IB Mathematics Applications & Interpretation Topic…

Turning Multiplication Into Rotation

In Cartesian form $a + bi$, adding complex numbers is easy but multiplying is fiddly.
There is a second way to describe the same number, by its distance from the origin and its angle, and in this form multiplication becomes almost effortless.
Multiplying two complex numbers simply multiplies their lengths and adds their angles, which is the same as scaling and rotating.

Note

This polar view is what makes complex numbers the natural language for waves, rotations, and alternating current.

Modulus-Argument (Polar) Form

Any complex number can be written using its modulus $r = |z|$ and argument $\theta = \arg(z)$.
From the diagram, the real part is $r\cos\theta$ and the imaginary part is $r\sin\theta$. $$z = r\cos\theta + i\,r\sin\theta = r(\cos\theta + i\sin\theta)$$
This is abbreviated to $r\,\mathrm{cis}\,\theta$, and it is given in your formula booklet under AHL 1.13. $$z = r(\cos\theta + i\sin\theta) = r\,\mathrm{cis}\,\theta = re^{i\theta}$$

Note

In polar and exponential form the angle is in radians, so put your GDC in radian mode.
The exponential form $re^{i\theta}$ is Euler's form, and it behaves like an ordinary power.

Converting Between Cartesian And Polar Form

To go from Cartesian to polar, find $r = \sqrt{a^2 + b^2}$ and $\theta = \arg(z)$, checking the quadrant.
To go from polar to Cartesian, evaluate $a = r\cos\theta$ and $b = r\sin\theta$.

Example question

Write $z = 1 + \sqrt{3}\,i$ in the form $re^{i\theta}$, with $\theta$ in radians.

Solution

Modulus: $$r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2$$
The point $(1, \sqrt{3})$ is in the first quadrant, so $\theta = \arctan\!\left(\dfrac{\sqrt{3}}{1}\right) = \dfrac{\pi}{3}$.
So $z = 2e^{i\pi/3}$.

Multiplying And Dividing In Polar Form

To multiply, multiply the moduli and add the arguments. $$z_1 z_2 = r_1 r_2\,\mathrm{cis}(\theta_1 + \theta_2)$$
To divide, divide the moduli and subtract the arguments. $$\frac{z_1}{z_2} = \frac{r_1}{r_2}\,\mathrm{cis}(\theta_1 - \theta_2)$$

Example question

Given $z_1 = 2\,\mathrm{cis}\dfrac{\pi}{3}$ and $z_2 = 3\,\mathrm{cis}\dfrac{\pi}{6}$, find $z_1 z_2$.

Solution

Multiply the moduli: $$2 \times 3 = 6$$
Add the arguments: $$\dfrac{\pi}{3} + \dfrac{\pi}{6} = \dfrac{\pi}{2}$$
So $z_1 z_2 = 6\,\mathrm{cis}\dfrac{\pi}{2} = 6i$

Common Mistake

Multiply the moduli, but add the arguments; do not multiply the angles.
Do addition and subtraction in Cartesian form, and multiplication, division, and powers in polar or exponential form.

De Moivre's Theorem: Powers Made Easy

Raising to a power just repeats multiplication, so the modulus is raised to the power and the argument is multiplied by it.
This is De Moivre's theorem, also given in your formula booklet under AHL 1.13.

$$z^n = r^n(\cos n\theta + i\sin n\theta) = r^n\,\mathrm{cis}(n\theta) = r^n e^{in\theta}$$

Example question

Use De Moivre's theorem to evaluate $(1 + i)^8$.

Solution

Write $1 + i$ in polar form: $r = \sqrt{2}$ and $\theta = \dfrac{\pi}{4}$.
Apply De Moivre: $$(1+i)^8 = (\sqrt{2})^8\,\mathrm{cis}\!\left(8 \times \dfrac{\pi}{4}\right)$$
Simplify: $(\sqrt{2})^8 = 2^4 = 16$ and $8 \times \dfrac{\pi}{4} = 2\pi$, so the result is $16\,\mathrm{cis}(2\pi) = 16$.

