Porisma

Complex numbers and the Argand plane

Calculate with real and imaginary parts.

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By the end, you can

  • Explain the extension from real to complex numbers
  • Find complex roots of quadratic equations
  • Use modulus, argument and conjugates
  • Perform the four operations with complex numbers
  • Use equality of complex numbers
  • Use conjugate roots for polynomial equations with real coefficients
  • Represent complex numbers on an Argand diagram
  • Interpret conjugation, negation, addition, subtraction and multiplication by i geometrically

about 45 minutes for the lesson · practise at your own pace.

The central idea

Complex numbers extend the real numbers with i²=−1. For z=a+bi, its conjugate is a−bi and its modulus is a2+b2\sqrt{a^2+b^2}. An argument gives direction in the Argand plane, with the correct quadrant essential. Multiplication adds arguments and multiplies moduli; multiplying by i rotates by π/2 anticlockwise.

The power cycle of i

Define i by i2=−1i^2=-1. Then i3=−ii^3=-i, i4=1i^4=1, i5=ii^5=i, i6=−1i^6=-1; the cycle repeats every four powers. For positive integers k, i2k=(−1)ki^{2k}=(-1)^k and i2k+1=(−1)kii^{2k+1}=(-1)^ki.

The equation x2+1=0x^2+1=0 has roots ±i\pm i. Write a complex number as a+iba+ib with real a,b: the real part is a and the imaginary part is b, not ib. For positive real a the principal root is −a=ia\sqrt{-a}=i\sqrt a. This does not license multiplying principal roots of arbitrary complex values without checking branches. “Complex” refers to the connected real and imaginary parts.

The source circuit motivation

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Source circuit motivation: rectangular loop a–b–c–d–a, alternating voltage source v0 cos(omega t) on ab, resistor R on bc with a rightward current arrow, capacitor C on cd with charge Q marked, and inductor L on da. This preserves the source component roles without posing a new circuit calculation.
Source circuit motivation: rectangular loop a–b–c–d–a, alternating voltage source v0 cos(omega t) on ab, resistor R on bc with a rightward current arrow, capacitor C on cd with charge Q marked, and inductor L on da. This preserves the source component roles without posing a new circuit calculation.
Source circuit motivation: rectangular loop a–b–c–d–a, alternating voltage source v0 cos(omega t) on ab, resistor R on bc with a rightward current arrow, capacitor C on cd with charge Q marked, and inductor L on da. This preserves the source component roles without posing a new circuit calculation.
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The nested number sets

Read from the smallest set to the largest. Every number in one row also belongs to every row below it. Each next set includes additional numbers.

  1. Natural numbers

    Counting numbers 1,2,3,…1,2,3,\ldots. This source starts at 1, so zero is not included in N\mathbb N here.

  2. Integers

    The counting numbers, their negatives and zero: …,−1,0,1,…\ldots,-1,0,1,\ldots.

  3. Rational numbers

    Numbers p/qp/q, where p and q are integers and q≠0q\ne0. Every integer n is rational because n=n/1n=n/1.

  4. Real numbers

    Rational and irrational numbers together. Irrationals such as 2\sqrt2, π\pi and ee are real but not rational; their decimal representations neither terminate nor recur.

  5. Complex numbers

    Numbers a+iba+ib, with a and b real and i2=−1i^2=-1. Every real a is also complex: a=a+0ia=a+0i. A complex number need not have a nonzero imaginary part.

Why is each inclusion strict?

A proper subset is contained in a larger set without being equal to it. These examples show what each larger set adds.

  • Natural numbers are a proper subset of integers

    N⊊Z\mathbb N\subsetneq \mathbb Z

    00 — Zero is an integer, but it is not natural under this source’s convention.

  • Integers are a proper subset of rational numbers

    Z⊊Q\mathbb Z\subsetneq \mathbb Q

    12\tfrac12 — One half is rational, but it is not an integer.

  • Rational numbers are a proper subset of real numbers

    Q⊊R\mathbb Q\subsetneq \mathbb R

    2\sqrt2 — The positive square root of 2 is real and irrational, so it is not rational.

  • Real numbers are a proper subset of complex numbers

    R⊊C\mathbb R\subsetneq \mathbb C

    ii — The imaginary unit is complex but not real: no real number has square −1.

Compare with the nested regions

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Nested regions show N strictly inside Z, Z inside Q, Q inside R, and R inside C. The real region outside Q represents irrational numbers. This source uses natural numbers starting at 1; every real a is complex as a+0i.
Nested regions show N strictly inside Z, Z inside Q, Q inside R, and R inside C. The real region outside Q represents irrational numbers. This source uses natural numbers starting at 1; every real a is complex as a+0i.
Nested regions show N strictly inside Z, Z inside Q, Q inside R, and R inside C. The real region outside Q represents irrational numbers. This source uses natural numbers starting at 1; every real a is complex as a+0i.
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Point, position vector and direction in the Argand plane

Illustration: one complex number as a point and a position vector.

z = 1 + 3i

Grid spacing is 1 unit. P = (1, 3). The arrow OP has components (1, 3); its length is |z|.

