Menu
Sign up Sign in

AI HL · AHL 6

Complex Numbers

0 of 66 skills done

0
AHL 6.7

Products, quotients and powers in cis form

AHL 6.7

Products, quotients and powers in cis form

Target: I can multiply and divide complex numbers in the form \(r\,\text{cis}\,\theta\). (+4 more)

By the end of this section

Key points

The three rules

  • To multiply, multiply the moduli and add the arguments: \(|zw| = |z||w|\) and \(\arg(zw) = \arg z + \arg w\). In this form, \(r\,\text{cis}\,\alpha \times s\,\text{cis}\,\beta = rs\,\text{cis}(\alpha + \beta)\).
  • To divide, divide the moduli and subtract the arguments: \(\left|\dfrac{z}{w}\right| = \dfrac{|z|}{|w|}\) and \(\arg\left(\dfrac{z}{w}\right) = \arg z - \arg w\).
  • To raise to a power, raise the modulus to the power and multiply the argument by it: \(|z^n| = |z|^n\) and \(\arg(z^n) = n\arg z\). In this form, \((r\,\text{cis}\,\theta)^n = r^n\,\text{cis}(n\theta)\).
  • Learn the rules, not only the calculator. You need them to work with expressions that have letters in them.

Giving the answer

  • Adding or subtracting can take the argument out of range. Add or subtract \(2\pi\) until it is in the range the question gives, usually \(0 \le \theta < 2\pi\). The point on the Argand diagram does not change.
  • In \(\dfrac{z^2}{w^3}\), work out \(z^2\) and \(w^3\) first, with moduli and arguments. Then divide.
  • To give an answer in the form \(x + iy\), use \(x = r\cos\theta\) and \(y = r\sin\theta\). Use exact values when the question says exact.

Mistakes to avoid

  • Do not add the moduli when you multiply. The moduli multiply and the arguments add.
  • Do not multiply the arguments of two numbers. Only a power multiplies the argument, by the whole number \(n\).
  • The power applies to the modulus as well: \(|z^3| = |z|^3\), not \(3|z|\).

On the calculator

Checking a product or a quotient

  • In Complex mode with the angle set to radians, type each number in polar form and multiply or divide. The calculator returns the answer in the form \(x + iy\).
  • Use the calculator's tools for the polar angle and the magnitude of the result to read off the modulus and the argument.
  • Your working is what is marked. Use the calculator to check it.

Screenshot and video to come

Explore: multiplying turns and scales

Drag z (black) and w (teal). The orange arrow is zw. Predict its distance from O and its angle before you read the working. Or press a button to see what multiplying does to a whole shape.

The points snap to whole and half numbers. Select one (Tab) and use the arrow keys to move it. Angles are in radians, to 3 significant figures; the sums are worked from the unrounded values. The dashed circle has radius 1.

  1. Press "Multiply by i". Where do A, B and C go? What happens to the size of the triangle, and to its angles?
  2. Drag w onto the dashed circle. What does multiplying by w do to the distance of every point from O? What does it still do to the angle?
  3. Press "Multiply by i" twice in your head: i × i. Which button gives the same result? Check with the arguments: π/2 + π/2.
  4. Show z ÷ w. Predict its distance and angle first. Why is dividing by w the same as turning back and shrinking?

Explore: the powers of z

Drag z (orange) or use the sliders for its modulus and argument. The teal points are z, z², z³ and so on, up to the power you choose. Predict where the next point goes before you move the slider for n.

The argument goes up in steps of π/24. Select z (Tab): the left and right arrow keys change its argument, the up and down keys change its modulus. The dashed circle has radius 1. The black arrow goes to the power you chose; the gold arc is its argument.

  1. Set r = 1. Where do the powers go as n grows? What happens when θ = π/4 and you reach n = 8?
  2. Set r below 1, then above 1, with the same θ. What changes about the spiral, and what stays the same?
  3. Between one power and the next, by what factor does the distance from O change? By what angle does the point turn? Check against the rule |zn| = |z|n and arg(zn) = n arg z.
  4. Choose θ = π/3 and r = 1. Which power of z is equal to 1? What does that tell you about the points?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A product in cis form

\(z = 4\,\text{cis}\left(\frac{\pi}{6}\right)\) and \(w = 7\,\text{cis}\left(\frac{\pi}{4}\right)\). Write \(zw\) in the form \(r\,\text{cis}\,\theta\). The argument of \(zw\) is \(\frac{k\pi}{12}\).

Working
  1. Multiply the moduli

    \(|zw| = 4 \times 7 = 28\)

    Moduli multiply: \(4 \times 7\), not \(4 + 7\).

  2. Add the arguments

    \(\arg(zw) = \frac{\pi}{6} + \frac{\pi}{4} = \frac{5\pi}{12}\)

    Write both over 12: \(\frac{2\pi}{12} + \frac{3\pi}{12}\).

  3. Write the answer

    \(zw = 28\,\text{cis}\left(\frac{5\pi}{12}\right)\)

    The argument is already between 0 and \(2\pi\).

Video walkthrough coming
A quotient of powers in Cartesian form

\(z = 6\,\text{cis}\left(\frac{\pi}{3}\right)\) and \(w = 2\,\text{cis}\left(\frac{\pi}{9}\right)\). Write \(\dfrac{z^2}{w^3}\) in the form \(x + iy\). Give the real part \(x\).

Working
  1. Find \(z^2\)

    \(|z^2| = 6^2 = 36,\quad \arg(z^2) = 2 \times \frac{\pi}{3} = \frac{2\pi}{3}\)

    Square the modulus. Double the argument.

  2. Find \(w^3\)

    \(|w^3| = 2^3 = 8,\quad \arg(w^3) = 3 \times \frac{\pi}{9} = \frac{\pi}{3}\)

    Cube the modulus. Treble the argument.

  3. Divide

    \(\left|\frac{z^2}{w^3}\right| = \frac{36}{8} = 4.5,\quad \arg = \frac{2\pi}{3} - \frac{\pi}{3} = \frac{\pi}{3}\)

    Divide the moduli. Subtract the arguments.

  4. Convert to Cartesian

    \(\frac{z^2}{w^3} = 4.5\,\text{cis}\left(\frac{\pi}{3}\right) = \frac{9}{4}+\frac{9 \sqrt{3}}{4}i\)

    From the calculator: the exact values of cos and sin, or decimals. \(x = \frac{9}{4} = 2.25\).

Video walkthrough coming
An argument outside the range

\(z = 3\,\text{cis}\left(\frac{11\pi}{6}\right)\) and \(w = 4\,\text{cis}\left(\frac{\pi}{2}\right)\). Write \(zw\) in the form \(r\,\text{cis}\,\theta\) with \(0 \le \theta < 2\pi\). The argument is \(\frac{k\pi}{3}\).

Working
  1. Multiply the moduli

    \(|zw| = 3 \times 4 = 12\)

    Moduli multiply.

  2. Add the arguments

    \(\arg(zw) = \frac{11\pi}{6} + \frac{\pi}{2} = \frac{7\pi}{3}\)

    The sum is bigger than \(2\pi\): one full turn too far.

  3. Take off a full turn

    \(\frac{7\pi}{3} - 2\pi = \frac{\pi}{3}\)

    Subtract \(2\pi\) so that \(0 \le \theta < 2\pi\).

  4. Write the answer

    \(zw = 12\,\text{cis}\left(\frac{\pi}{3}\right)\)

    For a quotient the same happens the other way: a negative argument gets \(2\pi\) added.

Video walkthrough coming