How Do You Write a Complex Number in Cartesian Form?

A complex number is in Cartesian form when it is written as $z=a+bi$, where $a,b\in\mathbb{R}$ and $i^2=-1$. To convert an expression, simplify it using $i^2=-1$, then collect the real terms and the coefficients of $i$ separately.

The Method

In $z=a+bi$, $a$ is the real part and $b$ is the imaginary part:

• $\operatorname{Re}(z)=a$
• $\operatorname{Im}(z)=b$

For example, $z=5-3i$ is already in Cartesian form. Its real part is $5$, while its imaginary part is $-3$.

When adding or subtracting complex numbers, combine real parts with real parts and imaginary terms with imaginary terms. When multiplying, expand the brackets before replacing every occurrence of $i^2$ with $-1$.

Signs matter when subtracting brackets. For instance, $(4+2i)-(1-5i)=4+2i-1+5i=3+7i$; the minus sign changes both terms in the second bracket. Cartesian form also allows zero coefficients: a real number such as may be viewed as , and a purely imaginary number such as may be viewed as . Normally, however, write the shorter forms and . This makes the values of and immediately visible.

For division, multiply the numerator and denominator by the denominator's complex conjugate. This produces a real denominator. For example,

$$\frac{1+i}{1-i}\times\frac{1+i}{1+i} =\frac{(1+i)^2}{1-i^2} =\frac{2i}{2}=i.$$

Therefore, the Cartesian form is $0+i$, conventionally written as $i$. For higher powers, use the repeating cycle $i^1=i$, $i^2=-1$, , and ; the cycle repeats every four powers.

A common misconception is that the imaginary part of $5-3i$ is $-3i$. This is incorrect: the imaginary part is the real coefficient $-3$, not the complete imaginary term.

Exam Technique

This is an HL-only skill from AHL 1.12, Introduction to complex numbers. In an IB Math AI response, show the expansion and substitution $i^2=-1$, then finish explicitly in the form $a+bi$. Do not leave powers of $i$, fractions with complex denominators, or unexpanded products in the final answer. A GDC may check arithmetic, but clear algebraic working is still expected.