also shows mathematical creativity: this connection isn’t obvious when real geometry, trigonometry and exponentials are first encountered as separate topics. From a TOK perspective, beauty may lead mathematicians towards fruitful ideas, but it cannot replace proof.

As an enrichment connection, the exponential expression also satisfies

Both $y=e^{i\theta}$ and $y=\cos\theta+i\sin\theta$ satisfy this differential equation, connecting complex forms with differential equations solved by separation of variables.