Complex numbers
in action.
How do complex numbers describe a transformation
or combine two waves?
Explore powers and rotations, then use complex addition to combine signals.
Write your working as you go. Try each question before opening its solution.
What do you remember?
Find each answer. Use radians and keep exact values.
- For −√3 + i, find the modulus and principal argument. Write the number in Euler form.
- Give the principal argument of e−7πi/4.
- Find the modulus and principal argument of (2e−πi/5)³.
Need a reminder?
The modulus is the distance from the origin. The argument is the angle from the positive real axis.
Use −π < Arg(z) ≤ π for the principal argument. Euler form is reiθ = r(cos θ + i sin θ), where r > 0.
Check your answers
- r = √(3 + 1) = 2. The point is in quadrant II, so θ = 5π/6 and z = 2e5πi/6.
- −7π/4 + 2π = π/4.
- Cube the modulus and triple the angle: 8 and −3π/5.
Your calculator and the mathematics
The modulus gives a length. The argument gives a direction.
- In MODE, select Radian and a+bi. Return home with 2nd → MODE.
- Enter 3+4i, using 2nd → . for i. Press STO→, choose Z with ALPHA, then ENTER.
- Open MATH → CPX. Choose 5: abs( for the modulus and 4: angle( for the argument.
- In Doc → Settings & Status → Document Settings, select Radian and Rectangular.
- On a Calculator page, enter 3+4i. Choose the imaginary i from the π symbol palette. Press Ctrl + Var, type z, then ENTER to store it.
- Insert abs( and angle( from the Catalog. Use Ctrl + Enter for decimals.
Find the modulus and argument of 3 + 4i. Explain what each value tells you.
These methods work without CAS. Keep stored values at full precision and round your final answer.
A power changes both parts
Raise the modulus to the power. Multiply the angle by that power.
This is de Moivre’s rule in Euler form. Here r > 0 and n is an integer.
For example, find (2eπi/3)² in Cartesian form.
- 2² = 4 and 2 × π/3 = 2π/3.
- So (2eπi/3)² = 4e2πi/3.
- 4(cos(2π/3) + i sin(2π/3)) = −2 + 2√3i.
Try it on your calculator
Use MODE: Radian, a+bi. Insert i with 2nd → ..
Enter the whole expression, including its brackets.
−2 + 3.464101615…i, which agrees with −2 + 2√3i.
Use a Calculator page with Radian and Rectangular settings. Insert the imaginary i from the symbol palette.
Enter the whole expression, including its brackets.
−2 + 3.464101615…i, which agrees with −2 + 2√3i.
Use Ctrl + Enter for a decimal result.
Your turn: a square
Find the result in Cartesian form. Show what happens to the modulus and the angle.
Hint
Square 2 and double 5π/12. Use the new angle to find the real and imaginary components.
Show the solution
(2e5πi/12)² = 4e5πi/6
= 4(−√3/2 + i/2) = −2√3 + 2i
The principal argument is 5π/6.
Another challenge
Find (√3 − i)³ in Cartesian form.
Check the challenge
√3 − i = 2e−πi/6
(2e−πi/6)³ = 8e−πi/2 = −8i
Negative powers follow the same rule
A reciprocal inverts the modulus and reverses the angle.
- The modulus becomes 2−1 = ½. The angle becomes −2π/3.
- ½e−2πi/3 = ½(cos(−2π/3) + i sin(−2π/3)).
- The result is −¼ − (√3/4)i.
Why does the angle change sign?
A number multiplied by its reciprocal gives 1. Their moduli multiply to 1 and their angles add to 0, modulo 2π.
A negative power requires a non-zero complex number.
Try it on your calculator
Use MODE: Radian, a+bi. Insert i with 2nd → ..
Enter the whole expression, including its brackets.
−0.25 − 0.433012702…i. The angle is negative and the modulus is 0.5.
Use a Calculator page with Radian and Rectangular settings. Insert the imaginary i from the symbol palette.
Enter the whole expression, including its brackets.
−0.25 − 0.433012702…i. The angle is negative and the modulus is 0.5.
Use Ctrl + Enter for a decimal result.
Your turn: a negative power
Find the result in Cartesian form, then state its principal argument.
