What's the Difference Between the Real Part, Imaginary Part, and Conjugate of a Complex Number?

For a complex number $z=a+bi$, the real part is $a$, the imaginary part is $b$, and the complex conjugate is the new complex number (a-bi). The first two are components of $z$, while the conjugate is a transformation of the entire number.

This is IB Mathematics: Analysis and Approaches HL, AHL 1.12: Complex numbers in Cartesian form and the Argand diagram. It may be tested analytically on Paper 1 or Paper 2.

Here $a$ and $b$ are real numbers, while $i^2=-1$. If $b=0$, then $$ is real and equals its conjugate. If , then is purely imaginary, but its imaginary part is still .

For example, let

$$z=-3+4i.$$

Then

$$\operatorname{Re}(z)=-3, \qquad \operatorname{Im}(z)=4, \qquad \overline{z}=-3-4i.$$

Taking the conjugate keeps the real part unchanged but reverses the sign of the imaginary part. This gives

$$z\overline{z}=(a+bi)(a-bi)=a^2+b^2=|z|^2,$$

which is real. It also follows that conjugation preserves modulus: $|\overline{z}|=|z|$, and applying it twice returns the original number, $\overline{\overline{z}}=z$. These properties provide quick checks in algebraic manipulation: if only the sign of changes, the conjugate has been formed correctly. The real-axis coordinate must remain unchanged throughout the process.

The conjugate is especially useful for division because multiplying by the denominator's conjugate removes its imaginary term. For example,

$$\frac{3+i}{1+2i}\times\frac{1-2i}{1-2i}=\frac{5-5i}{5}=1-i.$$

Since $i^2=-1$, the denominator becomes real. This method uses $w\overline{w}=|w|^2$ and converts a complex quotient into Cartesian form without changing its value.

A common misconception is writing $\operatorname{Im}(z)=bi$. This is incorrect: the imaginary part is the real coefficient $b$, not the term (bi). Also, $\overline{z}$ is a complex number, not merely the negative imaginary part.

Exam technique: For “state,” give only the relevant coefficient. For the conjugate, keep the real part and change only the imaginary part's sign. On an Argand diagram, describe conjugation as reflection in the real axis.