Common Mistake

Raise the modulus to the power too: $z^n$ has modulus $r^n$, not $r$.
Keep angles in radians when using the exponential form.

Adding Waves Of The Same Frequency

Two sine waves of the same frequency but different amplitudes and phases add to a single sine wave of that same frequency.
Representing each wave as a complex number, called a phasor, turns this hard trig sum into simple complex addition.
Add the phasors in Cartesian form, then convert back to modulus and argument to read off the combined amplitude and phase.
This is exactly how engineers combine alternating signals.

Example question

Two signals are modelled by phasors $z_1 = 3\,\mathrm{cis}\,0$ and $z_2 = 4\,\mathrm{cis}\dfrac{\pi}{2}$. Find the amplitude of their sum.

Solution

Convert to Cartesian: $z_1 = 3$ and $z_2 = 4i$.
Add them: $$z_1 + z_2 = 3 + 4i$$
The amplitude is the modulus: $$|3 + 4i| = \sqrt{3^2 + 4^2} = 5$$

Active recall

Write the polar and exponential forms of a complex number with modulus $r$ and argument $\theta$.
How do the modulus and argument change when you multiply two complex numbers?
State De Moivre's theorem.
Which operations are easier in Cartesian form, and which in polar form?
Why must the GDC be in radian mode for exponential form?

Turning Multiplication Into Rotation

In Cartesian form $a + bi$, adding complex numbers is easy but multiplying is fiddly.
There is a second way to describe the same number, by its distance from the origin and its angle, and in this form multiplication becomes almost effortless.
Multiplying two complex numbers simply multiplies their lengths and adds their angles, which is the same as scaling and rotating.

Note

This polar view is what makes complex numbers the natural language for waves, rotations, and alternating current.

Modulus-Argument (Polar) Form

Any complex number can be written using its modulus $r = |z|$ and argument $\theta = \arg(z)$.
From the diagram, the real part is $r\cos\theta$ and the imaginary part is $r\sin\theta$. $$z = r\cos\theta + i\,r\sin\theta = r(\cos\theta + i\sin\theta)$$
This is abbreviated to $r\,\mathrm{cis}\,\theta$, and it is given in your formula booklet under AHL 1.13. $$z = r(\cos\theta + i\sin\theta) = r\,\mathrm{cis}\,\theta = re^{i\theta}$$

Note

In polar and exponential form the angle is in radians, so put your GDC in radian mode.
The exponential form $re^{i\theta}$ is Euler's form, and it behaves like an ordinary power.

Converting Between Cartesian And Polar Form

To go from Cartesian to polar, find $r = \sqrt{a^2 + b^2}$ and $\theta = \arg(z)$, checking the quadrant.
To go from polar to Cartesian, evaluate $a = r\cos\theta$ and $b = r\sin\theta$.

Example question

Write $z = 1 + \sqrt{3}\,i$ in the form $re^{i\theta}$, with $\theta$ in radians.

Solution

Modulus: $$r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2$$
The point $(1, \sqrt{3})$ is in the first quadrant, so $\theta = \arctan\!\left(\dfrac{\sqrt{3}}{1}\right) = \dfrac{\pi}{3}$.
So $z = 2e^{i\pi/3}$.

Multiplying And Dividing In Polar Form

To multiply, multiply the moduli and add the arguments. $$z_1 z_2 = r_1 r_2\,\mathrm{cis}(\theta_1 + \theta_2)$$
To divide, divide the moduli and subtract the arguments. $$\frac{z_1}{z_2} = \frac{r_1}{r_2}\,\mathrm{cis}(\theta_1 - \theta_2)$$

Example question

Given $z_1 = 2\,\mathrm{cis}\dfrac{\pi}{3}$ and $z_2 = 3\,\mathrm{cis}\dfrac{\pi}{6}$, find $z_1 z_2$.

