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Conjugation — Definition, Formula & Examples

Conjugation is the process of forming a conjugate by changing the sign between two terms in a complex number or radical expression. For example, the conjugate of a+bia + bi is a−bia - bi, and the conjugate of 3+53 + \sqrt{5} is 3−53 - \sqrt{5}.

Given a complex number z=a+biz = a + bi where a,b∈Ra, b \in \mathbb{R}, conjugation maps zz to zˉ=a−bi\bar{z} = a - bi. More generally, for any binomial expression a+ba + b, its conjugate is a−ba - b. Conjugation preserves the real part while negating the imaginary (or radical) part, and the product of an expression with its conjugate yields a real, rational result.

Key Formula

z⋅zˉ=(a+bi)(a−bi)=a2+b2z \cdot \bar{z} = (a + bi)(a - bi) = a^2 + b^2
Where:
  • zz = A complex number $a + bi$
  • zˉ\bar{z} = The conjugate of $z$, equal to $a - bi$
  • aa = The real part of $z$
  • bb = The imaginary part of $z$ (a real number)

How It Works

To conjugate an expression, keep the first term the same and flip the sign of the second term. When you multiply an expression by its conjugate, the cross terms cancel, eliminating the imaginary part or the radical. This works because (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2, which is the difference of squares pattern. Conjugation is the standard technique for rationalizing denominators and for dividing complex numbers.

Worked Example

Problem: Simplify 2+3i1−4i\dfrac{2 + 3i}{1 - 4i} by multiplying by the conjugate.
Identify the conjugate: The denominator is 1−4i1 - 4i, so its conjugate is 1+4i1 + 4i.
zˉ=1+4i\bar{z} = 1 + 4i
Multiply numerator and denominator: Multiply both top and bottom by 1+4i1 + 4i.
(2+3i)(1+4i)(1−4i)(1+4i)\frac{(2 + 3i)(1 + 4i)}{(1 - 4i)(1 + 4i)}
Expand and simplify: The denominator becomes 12+42=171^2 + 4^2 = 17. The numerator expands to 2+8i+3i+12i2=2+11i−12=−10+11i2 + 8i + 3i + 12i^2 = 2 + 11i - 12 = -10 + 11i.
−10+11i17=−1017+1117i\frac{-10 + 11i}{17} = -\frac{10}{17} + \frac{11}{17}i
Answer: −1017+1117i-\dfrac{10}{17} + \dfrac{11}{17}i

Why It Matters

Conjugation appears throughout algebra and precalculus whenever you need to eliminate radicals or imaginary numbers from a denominator. In electrical engineering and signal processing, complex conjugates are used to compute power and impedance in AC circuits.

Common Mistakes

Mistake: Negating both terms instead of only the sign between them, turning 3+2i3 + 2i into −3−2i-3 - 2i.
Correction: Only change the sign of the imaginary (or radical) part. The conjugate of 3+2i3 + 2i is 3−2i3 - 2i, not −3−2i-3 - 2i. The real part stays the same.