Hint
The new modulus is 3−2. Multiply −π/3 by −2.
Show the solution
(3e−πi/3)−2 = (1/9)e2πi/3 = −1/18 + (√3/18)i
The principal argument is 2π/3.
Spot the error
A student writes (2e3πi/4)² = 4e3πi/4. Explain the first error and correct it.
Check the correction
The angle must double too: 4e3πi/2 = 4e−πi/2 = −4i.
One multiplication, repeated
Start at z₀ = 2. Each step multiplies by 1 + i = √2eπi/4.
Predict the next coordinate in your notebook, then reveal the point.
Each step scales the distance by √2 and rotates anticlockwise by π/4.
The starting point is z₀ = 2.
Hint
Use zₙ = 2(√2)ⁿenπi/4, or multiply the previous coordinate by 1 + i.
How much has the point changed?
Sketch the five points. Describe the total scale factor and rotation from z₀ to z₄.
Is the overall scale factor 4 or 8? Explain your choice.
Check your explanation
The distance changes from 2 to 8, so the scale factor is 8/2 = 4.
Four rotations of π/4 give an anticlockwise rotation of π.
z₄/z₀ = (√2eπi/4)⁴ = 4eπi
Further: what happens after eight steps?
Predict z₈ without finding the intermediate points. Does the point return to where it started?
Check the prediction
z₈ = 2(√2)⁸e2πi = 32
It returns to the same direction, but is 16 times as far from the origin. The least positive n for which (1+i)ⁿ is positive real is 8.
Addition needs a different approach
For addition, combine the real and imaginary components.
Find this sum. Explain why the answer is not 6.
Hint
Both real components are 3/2. The imaginary components have opposite signs.
Show the solution
(3/2 + 3√3i/2) + (3/2 − 3√3i/2) = 3
The imaginary components cancel. Moduli add directly only when the non-zero vectors point in the same direction.
Euler form is useful for multiplication and powers. Cartesian form makes addition straightforward.
Calculator entry
Use MODE: Radian, a+bi. Insert i with 2nd → ..
Enter the whole expression, including its brackets.
3. The imaginary parts cancel.
Use a Calculator page with Radian and Rectangular settings. Insert the imaginary i from the symbol palette.
Enter the whole expression, including its brackets.
3. The imaginary parts cancel.
Use Ctrl + Enter for a decimal result.
Check your understanding
Use your calculator to check the values. Show the reasoning behind your answer.
- Use Euler form to find (√3 + i)² in Cartesian form.
- State the modulus and principal argument of your result.
- Explain why reiα + seiβ does not generally equal (r+s)ei(α+β).
Check the numbers
Show the reasoning
√3+i = 2eπi/6, so (√3+i)² = 4eπi/3 = 2+2√3i.
The modulus is 4 and the principal argument is π/3.
Arguments add when multiplying. Addition combines Cartesian components. The previous sum gives 3, whereas the proposed rule would give 6.
Ready to continue? Next, use complex addition to combine waves.
A rotating point draws a cosine
The point’s horizontal coordinate becomes the height of the cosine graph.
Re(3ei(2t+π/3)) = 3cos(2t+π/3)
Explain why the wave starts at 1.5. Find the two times when it crosses zero.
Check your explanation
At t=0, the value is 3cos(π/3)=1.5. It is zero whenever the rotating point crosses the imaginary axis. In this cycle, the times are π/12 and 7π/12 seconds.
The circle has radius 3, so the wave ranges from −3 to 3. The angle increases by 2t, giving a period 2π/2=π seconds.
What does the sum look like?
At each time, add the heights of the two waves.
Predict the largest value of the sum. Then show the combined wave.
Both waves complete one cycle in π seconds.
We want one cosine that gives the same values as this whole expression.
Where do 3 and 4i come from?
The shared 2t describes how both signals move. Their amplitudes and starting angles determine the combination.
Amplitude 3, starting angle 0
3(cos 0 + i sin 0) = 3
Amplitude 4, starting angle π/2
4(cos(π/2) + i sin(π/2)) = 4i
Z = 3 + 4i
Explain why the second signal contributes 4i rather than 4.
