Year 12 Complex Numbers: Addition and subtraction

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Addition and subtraction are the cleanest place to begin with complex numbers because they show the basic structure of the topic without too much algebraic complication. A complex number in the form $a+bi$ carries two pieces of information: a real part and an imaginary part. Adding or subtracting complex numbers means combining like with like. Real parts combine with real parts, and imaginary parts combine with imaginary parts. If you keep those roles separate, the working stays neat and the topic becomes much less intimidating.

Subtopic 1: Adding complex numbers

When two complex numbers are added, their real parts add and their imaginary parts add. There is no special new law beyond ordinary collection of like terms. The only thing that matters is keeping the $i$ terms aligned. This is why presentation matters so much in complex-number algebra. If you write everything in standard form at each step, you can see immediately whether the result is organised correctly.

Worked example 1

Subtopic 2: Subtracting complex numbers

Subtraction works the same way, except that you must distribute the negative sign across the entire second complex number. This is where most early errors occur. Students often subtract the real part correctly but forget to reverse the sign of the imaginary part. If you rewrite the subtraction as addition of the opposite, the structure becomes safer.

Worked example 2

Why standard form matters

Standard form means writing the complex number as $a+bi$. Even if the arithmetic is simple, forcing each answer into that form helps you check whether the real and imaginary parts have been handled correctly. It also prepares the expression for later topics such as multiplication, division, powers, and equation solving. The more consistently you rewrite into standard form, the more stable the whole topic becomes.

Worked example 3

What this teaches for later work

Addition and subtraction show the basic bookkeeping pattern of complex numbers. They teach you to preserve the distinction between the two parts of the number, and that same discipline carries directly into multiplication and powers. In more advanced algebra, the calculations become longer, but the same principle remains: if the working stays organised, the topic usually stays manageable.

Common traps

• Combining a real term with an imaginary term as if they were like terms.
• Forgetting to distribute a subtraction sign across the entire second bracket.
• Leaving the result unsimplified instead of writing it as $a+bi$.
• Losing the sign of the imaginary part in the final answer.

Revision focus

In revision, rewrite every subtraction question as addition of the opposite before simplifying. This is a very reliable way to reduce sign mistakes. It also helps to mark the real and imaginary parts separately in the working until the structure feels completely natural.

Another useful check is to ask whether the final result still has exactly one real part and one imaginary part. If the expression ends with several loose terms, the simplification is not finished. Complex-number fluency starts with clean standard form.

Practice links

• Basic operations on complex numbers