Conjugate Calculator

What is a Mathematical Conjugate?

In mathematics, a conjugate refers to a pair of binomial or complex terms that share the exact same components but differ in the sign connecting them. The two most common types of conjugates in algebra and analysis are complex conjugates and radical (binomial) conjugates.

Our Conjugate Calculator computes both complex conjugates ($a - bi$) and radical binomial conjugates ($a - b\sqrt{c}$), providing immediate calculation of their products, sums, differences, and rationalized reciprocals. For related algebra and complex arithmetic, try our Complex Calculator, Complex Root Calculator, and a+bi Form Calculator.

1. Complex Conjugates ($z = a + bi$)

For any complex number $z = a + bi$ where $a$ and $b$ are real numbers and $i = \sqrt{-1}$:

$$\bar{z} = z^* = a - bi$$

Geometrically, taking the conjugate corresponds to reflecting the point across the real axis on the Argand plane.

Fundamental Properties of Complex Conjugates

Product is Real and Non-Negative: $$z \cdot \bar{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2 = |z|^2$$
Sum is Twice the Real Part: $$z + \bar{z} = (a + bi) + (a - bi) = 2a = 2\operatorname{Re}(z)$$
Difference is Pure Imaginary: $$z - \bar{z} = (a + bi) - (a - bi) = 2bi = 2i\operatorname{Im}(z)$$
Complex Division / Reciprocal: $$\frac{1}{z} = \frac{\bar{z}}{|z|^2} = \frac{a - bi}{a^2 + b^2}$$

2. Radical Binomial Conjugates ($a + b\sqrt{c}$)

For a binomial containing a square root radical $u + v = a + b\sqrt{c}$, the radical conjugate is:

$$\text{Conjugate} = a - b\sqrt{c}$$

Rationalizing Denominators using Difference of Squares

Multiplying an expression by its radical conjugate eliminates the square root through the algebraic identity $(u + v)(u - v) = u^2 - v^2$:

$$(a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - (b\sqrt{c})^2 = a^2 - b^2 c$$

This property is used to rationalize denominators in algebraic fractions:

$$\frac{1}{a + b\sqrt{c}} = \frac{1 \cdot (a - b\sqrt{c})}{(a + b\sqrt{c})(a - b\sqrt{c})} = \frac{a - b\sqrt{c}}{a^2 - b^2 c}$$

Step-by-Step Examples

Example 1: Complex Conjugate

Find the conjugate and product for $z = 3 + 4i$:

Conjugate: $\bar{z} = 3 - 4i$
Product: $z \cdot \bar{z} = 3^2 + 4^2 = 9 + 16 = 25$
Reciprocal: $\frac{1}{3 + 4i} = \frac{3 - 4i}{25} = 0.12 - 0.16i$

Example 2: Radical Conjugate & Rationalization

Rationalize the reciprocal of $2 + 3\sqrt{5}$:

Conjugate: $2 - 3\sqrt{5}$
Product: $(2 + 3\sqrt{5})(2 - 3\sqrt{5}) = 2^2 - (3\sqrt{5})^2 = 4 - (9 \times 5) = 4 - 45 = -41$
Rationalized Fraction: $\frac{1}{2 + 3\sqrt{5}} = \frac{2 - 3\sqrt{5}}{-41} = \frac{3\sqrt{5} - 2}{41}$

Frequently Asked Questions

What is the conjugate of a pure real number?

A pure real number has an imaginary part of 0 ($z = a + 0i$). Its conjugate is $\bar{z} = a - 0i = a$. Thus, a number equals its own complex conjugate if and only if it is real.

What is the conjugate of a pure imaginary number?

For a pure imaginary number $z = 0 + bi$, the conjugate is $\bar{z} = 0 - bi = -bi$. A number equals the negative of its conjugate if and only if it is purely imaginary.

Why is the conjugate notation sometimes written as z* and sometimes as z̄?

Both notations are standard. In pure mathematics, the overline ($\bar{z}$) is customary. In physics, quantum mechanics, and electrical engineering, the asterisk ($z^*$) is preferred to avoid confusing with spatial averages or vector notations.

How does conjugation affect roots of polynomial equations?

According to the Complex Conjugate Root Theorem, if a polynomial with real coefficients has a complex root $a + bi$ (where $b \neq 0$), then its complex conjugate $a - bi$ must also be a root of the polynomial.

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