MATH1061 Chap.5 Complex Numbers
Complex Numbers
Complex numbers combine a real coordinate and an imaginary coordinate, so algebra and geometry support one another. This chapter moves between Cartesian and polar forms, interprets modulus as distance and argument as direction, and uses conjugation to simplify quotients. Multiplication becomes scaling plus rotation in polar form, which makes powers and roots systematic.
Argument conventions are stated explicitly because two expressions can describe the same direction while using different angle ranges. The Argand diagram is not optional decoration: it predicts the quadrant, the sign of each component and the approximate argument before any inverse trigonometric calculation. Cartesian form is efficient for addition and subtraction, while polar form exposes multiplication, division and powers.
Roots require all directions separated by equal angular increments, so a single principal value is not the full solution. Each result can be checked by comparing its plotted location, modulus and argument with the transformation that produced it.
What this chapter covers
- 01
Plot complex numbers on an Argand diagram
- 02
Compute modulus and argument
- 03
Convert between Cartesian and polar forms
- 04
Use conjugates to divide complex numbers
- 05
Interpret multiplication as scaling and rotation
- 06
Apply De Moivre's formula to powers
Worked example · free
Convert and square a complex number
- 1The modulus is the square root of 1 + 3, so r = 2.
- 1The point is in the first quadrant with argument pi/3.
- 1Thus z = 2(cos(pi/3) + i sin(pi/3)).
- 1Squaring gives 4(cos(2pi/3) + i sin(2pi/3)) = -2 + 2 square root of 3 times i.
Key terms
- Argand diagram
- A coordinate plane with real and imaginary axes for plotting complex numbers.
- Complex conjugate
- The number obtained by reversing the sign of the imaginary part.
- Principal argument
- A chosen representative angle from a specified interval.
- Polar form
- A representation using a modulus and an argument.
- Imaginary unit
- The number i defined by the relation that i squared equals negative one.
- Cartesian form
- A complex representation separating real and imaginary components.
- De Moivre formula
- A rule that raises a modulus to a power and multiplies its argument.
- Complex root
- A complex number whose stated integer power equals the target number.
Complex Numbers FAQ
Why can a complex number have many arguments?
A full rotation returns to the same ray, so adding any integer multiple of two pi gives another valid argument. A principal argument selects one representative from a declared interval. When comparing answers, first check whether the angles differ by a whole number of turns before concluding that the directions are different.
When is polar form more useful than Cartesian form?
Cartesian form is convenient for addition, subtraction and reading real or imaginary parts. Polar form is usually cleaner for multiplication, division, powers and roots because moduli multiply or divide while arguments add or subtract. Converting is worthwhile when the operation matches that geometry, not simply because polar notation is available.
How can I check a complex calculation?
Use both representations. Plot the expected quadrant, compare the modulus before and after the operation and convert back to Cartesian form when practical. Under multiplication, the output modulus should be the product of the input moduli and the output argument should be their sum modulo a full turn.
How many roots should a complex power equation have?
A nonzero complex number has the expected number of distinct roots for the stated positive integer power. Their moduli are equal and their arguments are evenly spaced around a full turn. Generate them with consecutive integer angle choices, then check that no endpoint duplicates the first root and that raising each candidate restores the original number.
Exam move
Draw the Argand diagram before computing an argument. Mark the signs of the real and imaginary parts, estimate the quadrant and only then use a trigonometric ratio. Build conversion fluency in both directions and annotate multiplication problems with the words scale and rotate. Use a two-column drill: solve in the form best suited to the operation, then convert the result to the other form for checking.
For division, predict whether the modulus should grow or shrink and subtract arguments in the correct order. For powers, predict the new modulus and quadrant before expanding. For roots, place equally spaced rays around the circle and count distinct solutions before calculating coordinates. Record the principal-argument interval whenever a convention matters.
If two answers appear different, compare their moduli and reduce the angle difference by full turns before deciding. The geometric prediction is a fast independent check on sign errors in Cartesian algebra, while direct multiplication verifies a proposed conjugate, power or root exactly.
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