Complex numbers · Definition

Complex conjugate

complex analysiscomplex numbers

Before you start you need to know

It is the reflection of the complex plane across the real axis, and the tool for dividing, extracting real and imaginary parts, and later measuring distances.

Definition

The conjugate of a complex number \(z\), with real part \(a\) and imaginary part \(b\), is the complex number \(\bar z\) that keeps the real part and flips the sign of the imaginary part.

\[z=a+bi\ \Longrightarrow\ \bar z=a-bi\]

Conjugating means reflecting the point \((a,b)\) across the horizontal axis. Sum and product are preserved: \(\overline{z+w}=\bar z+\bar w\), \(\overline{zw}=\bar z\,\bar w\), and \(\overline{\bar z}=z\). Hence if a polynomial with real coefficients has a complex root, its conjugate is a root too.

The product \(z\bar z=a^2+b^2\) is always a nonnegative real; it is the key to dividing: \(\frac{w}{z}=\frac{w\bar z}{z\bar z}\) makes the denominator real. For instance, \(\frac{1}{1+i}=\frac{1-i}{2}\). The number \(a^2+b^2\) will be the square of the modulus, defined next.

The parts of \(z\) are recovered with the conjugate: \(\operatorname{Re}z=\frac{z+\bar z}{2}\) and \(\operatorname{Im}z=\frac{z-\bar z}{2i}\). Also, \(z=\bar z\) means \(z\) is real, and \(\bar z=-z\) means it is pure imaginary.

A typical error is to conjugate only part of an expression or to flip the sign of the real part. Keep in mind too that conjugation is not an analytic operation: \(z\mapsto\bar z\) will not be differentiable, which matters when we study the Cauchy-Riemann conditions.

Think about it

If \(p\) has real coefficients and \(p(z_0)=0\), what else vanishes?

\(p(\bar z_0)=0\), since \(\overline{p(z_0)}=p(\bar z_0)\) when the coefficients are real.

What geometric transformation is \(z\mapsto\bar z\)?

The reflection across the real axis.

Exercises 10

  1. Find the conjugate of \(3-4i\).
  2. If \(z=2+3i\), compute \(z\bar z\).
  3. If \(z=5-2i\), compute \(z+\bar z\).
  4. If \(z=2-6i\), compute \(\dfrac{z-\bar z}{2i}\).
  5. Write \(\dfrac{1+2i}{3-i}\) in the form \(a+bi\).
  6. Compute \(\overline{(1+i)(2-i)}\).
  7. If \(z=1+2i\), compute \((\bar z)^2\).
  8. Find the complex number \(z\) with \(z+2\bar z=6-3i\).
  9. For which complex numbers does \(z=\bar z\) hold?
  10. If \(z\neq0\) and \(\bar z=-z\), what is \(z\)?

References

  • Elias M. Stein, Rami Shakarchi, Complex Analysis, 1st edition, Princeton University Press, 2003. Cap. 1, §1.1
  • James Ward Brown, Ruel V. Churchill, Complex Variables and Applications, 9th edition, McGraw-Hill, 2014. §5
  • Lars V. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, 1979. Cap. 1, §1.1