Solution

Multiply the moduli: $$2 \times 3 = 6$$
Add the arguments: $$\dfrac{\pi}{3} + \dfrac{\pi}{6} = \dfrac{\pi}{2}$$
So $z_1 z_2 = 6\,\mathrm{cis}\dfrac{\pi}{2} = 6i$

Common Mistake

Multiply the moduli, but add the arguments; do not multiply the angles.
Do addition and subtraction in Cartesian form, and multiplication, division, and powers in polar or exponential form.

De Moivre's Theorem: Powers Made Easy

Raising to a power just repeats multiplication, so the modulus is raised to the power and the argument is multiplied by it.
This is De Moivre's theorem, also given in your formula booklet under AHL 1.13.

$$z^n = r^n(\cos n\theta + i\sin n\theta) = r^n\,\mathrm{cis}(n\theta) = r^n e^{in\theta}$$

Example question

Use De Moivre's theorem to evaluate $(1 + i)^8$.

Solution

Write $1 + i$ in polar form: $r = \sqrt{2}$ and $\theta = \dfrac{\pi}{4}$.
Apply De Moivre: $$(1+i)^8 = (\sqrt{2})^8\,\mathrm{cis}\!\left(8 \times \dfrac{\pi}{4}\right)$$
Simplify: $(\sqrt{2})^8 = 2^4 = 16$ and $8 \times \dfrac{\pi}{4} = 2\pi$, so the result is $16\,\mathrm{cis}(2\pi) = 16$.

Common Mistake

Raise the modulus to the power too: $z^n$ has modulus $r^n$, not $r$.
Keep angles in radians when using the exponential form.

Adding Waves Of The Same Frequency

Two sine waves of the same frequency but different amplitudes and phases add to a single sine wave of that same frequency.
Representing each wave as a complex number, called a phasor, turns this hard trig sum into simple complex addition.
Add the phasors in Cartesian form, then convert back to modulus and argument to read off the combined amplitude and phase.
This is exactly how engineers combine alternating signals.

Example question

Two signals are modelled by phasors $z_1 = 3\,\mathrm{cis}\,0$ and $z_2 = 4\,\mathrm{cis}\dfrac{\pi}{2}$. Find the amplitude of their sum.

Solution

Convert to Cartesian: $z_1 = 3$ and $z_2 = 4i$.
Add them: $$z_1 + z_2 = 3 + 4i$$
The amplitude is the modulus: $$|3 + 4i| = \sqrt{3^2 + 4^2} = 5$$

Active recall

Write the polar and exponential forms of a complex number with modulus $r$ and argument $\theta$.
How do the modulus and argument change when you multiply two complex numbers?
State De Moivre's theorem.
Which operations are easier in Cartesian form, and which in polar form?
Why must the GDC be in radian mode for exponential form?

IB Mathematics Applications & Interpretation AHL 1.13—complex Numbers Continued Notes

IB Mathematics Applications & Interpretation AHL 1.13—complex Numbers Continued Notes

Turning Multiplication Into Rotation

Modulus-Argument (Polar) Form

Converting Between Cartesian And Polar Form

Multiplying And Dividing In Polar Form

De Moivre's Theorem: Powers Made Easy

Adding Waves Of The Same Frequency

Turning Multiplication Into Rotation

Modulus-Argument (Polar) Form

Converting Between Cartesian And Polar Form

Multiplying And Dividing In Polar Form

De Moivre's Theorem: Powers Made Easy

Adding Waves Of The Same Frequency

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

IB Mathematics Applications & Interpretation AHL 1.13—complex Numbers Continued Notes

IB Mathematics Applications & Interpretation AHL 1.13—complex Numbers Continued Notes

Turning Multiplication Into Rotation

Modulus-Argument (Polar) Form

Converting Between Cartesian And Polar Form

Multiplying And Dividing In Polar Form

De Moivre's Theorem: Powers Made Easy

Adding Waves Of The Same Frequency

Turning Multiplication Into Rotation

Modulus-Argument (Polar) Form

Converting Between Cartesian And Polar Form

Multiplying And Dividing In Polar Form

De Moivre's Theorem: Powers Made Easy

Adding Waves Of The Same Frequency