Eclatech Solutions | Interactive Pre-Calculus ToolsFree for educational use
The Complex Number Visualizer
Operations on complex numbers, polar form, products, quotients, powers, and roots
"Imaginary" is the worst name in mathematics. These numbers describe AC electricity, signal processing, and control systems for real physical machines. What multiplying by i actually does is rotate you 90 degrees. That is not imaginary. That is geometry.
Step 1 of 8: Plotting a Complex Number
Here is \(3 + 4i\) plotted as a point. Real part on the horizontal axis, imaginary part on the vertical. How far is this point from the origin?
The point 3 plus 4i is plotted at coordinates (3, 4) on the complex plane.
The distance is 5, by the Pythagorean theorem: \(\sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5\).
That distance is called the modulus. The angle from the positive real axis is \(\arctan(4/3) \approx 53.1°\). That is called the argument. You have seen this before. It is polar coordinates from Simulation 21 with new vocabulary.
Step 2 of 8: Multiplying by i
Take \(3 + 4i\) and multiply it by \(i\). The result is \(i(3 + 4i) = 3i + 4i^2 = -4 + 3i\). Both points are plotted below.
Two points on the complex plane: 3 plus 4i (original) and negative 4 plus 3i (after multiplying by i).
What happened geometrically?
Now watch it happen again. Each click multiplies by \(i\) one more time.
\(3+4i\)→\(-4+3i\)
Four multiplications by \(i\) bring you back to the start. Four 90-degree rotations make a full 360-degree turn.
\(i^4 = 1\), and now you can see why. Multiplying by \(i\) is a 90-degree counterclockwise rotation. It is geometry, not magic.
Step 3 of 8: Multiply the Hard Way (FOIL)
Multiply \((1 + i)(\sqrt{3} + i)\) using FOIL. What is the result?
Complex plane showing 1 plus i, square root of 3 plus i, and their product (square root of 3 minus 1) plus (square root of 3 plus 1) times i.
Step 4 of 8: Multiply the Polar Way
Convert both numbers to polar form:
\(1 + i = \sqrt{2}\,\text{cis}\,45°\) \(\sqrt{3} + i = 2\,\text{cis}\,30°\)
To multiply in polar form, what do you do with the moduli and arguments?
Complex plane showing two factors in polar form and their product. Moduli multiply and arguments add.
Step 5 of 8: See It Move
One number is fixed at \(2\,\text{cis}\,0° = 2\). The other number rotates around the unit circle. Watch what happens to their product.
Animation: a point rotates on the unit circle while the product traces a circle of radius 2.
Press Play to watch multiplication in action. The fixed number (green, modulus 2) scales the rotating number (blue, on the unit circle). The product (purple) traces a circle of radius 2, always at the sum of the two arguments.
Multiply the distances. Add the angles. That is all multiplication does.
The product of two complex numbers has modulus equal to the product of the moduli, and argument equal to the sum of the arguments. Multiplication is scaling and rotation combined.
Step 6 of 8: Division
Multiplication multiplies moduli and adds arguments. Based on that pattern, what should division do?
Test it. Divide \(6\,\text{cis}\,120°\) by \(3\,\text{cis}\,45°\).
DeMoivre example: z equals square root 2 cis 30 degrees. z to the fifth equals 4 square root 2 cis 150 degrees.
You did not learn DeMoivre's theorem. You noticed it.
\((r\,\text{cis}\,\theta)^n = r^n\,\text{cis}(n\theta)\). Raise the modulus to the power. Multiply the argument by the power. That is the whole theorem, and it follows directly from what multiplication does.
Step 8 of 8: Roots, the Payoff
Find the cube roots of 8. You know one: \(2\). But there are three cube roots. How many complex cube roots does 8 have in total?
The modulus of each root is \(\sqrt[3]{8} = 2\), so all three sit on a circle of radius 2. The argument of 8 is 0, but angles wrap: 0, \(2\pi\), \(4\pi\) are all valid. Divide each by 3:
\(\theta = 0°, \quad 120°, \quad 240°\)
Three roots, evenly spaced 120 degrees apart:
Three cube roots of 8 on a circle of radius 2: at 0 degrees (which is 2), at 120 degrees (negative 1 plus square root 3 times i), and at 240 degrees (negative 1 minus square root 3 times i). They form an equilateral triangle.
The \(n\)th roots of any complex number always sit at the vertices of a regular \(n\)-gon on a circle.