Why the full expression is preserved
3cos(2t)+4cos(2t+π/2)
= Re(3e2ti+4ei(2t+π/2))
= Re((3+4i)e2ti)
The common rotation factors out, so we can add the fixed complex numbers first. These fixed numbers are called phasors.
This method needs the same angular frequency. A sum such as 3cos(2t)+4cos(3t) does not generally reduce to one cosine with constant amplitude and phase.
From the calculator to one cosine
Store 3+4i as Z using STO→.
From MATH → CPX, evaluate:
5: the combined amplitude
0.927295218…: the phase in radians
Store 3+4i as z using Ctrl + Var.
Insert these functions from the Catalog:
5: the combined amplitude
0.927295218…: the phase in radians
Keep the shared 2t.
5cos(2t + 0.927295218…)
The amplitude is 5. The phase is about 0.927 radians.
The exact phase is tan⁻¹(4/3). The dots mean we keep the calculator’s full value.
Write the combined cosine. Label its amplitude, angular frequency and phase.
Do the two expressions agree?
Graph the original sum and the combined cosine together.
Open Y=. Use X for time.
Use Z = 3 + 4i. In WINDOW set Xmin = 0, Xmax = 2π, Ymin = −6, Ymax = 6. Press GRAPH, then TRACE to compare values.
Add a Graphs page in the same problem. Use x for time.
Use z = 3 + 4i. Choose Menu → Window/Zoom → Window Settings: x from 0 to 2π, y from −6 to 6. Use Menu → Trace → Graph Trace.
Check that the curves overlap. Confirm the amplitude of 5 and period of π seconds.
What the check tells us
Matching graphs help catch an entry or rounding error. The complex-number working explains why the expressions agree for every time.
The phase is 0.927295… radians. The horizontal shift is −0.927295…/2 ≈ −0.464 seconds, a shift to the left.
The same method with other angles
Both signals share 3t. Use each amplitude and starting angle to form the complex sum.
In Radian and a+bi modes, enter the following and store it as Z with STO→.
On a Calculator page in Radian and Rectangular modes, enter the following and store it as z with Ctrl + Var.
Enter the calculation. Then use abs and angle to find the combined amplitude and phase.
Check the complex sum
Z ≈ 4.882143 + 7.294834i
Keep the stored value at full precision for the next step.
What do these results mean?
Use MATH → CPX for both commands.
Use the Catalog for both commands.
8.78cos(3t + 0.981)
This is the combined expression to three significant figures.
The amplitude is about 8.78. The period remains 2π/3 seconds.
Graph the original sum and your combined cosine. Explain why the amplitude is less than 11.
Graph entries and explanation
Use a window from 0 to 4.2 horizontally and −12 to 12 vertically. Use the stored complex sum from the previous slide.
The peaks of the two original signals occur at different times, so they never reach 6 and 5 together.
Your turn: combine and interpret
Write the sum as one cosine. State its amplitude and period, then check it by graphing both expressions.
Time is in seconds. Give the amplitude and phase to three significant figures.
Help with the first entry
Use MODE: Radian, a+bi. Insert i with 2nd → ..
Enter the whole expression, including its brackets.
Store this complex sum. Use abs for the amplitude and angle for the phase.
Use a Calculator page with Radian and Rectangular settings. Insert the imaginary i from the symbol palette.
Enter the whole expression, including its brackets.
Store this complex sum. Use abs for the amplitude and angle for the phase.
Use Ctrl + Enter for a decimal result.
Show the solution
Z ≈ 5.391463 + 0.106502i
|Z| ≈ 5.392514, Arg(Z) ≈ 0.0197512
The result is approximately 5.39cos(2t+0.0198). Its period is π ≈ 3.14 seconds.
For the graph check, use the stored values of |Z| and Arg(Z).
When do the amplitudes add?
Predict the amplitude. Graph the full sum on your calculator, then check your prediction here.
Record and explain
| Phase difference | My prediction | Checked amplitude |
|---|---|---|
| 0 | — | — |
| π | — | — |
| π/2 | — | — |
When does the combined amplitude equal 3 + 4? Explain using the positions of the wave peaks.
