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AI HL · AHL 6

Complex Numbers

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AHL 6.1

The number i and Cartesian form

AHL 6.1

The number i and Cartesian form

Target: I can write the square root of a negative number as a multiple of i, and state the real and imaginary parts of a complex number. (+5 more)

By the end of this section

Key points

The number i and Cartesian form

  • \(i = \sqrt{-1}\), so \(i^2 = -1\). Treat i like a constant, such as \(\sqrt{-36} = \sqrt{-1}\sqrt{36} = 6i\).
  • A complex number in Cartesian form is \(z = x + iy\) with \(x, y \in \mathbb{R}\). The set of complex numbers is \(\mathbb{C}\).
  • \(\text{Re}(z) = x\) and \(\text{Im}(z) = y\). Both are real numbers. For \(z = 3 - 2i\), \(\text{Re}(z) = 3\) and \(\text{Im}(z) = -2\): the sign stays and the i does not.

Powers of i

  • Split the power using \(i^2 = -1\): \(i^3 = i^2 \times i = -i\) and \(i^4 = (i^2)^2 = 1\).
  • The powers repeat in a cycle of four: \(i, -1, -i, 1\). Divide the power by 4 and use the remainder.
  • \((ki)^2 = k^2i^2 = -k^2\). Square the number and the i.

Adding, subtracting, multiplying

  • Add and subtract by grouping the real parts and the imaginary parts.
  • When you subtract, the minus applies to every term of the second bracket.
  • Multiply by expanding the brackets, replace \(i^2\) by \(-1\), then group the real and imaginary parts.
  • To square, write the bracket twice. \((x + iy)^2\) is not \(x^2 + (iy)^2\): there is a middle term.

Complex algebra and real algebra

  • The rules are the ones you know. Expand, group like terms, take care with signs.
  • The one extra step: whenever \(i^2\) appears, replace it by \(-1\). This is why a product of complex numbers comes back in the form \(x + iy\).
  • Subtraction and addition are identical to real algebra: no \(i^2\) arises.

On the calculator

Complex mode

  • In Document Settings, the Real or Complex setting decides how the calculator treats \(\sqrt{-1}\).
  • In Real mode, \(\sqrt{-1}\) gives a non-real calculation message. In Complex mode (Rectangular), it gives \(i\).
  • Put the calculator in Press to Test mode as the exam requires, then check that it is still in Complex mode before you rely on it.

Screenshot and video to come

Checking complex arithmetic

  • In Complex mode, type the sum, difference, product or power exactly as you wrote it.
  • The answer comes back in the form \(x + iy\). Compare it with your hand working.
  • Use the calculator to check, not to replace the working: the exam asks for method.

Screenshot and video to come

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Powers of i and products of imaginary numbers

Simplify \(i^7\), \(i^{10}\) and \((-3i)(5i)\).

Working
  1. Split off i²

    \(i^7 = (i^2)^3 \times i = (-1)^3 \times i = -i\)

    Write the power as a power of i² times a left-over i. Then i² = −1.

  2. An even power

    \(i^{10} = (i^2)^5 = (-1)^5 = -1\)

    An even power of i is always 1 or −1.

  3. Multiply the numbers, then the i's

    \((-3i)(5i) = (-3 \times 5)(i \times i) = -15i^2 = 15\)

    Treat i like a constant: group the numbers, group the i's, then use i² = −1.

Video walkthrough coming
Square roots of negative numbers

Write \(\sqrt{-49}\) and \(\sqrt{-20}\) in the form \(ki\) with \(k\) real. The number \(z\) is \(6 - \sqrt{-25}\). Write \(z\) in the form \(x + iy\) and state \(\text{Im}(z)\).

Working
  1. Take out √−1

    \(\sqrt{-49} = \sqrt{-1}\sqrt{49} = 7i\)

    Split off \(\sqrt{-1} = i\). The rest is an ordinary square root.

  2. A surd

    \(\sqrt{-20} = \sqrt{20}\,i = 2\sqrt{5}\,i\)

    Leave \(\sqrt{20}\) as an exact surd, \(2\sqrt{5}\), or use \(4.47i\) to 3 significant figures.

  3. Write z

    \(z = 6 - \sqrt{-25} = 6 - 5i\)

    First turn the root into a multiple of i.

  4. Read off

    \(\text{Re}(z) = 6,\ \text{Im}(z) = -5\)

    Im(z) is the coefficient of i, with its sign. It is a real number: no i.

Video walkthrough coming
Adding, subtracting and multiplying

\(z = 4 + 3i\) and \(w = 1 - 2i\). Find \(z + w\), \(z - w\) and \(zw\). Give your answers in the form \(x + iy\).

Working
  1. Add

    \(z + w = (4 + 1) + (3 - 2)i = 5 + i\)

    Add the real parts, then add the imaginary parts.

  2. Subtract

    \(z - w = 4 + 3i - 1 + 2i = 3 + 5i\)

    The minus changes the sign of both parts of w. Then group as before.

  3. Expand

    \(zw = (4 + 3i)(1 - 2i) = 4 - 8i + 3i - 6i^2\)

    Expand the brackets exactly as you would with x.

  4. Use i² = −1

    \(zw = 4 - 8i + 3i + 6 = 10 - 5i\)

    −6i² becomes +6. Then group real and imaginary parts.

  5. Check

    From the calculator (complex mode): \(10 - 5i\)

    Type the product and compare. The calculator checks your working; it does not replace it.

Video walkthrough coming
Squaring a complex number

Find \((3 - 5i)^2\) in the form \(x + iy\).

Working
  1. Write the bracket twice

    \((3 - 5i)(3 - 5i) = 9 - 15i - 15i + 25i^2\)

    Squaring a bracket is not squaring each term. The middle terms matter.

  2. Use i² = −1

    \(= 9 - 30i - 25 = -16 - 30i\)

    25i² = −25. The two middle terms give −30i.

Video walkthrough coming
A mixed expression

Simplify \((2 + i)^2 - 3i(1 - 4i)\).

Working
  1. First bracket

    \((2 + i)^2 = 4 + 4i + i^2 = 3 + 4i\)

    Square first, as in any order of operations.

  2. Second term

    \(3i(1 - 4i) = 3i - 12i^2 = 12 + 3i\)

    Expand, then i² = −1.

  3. Subtract the whole term

    \((3 + 4i) - (12 + 3i) = -9 + i\)

    The minus applies to both parts of 12 + 3i.

Video walkthrough coming
Complex algebra and real algebra

Expand \((1 + x)(4 - 3x)\) and \((1 + i)(4 - 3i)\), and say what is different about the second. Give the real part of \((1 + i)(4 - 3i)\).

Working
  1. Real algebra

    \((1 + x)(4 - 3x) = 4 + x - 3x^2\)

    Four terms collect to three. The \(x^2\) term stays.

  2. Complex algebra

    \((1 + i)(4 - 3i) = 4 + i - 3i^2\)

    Same expansion, same rules.

  3. The extra step

    \(-3i^2 = +3, \text{ so } 4 + i + 3 = 7 + i\)

    i² is a real number, −1. So the answer goes back to the form x + iy. That is the only difference. Subtraction has no extra step.

Video walkthrough coming