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Complex numbers and the Argand plane

Calculate with real and imaginary parts.

Learn one idea at a time

Choose a unit. Each has worked examples and practice.

  1. Read complex numbers in the plane

    Connect real and imaginary components with points, vectors and operations.

  2. Use square roots and conjugates

    Distinguish both roots from a principal root and interpret conjugation.

  3. Divide and solve equations

    Rationalise a complex denominator and solve simultaneous equations.

  4. Recover and factor polynomials

    Check real coefficients before pairing conjugate roots.

  5. Measure modulus and argument

    Choose the principal argument from the correct quadrant.

  6. Explore the historical appendices

    Read the Euler biography and the source’s seventeen-equation reading list.

Read or print the complete chapter

By the end, you can

  • Explain the extension from real to complex numbers
  • Find complex roots of quadratic equations
  • Use modulus, argument and conjugates
  • Perform the four operations with complex numbers
  • Use equality of complex numbers
  • Use conjugate roots for polynomial equations with real coefficients
  • Represent complex numbers on an Argand diagram
  • Interpret conjugation, negation, addition, subtraction and multiplication by i geometrically

6 focused units · about 45 minutes for the full chapter. Allow extra time for practice.

The central idea

Complex numbers extend the real numbers with i²=−1. For z=a+bi, its conjugate is a−bi and its modulus is a2+b2\sqrt{a^2+b^2}. An argument gives direction in the Argand plane, with the correct quadrant essential. Multiplication adds arguments and multiplies moduli; multiplying by i rotates by π/2 anticlockwise.

The power cycle of i

Define i by i2=−1i^2=-1. Then i3=−ii^3=-i, i4=1i^4=1, i5=ii^5=i, i6=−1i^6=-1; the cycle repeats every four powers. For positive integers k, i2k=(−1)ki^{2k}=(-1)^k and i2k+1=(−1)kii^{2k+1}=(-1)^ki.

The equation x2+1=0x^2+1=0 has roots ±i\pm i. Write a complex number as a+iba+ib with real a,b: the real part is a and the imaginary part is b, not ib. For positive real a the principal root is −a=ia\sqrt{-a}=i\sqrt a. This does not license multiplying principal roots of arbitrary complex values without checking branches. “Complex” refers to the connected real and imaginary parts.

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Worked example

(1+i)/(1−i)(1+i)/(1-i) becomes (1+i)2/2=i(1+i)^2/2=i. The numerator has modulus 2\sqrt2 and argument π/4\pi/4; the denominator has the same modulus and argument −π/4-\pi/4, so the quotient has modulus 11 and argument π/2\pi/2.

An inverse tangent value alone may select the wrong quadrant.

Check your understanding

Find the modulus and principal argument of −1+i-1+i using the interval (−π,π](-\pi,\pi].

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Practice questions

54 examples and practice questions, in 9 sections.

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