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Complex numbers — Mathematics, 14–17 years

Numbers that extend the number line so equations such as x² = −1 can have a meaningful solution.

A new direction for numbers

The symbol i is defined by i² = −1. A complex number has a real part and an imaginary part, such as 3 + 2i. You can add and multiply these numbers using ordinary algebra, while remembering that every i² can be replaced by −1.

The equation that had no real answer

The real numbers cannot solve x² = −1, because the square of every real number is zero or positive. Mathematicians wanted algebra to keep working instead of stopping at this obstacle. Introducing i creates a larger number system, where equations can have solutions that were impossible on the real number line.

Multiplying two complex numbers

Calculate (2 + 3i)(1 − 4i). First expand: 2 − 8i + 3i − 12i². Combine like terms: 2 − 5i − 12i². Since i² = −1, −12i² = 12, so the answer is 14 − 5i. The method is ordinary expansion, plus the special rule for i².

i is not a variable

A tempting mistake is to treat i as an unknown letter and leave i² unchanged, or to say that i is simply “the square root of −1” without using its rule carefully. The notation looks like an ordinary variable. But i has a fixed meaning: i² = −1, so powers of i cycle and must be simplified.

Signals and rotations

Complex numbers give a compact way to describe rotations and repeating waves. Engineers use them for alternating electrical current, radio signals and sound processing, where size and phase matter together. They are not claiming that electricity is imaginary; the number system is a useful tool for recording two linked quantities.

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