Real coordinate x1
Imaginary coordinate y3
Try an example

z=x+iyz=x+iy

OP→=(xy)\overrightarrow{OP}=\begin{pmatrix}x\\y\end{pmatrix}

Real part x
1
Imaginary part y
3
Modulus |z|
≈ 3.1623
Principal argument θ
≈ 1.249 rad

The imaginary part is the real number y, not iy. A point and its position vector describe the same two coordinates.

Directions use −π < θ ≤ π, measured from the positive real axis. Anticlockwise is positive; clockwise is negative.

Choose a principal argument

Compare with the three source views

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Three source views use the same first-quadrant point P(x,y). First shows dashed real/imaginary projections, second shows position vector OP, third labels its length r, its angle theta from the positive real axis and the x/y components. All axes are labelled Re and Im.
Three source views use the same first-quadrant point P(x,y). First shows dashed real/imaginary projections, second shows position vector OP, third labels its length r, its angle theta from the positive real axis and the x/y components. All axes are labelled Re and Im.
Three source views use the same first-quadrant point P(x,y). First shows dashed real/imaginary projections, second shows position vector OP, third labels its length r, its angle theta from the positive real axis and the x/y components. All axes are labelled Re and Im.
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Conjugation reflects in the real axis

Start with the source pair 1 + 3i and 1 − 3i, then change the coordinates.

z = 1 + 3i; z* = 1 − 3i

Grid spacing is 1 unit. Reflection pairs (1, 3) with (1, -3). The two distances from the origin are equal.

Real coordinate x1
Imaginary coordinate y3
Try an example

z∗=x−iyz^*=x-iy

∣z∣=x2+y2|z|=\sqrt{x^2+y^2}

Real part x
1
Imaginary part y
3
Common modulus |z| = |z*|
≈ 3.1623
Principal argument θ
≈ 1.249 rad
Conjugate z*
1 − 3i
Principal argument of z*
≈ -1.249 rad

Conjugation keeps the real part and changes the sign of the imaginary part. It reflects the point in the real axis.

Directions use −π < θ ≤ π, measured from the positive real axis. Anticlockwise is positive; clockwise is negative.

Choose a principal argumentPractise the source conjugates

Compare with the source reflection

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Equal-unit Argand axes show vectors from O to z1=(1,3) and its conjugate z1*=(1,-3), with dashed grid and equal distances above and below the real axis. The real coordinate remains 1; the imaginary coordinate changes from 3 to -3.
Equal-unit Argand axes show vectors from O to z1=(1,3) and its conjugate z1*=(1,-3), with dashed grid and equal distances above and below the real axis. The real coordinate remains 1; the imaginary coordinate changes from 3 to -3.
Equal-unit Argand axes show vectors from O to z1=(1,3) and its conjugate z1*=(1,-3), with dashed grid and equal distances above and below the real axis. The real coordinate remains 1; the imaginary coordinate changes from 3 to -3.
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Method map: Choose a principal argument

Plot z = x + iy. Check zero, then choose its axis or quadrant. Measure from the positive real axis: anticlockwise is positive; clockwise is negative.

  1. Check that the point has a direction

    Write z = x + iy with real x and y. The principal interval here includes π and excludes −π.

    −π<θ≤π-\pi<\theta\le\pi

    x = 0 and y = 0Stop: zero has no argument

    The zero vector has no direction. Its modulus is ∣0∣=0|0|=0, but an argument is undefined; 0/00/0 cannot supply an angle.

    At least one coordinate is nonzeroChoose the matching position case

    If either coordinate is zero, use the axis cases. If both are nonzero, use the quadrant cases. A zero real part does not make the argument undefined when the imaginary part is nonzero.

    Keep this interval throughout the calculation. The two directions π and −π reach the same negative real ray, but only π belongs to the chosen principal interval.

  2. Read axis directions directly

    Use this stage only for a nonzero point on an axis. One coordinate is zero, so an acute reference triangle is unnecessary.

    x=0x=0 or y=0y=0

    Positive real · x > 0, y = 0No turn

    θ=0\theta=0

    Negative real · x < 0, y = 0Half a turn anticlockwise

    θ=π\theta=\pi

    Use π\pi, not the excluded endpoint −π-\pi.

    Positive imaginary · x = 0, y > 0Quarter turn anticlockwise

    θ=π/2\theta=\pi/2

    Negative imaginary · x = 0, y < 0Quarter turn clockwise

    θ=−π/2\theta=-\pi/2

    Do not divide by x for a point on the imaginary axis. Its direction is read geometrically even though y/x is undefined.