Dividing a full turn (\(360°\)) by \(n\) is exactly what makes a regular polygon. The cube roots sit \(120°\) apart, forming a triangle. Fourth roots sit \(90°\) apart, forming a square. This pattern holds for every \(n\).
Explore Complex Operations
Enter two complex numbers or click on the plane to place them. Choose an operation to see the geometric result.
3
Click on the plane to place complex numbers, or use the input fields above.
\(z_1 = 3 + 2i\)
\(z_2 = 1 + 2i\)
Result: \(z_1 + z_2 = 4 + 4i\)
Addition: add the real parts and the imaginary parts separately. Geometrically, this is the parallelogram rule, just like adding vectors.
Try This
Polar Form and Multiplication
Write \(3 + 4i\) and \(-2 + 2i\) in polar form. Multiply them in polar form. Then convert back to rectangular and check against FOIL.
What is \((3+4i)(-2+2i)\) in rectangular form?
DeMoivre's Shortcut
Compute \((1 + i)^8\) using DeMoivre's theorem. Doing it by repeated multiplication would take seven FOILs.
First, \(1 + i\) in polar form is \(\sqrt{2}\,\text{cis}\,45°\).
What is \((1+i)^8\)?
Finding All the Roots
Find all four fourth roots of \(-16\). The modulus of \(-16\) is 16, and its argument is \(180°\). Each fourth root has modulus \(\sqrt[4]{16} = 2\).
The arguments of the four roots are:
Those four roots form a square. Now predict: where do the six sixth roots of 64 sit?
The four fourth roots of negative 16 form a square on a circle of radius 2. The six sixth roots of 64 form a regular hexagon on a circle of radius 2.
The six sixth roots of 64 sit on a circle of radius \(\sqrt[6]{64} = 2\), at arguments \(0°, 60°, 120°, 180°, 240°, 300°\). They form a regular hexagon. The pattern holds for every \(n\): \(n\)th roots always form a regular \(n\)-gon.
Plot complex numbers on the complex plane and identify the real and imaginary parts
Compute the modulus and argument of a complex number
Convert between rectangular form \(a + bi\) and polar form \(r\,\text{cis}\,\theta\)
Multiply and divide complex numbers in polar form
Explain why multiplying by \(i\) is a 90-degree rotation
Apply DeMoivre's theorem to compute powers of complex numbers
Find all \(n\)th roots of a complex number and recognize the regular polygon pattern
Quick Check
If \(z = 3\,\text{cis}\,40°\), what is \(z^3\)?
Instructor Notes
Teaching Notes
This simulation builds complex number operations from geometry rather than algebra. The key insight is that multiplication by \(i\) is a 90-degree rotation, which makes polar form natural rather than arbitrary. Students who struggle with the algebraic definition of complex multiplication often find the geometric view much more intuitive.
The polar multiplication rule (multiply moduli, add arguments) follows directly from the rotation interpretation. DeMoivre's theorem then becomes obvious rather than something to memorize. Let students discover this chain themselves.
Common Student Errors
Believing "imaginary" means "not real" or "made up." Address this directly with the hook.
Expecting only one root when there are \(n\) of them. Step 8 is designed to break this expectation.
Forgetting that \(i^2 = -1\) when doing FOIL. Step 3 catches this.
Using degrees in trig functions without converting. The simulation uses degrees for clarity but all internal computation is in radians.
Discussion Questions
Why does multiplying by \(i\) rotate 90 degrees? Can you explain this without polar form?
If the 4th roots of a number form a square, and the 6th roots form a hexagon, what shape do the 5th roots form?
In what real-world situations would you need all the complex roots, not just the real one?
Exam Connection
Typical exam questions: convert between rectangular and polar form, multiply/divide in polar form, apply DeMoivre's theorem, find all nth roots. The simulation covers all of these. The challenge problem (fourth roots of -16) is at exam level.
How to Use This Simulation
Discover: Work through 8 guided steps that build your understanding of complex numbers, from plotting to finding roots. Each step asks you to predict before it reveals.
Explore: Place complex numbers on the plane and apply operations (add, subtract, multiply, divide, power, roots). Watch the geometric transformations animate. Toggle between rectangular and polar display.
Apply: Solve problems at three difficulty levels using polar form, DeMoivre's theorem, and root finding.
Keyboard: Use Tab to move between controls, arrow keys to adjust sliders, and Enter or Space to select options.
Disclaimer
These simulations are educational tools provided free of charge. They are not a substitute for a textbook, an instructor, or professional mathematical software. While we strive for accuracy, users should verify critical calculations independently. Eclatech Solutions is not responsible for errors or for academic decisions made based on these tools.