Key takeaway: The complex conjugate of a + bi is a − bi, the number with the same real part and an imaginary part of opposite sign, and the product of the two is always the real number a² + b².
Grade 11 – Algebra 2N-CN.A.2Numbers and Operations
Definition of Complex Conjugate
The complex conjugate of a number is found by flipping the sign of its imaginary part. The conjugate of \(3 – 4i\) is \(3 + 4i\). On the complex plane the two points are mirror images across the horizontal axis.
The point of the conjugate is the product. Multiplying a complex number by its conjugate cancels the \(i\) terms and leaves a real number, the square of the distance from the origin. That trick is how to divide complex numbers: multiply the top and bottom by the conjugate of the bottom, which turns the denominator real.
Formula and notation
The complex conjugate of \(z = a + bi\) is written \(\,\overline{z}\):
\[ \,\overline{z} = a – bi, \qquad z\,\,\overline{z} = a^2 + b^2 \]
Also \(z + \,\overline{z} = 2a\) is real. For a quadratic with real coefficients, complex roots come in conjugate pairs.
Key facts
- Only the sign of the imaginary part changes; the real part stays.
- A number times its conjugate is real and never negative.
- The conjugate of a real number is the number itself.
- Complex roots of real-coefficient polynomials always occur in conjugate pairs.
Complex Conjugate example
Write \(\dfrac{5 + 2i}{3 – 4i}\) in the form \(a + bi\).
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The conjugate of the denominator \(3 – 4i\) is \(3 + 4i\). Multiply top and bottom by it.
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Top: \((5 + 2i)(3 + 4i) = 15 + 20i + 6i + 8i^2 = 15 – 8 + 26i = 7 + 26i\).
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Bottom: \((3 – 4i)(3 + 4i) = 9 + 16 = 25\). So the quotient is \(\frac{7 + 26i}{25}\).
The quotient is \(\frac{7}{25} + \frac{26}{25}i\), or \(0.28 + 1.04i\).
Common mistake
Watch out
Ira multiplies top and bottom by \(3 – 4i\), the denominator itself, and ends with \((3 – 4i)^2 = -7 – 24i\) still in the bottom.
Why it is wrong
Multiplying by the denominator does not remove \(i\). The multiplier must be the complex conjugate \(3 + 4i\), because \((a – bi)(a + bi) = a^2 + b^2\) is real. Only then does the denominator become the plain number 25.
Related terms and lessons
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