Mathematics · Complex numbers and conjugation
Edited by: Hitoiki Arcade / Hitoiki Learning team · Published: · Editorial policy
You can expand algebraic expressions, yet a calculation suddenly looks unfamiliar when i appears. Start with one rule, i² = −1, and follow the terms using the distributive law you already know.
This article builds on expressions and equalities from the prerequisite article, and checks expansion using the distributive law here. Work through a product, a conjugate and a reciprocal, then connect them to see why conjugates help with division. Each new symbol is introduced where it is needed.
Write a complex number as a + bi, where a and b are real and i² = −1. Its real part is a and its imaginary part is b. The imaginary part means the real coefficient of i, rather than the whole term bi.
The first example is (1 + 2i)(3 − i). Expanding gives 3 − i + 6i − 2i². Since −2i² = +2, the real terms give 3 + 2 = 5 and the i terms give −i + 6i = 5i. The answer is 5 + 5i. Leaving −2i² as −2 gives the wrong real part.
The conjugate of a + bi is a − bi. In the second example, 3 − 4i has real part 3 and imaginary part −4, so its conjugate is 3 + 4i. This differs from multiplying the whole original number by −1, which gives −3 + 4i.
Taking the conjugate twice returns the original number. Check that the real part stays unchanged and only the sign of the imaginary part reverses. Conjugation does not swap the components either.
The third example is 1/(1 + i). Multiply both numerator and denominator by the denominator’s conjugate, 1 − i. The denominator becomes (1 + i)(1 − i) = 1 − i² = 2, giving (1 − i)/2. Multiplying only the numerator changes the value of the fraction.
In general, (a + bi)(a − bi) = a² + b². The terms −abi and +abi cancel, and the remaining −b²i² becomes +b². If a + bi is nonzero, a² + b² is positive, so this method makes the denominator a nonzero real number.
Check the reciprocal by multiplying back: (1 + i) × (1 − i)/2 = 1. The option 1 − i would give a product of 2, so the denominator 2 cannot be dropped.
Explain the three examples in order: expansion, conjugation and making a denominator real. For an extra hand calculation, verify (2 + i)(2 − i) = 5 and use it to obtain 1/(2 + i) = (2 − i)/5. The conjugate 2 − i and the reciprocal (2 − i)/5 have different values.
The practice button selects high-school standard questions in Numbers and calculation. It also includes matching questions beyond complex numbers. Return to the prerequisite article for algebraic expressions, or follow the related article on exponents and logarithms to organize powers and inverse questions.
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Example 1 · Numbers and calculation
Answer: 5＋5i
Expand to 3 − i + 6i − 2i². Using i² = −1 gives 5 + 5i.
Example 2 · Numbers and calculation
Answer: 3＋4i
The imaginary part of 3 − 4i is the real coefficient −4; its real part is 3. Complex conjugation keeps the real part and reverses the sign of the imaginary part, giving 3 + 4i.
Example 3 · Numbers and calculation
Answer: （1−i）/2
Multiply numerator and denominator by 1 − i. The denominator is (1 + i)(1 − i) = 2, giving (1 − i)/2.
Source checked: 2026-10-30
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