What Is the Imaginary Unit \(i\) and Why Does \(i^2=-1\)?

In IB Mathematics: Analysis and Approaches HL, the imaginary unit $i$ is defined by the property $i^2=-1$. It extends the real number system so that equations such as $x^2=-1$, which have no real solutions, can be solved.

Why Is $i$ Needed?

For every real number $x$, $x^2\geq 0$. Therefore, no real number can satisfy $x^2=-1$. Mathematicians extend the real numbers by defining a new number:

$$i=\sqrt{-1}, \qquad i^2=-1.$$

The equation $i^2=-1$ is therefore not derived from real-number rules; it is the defining property of $i$. Equivalently, $i$ is a solution of $x^2+1=0$. Because squaring either or (-i) gives , that equation has the two complex roots . The notation conventionally identifies the principal square root, whereas solving an equation requires all roots.

This extension creates the complex numbers, written in Cartesian form as

$$z=a+bi,$$

where $a,b\in\mathbb{R}$. Here, $a$ is the real part and $b$ is the coefficient of the imaginary part. On an Argand diagram, $a+bi$ is represented by the point ((a,b)).

Consider $z^2+4=0$. Rearranging gives $z^2=-4$, so $z=$, since . Negative real quantities can therefore have complex square roots because the factor contributes the negative sign.

Algebra with complex numbers follows the usual distributive and associative laws, together with the substitution $i^2=-1$. This makes the extension consistent: addition and multiplication remain closed within expressions of the form $a+bi$. For example, the real and imaginary parts are collected separately. This is why calculations do not require a new multiplication rule beyond simplifying each occurrence of $i^2$, , or during expansion.

For example,

$$(2+3i)(1-i)=2-2i+3i-3i^2=5+i.$$

Powers of $i$ repeat every four powers:

$$i,\quad -1,\quad -i,\quad 1.$$

A common misconception is that $i$ is negative. It is not: complex numbers are not ordered as positive or negative like real numbers. Also, while $\sqrt{-9}=3i$, the equation $z^2=-9$ has two solutions, .

Exam Technique

This is AHL 1.12 content and may appear on AA HL Paper 1 or Paper 2. Replace every $i^2$ with $-1$ and give final answers in the form $a+bi$.