Complex Numbers

Have a go first. Then open the working one line at a time.

1. Write z and z*

   \(z = x + iy,\ z^* = x - iy\)

   Put both into the equation.

2. Expand

   \(x + iy + 5i = 3x - 3iy + 2\)

   The conjugate changes the sign of \(iy\), so \(3z^* = 3x - 3iy\).

3. Equate real parts and imaginary parts

   Real: \(x = 3x + 2\); imaginary: \(y + 5 = -3y\)

   Both parts of the left side must equal both parts of the right.

4. Solve

   \(x = -1,\ y = -\dfrac{5}{4}\), so \(z = -1 - \dfrac{5}{4}i\)

   Check in the calculator if you like: complex mode, with \(z\) typed in.

Have a go first. Then open the working one line at a time.

1. Expand each term

   \(zz^* = x^2 + y^2\), \(z^2 = x^2 - y^2 + 2ixy\)

   \(zz^*\) is always real. Square \(x + iy\) as in Lesson 1.

2. Add

   \(zz^* + z^2 = 2x^2 + 2ixy\)

   The \(y^2\) terms cancel.

3. Equate parts

   Real: \(2x^2 = 8\); imaginary: \(2xy = 12\)

   The real part gives \(x\) first: \(x^2 = 4\), so \(x = \pm 2\).

4. Find y for each x

   \(x = 2: y = 3\); \(x = -2: y = -3\)

   \(xy = 6\) fixes \(y\) from each value of \(x\).

5. State both solutions

   \(z = 2 + 3i\) or \(z = -2 - 3i\)

   A quadratic-type equation gives two solutions.

Have a go first. Then open the working one line at a time.

1. Work out each number

   \(-z = -3 - i,\ z^* = 3 - i,\ 2z = 6 + 2i\)

   Change signs, or double both parts.

2. Multiply by i

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

   Multiplying by \(i\) turns the point a quarter turn anticlockwise about the origin.

3. Add and subtract

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

   Add or subtract the real parts and the imaginary parts separately.

4. Plot the points

   \(z(3, 1),\ -z(-3, -1),\ z^*(3, -1),\ 2z(6, 2),\ iz(-1, 3),\ w(1, 2),\ z + w(4, 3),\ z - w(2, -1)\)

   \(0, z, z + w, w\) form a parallelogram: join them and shade it. \(z - w\) is \(z\) plus \(-w\), the reflection of \(w\) through the origin.

Have a go first. Then open the working one line at a time.

1. The discriminant

   \(b^2 - 4ac = 16 - 4(4)(5) = -64\)

   Negative: the roots are complex.

2. Solve

   From the calculator (complex mode): \(x = -0.5 \pm i\)

   By hand: \(x = \dfrac{-4 \pm \sqrt{-64}}{8} = \dfrac{-4 \pm 8i}{8}\).

3. Plot

   \((-0.5, 1)\) and \((-0.5, -1)\)

   A conjugate pair: reflections in the real axis, at the x-coordinate of the vertex, \(x = -0.5\).

Have a go first. Then open the working one line at a time.

1. Divide the fraction

   \(\dfrac{20i}{2 - i} \times \dfrac{2 + i}{2 + i} = \dfrac{40i + 20i^2}{5} = \dfrac{-20 + 40i}{5} = -4 + 8i\)

   Multiply by the conjugate of the denominator: \((2 - i)(2 + i) = 5\).

2. Add

   \(z = 5 - 3i + (-4 + 8i) = 1 + 5i\)

   Real parts together, imaginary parts together.

3. Conjugate

   \(z^* = 1 - 5i\)

   Simplify first. Take the conjugate of the answer, not of each piece.

Have a go first. Then open the working one line at a time.

1. Multiply by the conjugate

   \(\dfrac{3k - i}{k + 2i} \times \dfrac{k - 2i}{k - 2i}\)

   \(k\) is real, so the conjugate of \(k + 2i\) is \(k - 2i\).

2. Top

   \((3k - i)(k - 2i) = 3k^2 - 6ki - ki + 2i^2 = (3k^2 - 2) - 7ki\)

   \(2i^2 = -2\).

3. Bottom

   \((k + 2i)(k - 2i) = k^2 + 4\)

   Always real, and never 0 for real \(k\).

4. Write as x + iy

   \(\dfrac{3k^2 - 2}{k^2 + 4} - \dfrac{7k}{k^2 + 4}\,i\)

   Divide each part by \(k^2 + 4\).

Have a go first. Then open the working one line at a time.

1. Write z as x + iy

   \(\dfrac{(a + 4i)(a + i)}{(a - i)(a + i)} = \dfrac{a^2 - 4}{a^2 + 1} + \dfrac{5a}{a^2 + 1}\,i\)

   Multiply by the conjugate of the denominator.

2. Use the condition

   \(\dfrac{5a}{a^2 + 1} = 2\)

   Only the imaginary part is used.

3. Rearrange

   \(5a = 2a^2 + 2,\quad 2a^2 - 5a + 2 = 0\)

   Get a quadratic in \(a\).

4. Solve

   From the calculator: \(a = 2\) or \(a = 0.5\)

   Both are real, so both are allowed. Put each back in to check.

Have a go first. Then open the working one line at a time.

1. Calculate

   From the calculator (complex mode): \(z^2 = 2i,\ z^3 = -2 + 2i,\ z^4 = -4\)

   Type each power. Check one by hand: \((1 + i)^2 = 1 + 2i + i^2 = 2i\).

2. Plot

   \((1, 1),\ (0, 2),\ (-2, 2),\ (-4, 0)\)

   Each point is further from the origin and has turned anticlockwise from the one before.