Complex Conjugates

Every complex number written in the form \(a + bi\) has a partner number called its complex conjugate, written \(a - bi\). To form the conjugate, you keep the real part exactly the same and flip the sign of the imaginary part. That's the entire rule: change the sign in front of \(bi\), and leave everything else alone.

For example, the conjugate of \(3 + 5i\) is \(3 - 5i\), and the conjugate of \(-2 - 7i\) is \(-2 + 7i\). If you haven't worked with the imaginary unit \(i\) before, it helps to review introduction to imaginary numbers first, since conjugates only make sense once you're comfortable with \(i = \sqrt{-1}\).

A complex number \(a + bi\) can be plotted as the point \((a, b)\) on the complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis. Its conjugate \(a - bi\) sits at \((a, -b)\), which is exactly the mirror image across the real axis. This geometric picture is why the term "conjugate" fits so well, one number is simply the reflection of the other. For more on plotting complex numbers this way, see complex numbers and complex planes.

One of the most useful facts about conjugates is what happens when you multiply a complex number by its own conjugate:

\((a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2\)

Notice the imaginary part disappears completely, leaving a purely real result. This is the same pattern as the difference of squares, \((x+y)(x-y) = x^2 - y^2\), applied to complex numbers, and it works because \(i^2 = -1\) turns the subtraction into addition.

Example: Find the product of \(4 + 3i\) and its conjugate.

The conjugate of \(4 + 3i\) is \(4 - 3i\). Multiplying:

\((4 + 3i)(4 - 3i) = 4^2 + 3^2 = 16 + 9 = 25\)

This "always real" property is exactly why conjugates are used to simplify expressions with imaginary numbers in the denominator: multiplying the top and bottom of a fraction by the conjugate of the denominator clears the imaginary part from it. If you need a refresher on combining complex numbers before working through such problems, take a look at adding and subtracting complex numbers.

The complex conjugate root theorem states that if a polynomial has only real number coefficients, then any complex (non-real) roots must occur in conjugate pairs. In other words, if \(a + bi\) is a root of such a polynomial, then \(a - bi\) must also be a root.

Example: A quadratic with real coefficients has \(2 + i\) as one root. Find the other root.

Since the coefficients are real, the complex conjugate root theorem guarantees the other root is the conjugate, \(2 - i\).

This theorem is especially handy when factoring or solving higher-degree polynomials: as soon as you spot one complex root, you automatically know its partner without any extra calculation.

For a complex number \(a + bi\):

Conjugate: \(a - bi\)
Sum with conjugate: \((a+bi) + (a-bi) = 2a\)
Product with conjugate: \((a+bi)(a-bi) = a^2 + b^2\)

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