What's the Difference Between the Real Part, Imaginary Part, Conjugate, Modulus, and Argument?

The real part and imaginary part are the two numerical components of a complex number. The conjugate, modulus, and argument are related quantities describing its reflection, distance from the origin, and direction on an Argand diagram.

Definitions and Mechanisms

For a complex number $z=a+bi$, where $a,b\in\mathbb{R}$ and $i^2=-1$:

On an Argand diagram, $a$ is plotted horizontally and $b$ vertically, so $z$ corresponds to the point ((a,b)). This coordinate view explains all five quantities.

The conjugate changes only the sign of the imaginary component. This gives the useful result

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

To find the argument, first calculate a reference angle, then use the quadrant. For example, if $w=-3+4i$, the reference angle is $\tan^{-1}(4/3)$, but $w$ lies in quadrant II, so radians. A bare inverse-tangent output would give the wrong direction.

The argument must be placed in the correct quadrant according to the signs of $a$ and $b$. For $z=0$, the modulus is $0$, but the argument is undefined. The modulus is always non-negative, while an argument may be positive or negative depending on the chosen interval.

A common misconception is that the imaginary part of $3+4i$ is $4i$. It is actually the real coefficient $4$; $4i$ is the imaginary term.

Exam Technique

This is AHL 1.12: Introduction to complex numbers, so it is HL-only content. For a command such as find or determine, write each quantity using correct notation, show the modulus calculation, and verify the argument's quadrant rather than relying on an uncorrected inverse-tangent value. A GDC may be used on every Math AI paper, but the final answer must use mathematical notation.