Check the conclusion
The amplitudes are 7, 1 and 5. They add to 7 when the starting angles match modulo 2π, so the peaks coincide.
For the opposite case, the sum is −cos(2t)=cos(2t+π). Its amplitude is 1, with a phase of π.
Predict the amplitude of each sum. Check it with your calculator.
- 3cos(2t) + 4cos(2t)
- 3cos(2t) + 4cos(2t+π)
- 3cos(2t) + 4cos(2t+π/2)
When does the combined amplitude equal 3+4? Explain using the wave peaks.
A negative value needs interpretation
Combine the signals. Write the answer with a positive amplitude and check its value at t = 0.
Your complex sum is −4. What does that tell you about the amplitude and phase?
Calculator help
The storage arrow uses STO→ on the TI-84 and Ctrl + Var on the TI-Nspire.
Show the solution
2cos(2t)+6cos(2t+π)=−4cos(2t)=4cos(2t+π)
The amplitude is 4. The phase is π radians. At t=0, the value is −4.
An angle of −π gives the same cosine. Use π when the requested principal interval is −π < φ ≤ π.
Two voltage signals
Two ideal voltage signals add. Time t is in seconds and voltage is in volts.
Write V(t) as one cosine. Find its maximum voltage and period.
Use a graph of your answer to find when the first maximum occurs for t ≥ 0.
Help with the combination
Use MODE: Radian, a+bi. Insert i with 2nd → ..
Enter the whole expression, including its brackets.
8 − 6i. Store this result, then find its modulus and argument.
Use a Calculator page with Radian and Rectangular settings. Insert the imaginary i from the symbol palette.
Enter the whole expression, including its brackets.
8 − 6i. Store this result, then find its modulus and argument.
Use Ctrl + Enter for a decimal result.
Check the model
|8−6i|=10, Arg(8−6i)≈−0.643501109
V(t)=10cos(50t−tan⁻¹(3/4))
The maximum voltage is 10 V. The period is 2π/50=π/25≈0.126 seconds.
Finding a maximum with technology
Enter the original expression in Y₁, using X for time. Set WINDOW to X: 0 to 0.14 and Y: −12 to 12.
Choose 2nd → TRACE (CALC) → 4: maximum. Enter a left bound of 0, a right bound of 0.04 and a guess of 0.01, pressing ENTER after each.
Add a Graphs page and enter the original expression in f1(x). Set the window to x: 0 to 0.14 and y: −12 to 12.
Choose Menu → Analyze Graph → Maximum. Select the curve, then select a lower bound near 0 and an upper bound near 0.04 around the first peak.
First maximum ≈ (0.01287, 10)
Interpret both coordinates. Give the first maximum time to three significant figures.
Check the interpretation
The voltage first reaches 10 V at 0.0129 s. The horizontal coordinate is a time, and the vertical coordinate is a voltage.
Algebra confirms this: 50t−tan⁻¹(3/4)=0, so t=tan⁻¹(3/4)/50.
One final model
Combine the signals using your calculator. State the amplitude and period, then verify your model graphically.
Explain why 5cos(4t)+5cos(3t+π/2) needs a different approach.
Check your answer
Z=5+5i, |Z|=5√2, Arg(Z)=π/4
The combined signal is 5√2cos(4t+π/4), with amplitude 5√2≈7.07 and period π/2≈1.57 seconds.
For the second expression, the angular frequencies differ. The waves drift in and out of alignment, so the same constant-amplitude cosine method does not apply.
A method you can reuse
Powers and transformations
(reiθ)ⁿ = rⁿeinθ
Use the modulus and argument to explain the scale and rotation.
Check the Cartesian result on your calculator.
Combining cosine signals
Check that the angular frequencies match.
Z=A(cos α+i sin α)+B(cos β+i sin β)
Use abs(Z) and angle(Z) to find R and φ.
Rcos(ωt+φ)
Explain what each calculator result means. Check the whole expression on a graph.
Useful details
For ω>0, the period is 2π/ω. The phase φ is an angle. The horizontal shift is −φ/ω.
The argument command uses the correct quadrant. If the combined complex number is zero, the signals cancel and there is no unique phase or fundamental period.
For sine signals, first use sin u=cos(u−π/2), then follow the cosine method.