Conjugate of a complex number: uses and examples

Complex numbers extend ordinary arithmetic into a second dimension, and the conjugate is one of their most useful partners. It connects algebraic simplification, geometry and the roots of quadratic equations without requiring advanced machinery.

Understanding how to form and apply a conjugate also makes it easier to check answers. The examples below use the conventions common in Australian maths classes, including decimal points and the symbol (i) for the imaginary unit.

What the conjugate means

Start with a complex number (z=a+bi), where (a) and (b) are real numbers and (i^2=-1). Its conjugate is written (\overline{z}=a-bi). The real part stays the same, while the sign of the imaginary part changes.

For example, the conjugate of (3+2i) is (3-2i), and the conjugate of (-5-7i) is (-5+7i). A real number is unchanged because it has no imaginary part, while the conjugate of (4i) is (-4i).

Geometrically, conjugates are mirror images across the real axis. This reflection explains why their sum and product have useful real values, which is why the operation appears in so many algebraic methods.

Products, moduli and division

Multiplying a complex number by its conjugate produces a real result:

[ (a+bi)(a-bi)=a^2+b^2. ]

For (z=3+4i), this gives (3^2+4^2=25). This identity is also written (z\overline{z}=|z|^2), where the modulus is (|z|=\sqrt{a^2+b^2}).

Division works by multiplying the numerator and denominator by the conjugate of the denominator. For example,

[ \frac{3+4i}{2-i}\times\frac{2+i}{2+i} =\frac{(3+4i)(2+i)}{(2-i)(2+i)} =\frac{2+11i}{5}. ]

The result is (\frac25+\frac{11}{5}i). In general,

[ \frac{a+bi}{c+di} =\frac{(a+bi)(c-di)}{c^2+d^2}. ]

The same process gives the reciprocal of a complex number. Since (z\overline{z}=a^2+b^2), the inverse of (a+bi) is

[ \frac{1}{a+bi}=\frac{a-bi}{a^2+b^2}. ]

This is useful whenever a denominator contains (i).

Radicals, powers and reciprocal expressions

Taking square roots of negative real numbers also uses complex numbers. For instance, (\sqrt{-16}=4i) because ((4i)^2=-16). Its conjugate, (-4i), is the other square root of (-16).

Conjugation also respects addition and multiplication:

[ \overline{z+w}=\overline{z}+\overline{w}, \qquad \overline{zw}=\overline{z},\overline{w}. ]

These identities can reveal the form of an answer without repeating every calculation. For example, ((1+i)^2=2i), ((1+i)^4=-4), and ((1+i)^8=16).

A simple reciprocal illustrates the method:

[ \frac{1}{1+i} =\frac{1-i}{(1+i)(1-i)} =\frac{1-i}{2}. ]

Multiplying by (1-i), the conjugate of (1+i), has removed (i) from the denominator.

Complex numbers in Australian contexts

Complex numbers can give geometry problems a coordinate system. A student might place one end of Sydney Harbour Bridge at the origin, measure 120 metres east and 80 metres north, and represent the point as (120+80i). Its conjugate, (120-80i), reflects that point across the east–west axis.

The same idea works for a simplified map of Melbourne’s tram network. With Federation Square chosen as the origin, a stop two kilometres east and one kilometre north could be represented by (2+i). On the complex plane, (2-i) is its reflection across the horizontal axis.

Complex roots also appear in Australian secondary-school software, calculators and homework-support platforms. The local market for tutoring and maths tools is competitive, with many services priced in Australian dollars. Clear, step-by-step explanations are therefore more valuable than an unexplained decimal result, a principle reflected in CurryPlum’s approach to algebra support.

Quadratics, checks and vertex form

A quadratic with real coefficients has complex roots in conjugate pairs. If (2+3i) is one root of (x^2-4x+13=0), then (2-3i) is the other. Their sum is (4), matching the coefficient of (x), and their product is (13), matching the constant term.

The vertex form makes this symmetry especially clear:

[ x^2-4x+13=(x-2)^2+9. ]

Its vertex is ((2,9)), and solving ((x-2)^2+9=0) gives (x=2\pm3i). The real part of each root is the horizontal coordinate of the vertex, while the opposite imaginary signs show the roots’ symmetry around that axis.

A short routine can prevent avoidable errors:

Before calculating

• Label the real and imaginary parts.
• Reverse only the imaginary sign to form the conjugate.
• Identify whether the product, quotient or root relationship is needed.

After calculating

• Confirm that conjugate products are real and non-negative.
• Check that rationalised denominators contain (a^2+b^2).
• Substitute the final value into the original expression.

For extra guided practice, students can use additional algebra practice alongside these worked examples while checking each transformation by hand.

The idea worth keeping

The conjugate is not a separate calculation invented for difficult questions. It is the mirror partner of a complex number, preserving the real part and reversing the imaginary sign. That pattern supports division, reciprocal calculations, radical simplification, geometric reflection and conjugate roots of quadratic equations.

The rule worth remembering is simple: keep the real part, reverse the sign of the imaginary part, and use the resulting partner to simplify, divide or verify.