  3. Use the basic angle and both coordinate signs

    Use this stage only when both coordinates are nonzero. Calculate the positive acute basic angle α in radians, then match the quadrant.

    α=tan⁡−1∣y/x∣\alpha=\tan^{-1}|y/x|

    0<α<π/20<\alpha<\pi/2

    I · x > 0, y > 0Use the basic angle

    θ=α\theta=\alpha

    II · x < 0, y > 0Subtract from π

    θ=π−α\theta=\pi-\alpha

    III · x < 0, y < 0Start at −π and add the basic angle

    θ=−π+α\theta=-\pi+\alpha

    IV · x > 0, y < 0Use the negative basic angle

    θ=−α\theta=-\alpha

    The ratio y/xy/x alone cannot distinguish opposite quadrants. Check both signs on the sketch and confirm −π<θ≤π-\pi<\theta\le\pi. Other arguments differ by 2kπ2k\pi for an integer k; select the representative in the stated interval.

Try a worked example

Read a case, make a prediction, then reveal the reasoning. The examples keep their stated values.

A negative real number uses the included endpoint

In Example 15(b), z=−3z=-3. Find its modulus and choose its principal argument.

Show reasoning
  1. 1. Locate the nonzero point

    The point (−3, 0) is on the negative real axis. It is not the origin, so it has a direction.

  2. 2. Separate distance from direction

    The squared coordinates sum to 9, so the distance is 3. The principal direction uses the included endpoint π.

    ∣−3∣=9=3,arg⁡(−3)=π.\begin{aligned}|-3|&=\sqrt9\\&=3,\\\arg(-3)&=\pi.\end{aligned}
  3. 3. Check the endpoint

    A clockwise half-turn gives −π, which reaches the same ray but is excluded from this principal interval. A zero imaginary part does not imply zero argument.

Continue with the lesson question

Equal negative coordinates do not give a first-quadrant angle

In Example 15(h), z=−1−iz=-1-i. Find its modulus and choose its principal argument.

Show reasoning
  1. 1. Use both signs before the quotient

    The point (−1, −1) lies in quadrant III. Its y/x ratio is positive, just as in quadrant I, so the quotient alone cannot select the direction.

    ∣−1−i∣=1+1=2.\begin{aligned}|-1-i|&=\sqrt{1+1}\\&=\sqrt2.\end{aligned}
  2. 2. Calculate the acute basic angle

    The absolute ratio gives the acute angle to the nearest horizontal ray, not the final principal argument.

    α=tan⁡−1∣−1−1∣=π4.\alpha=\tan^{-1}\left|\frac{-1}{-1}\right|=\frac\pi4.
  3. 3. Choose the clockwise principal direction

    Quadrant III requires a negative angle in the chosen interval. The positive turn 5π/4 reaches the same ray but is outside it.

    arg⁡(−1−i)=−π+π4=−3π4.\begin{aligned}\arg(-1-i)&=-\pi+\frac\pi4\\&=-\frac{3\pi}{4}.\end{aligned}
Continue with the lesson question

Compare with the eight Argand positions

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Eight Argand panels retain the source axis and quadrant cases: positive real 0, negative real pi, positive imaginary pi/2, negative imaginary -pi/2, quadrant I alpha, II pi-alpha, III -pi+alpha, IV -alpha. Blue arrows measure signed theta from the positive real axis; purple basic-angle arcs lie beside the nearest horizontal direction. Zero has no argument.
Eight Argand panels retain the source axis and quadrant cases: positive real 0, negative real pi, positive imaginary pi/2, negative imaginary -pi/2, quadrant I alpha, II pi-alpha, III -pi+alpha, IV -alpha. Blue arrows measure signed theta from the positive real axis; purple basic-angle arcs lie beside the nearest horizontal direction. Zero has no argument.
Eight Argand panels retain the source axis and quadrant cases: positive real 0, negative real pi, positive imaginary pi/2, negative imaginary -pi/2, quadrant I alpha, II pi-alpha, III -pi+alpha, IV -alpha. Blue arrows measure signed theta from the positive real axis; purple basic-angle arcs lie beside the nearest horizontal direction. Zero has no argument.
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Explore

Use the worked example below, then change one input or assumption. Predict which part of the answer must change before calculating.

Related interactive example

Explore this related example, then return to the prediction task above.

Multiplying by i is a quarter-turn

4iz4iz is 4z4z turned through 90∘90^\circ anticlockwise. Slide to apply the turn, and read how far 4iz4iz lands from 3w3w.

Turn applied to 4z0°
Try an example
turned 4z
20 + 8i
turned 4z − 3w
29 − 7i
  • Moves as the slider changes; a faint path shows where it goes
Key idea

Equation (1) says 4iz4iz and 3w3w differ by exactly 1+5i1+5i. The quarter-turn turns 4z4z into 4iz4iz and closes that gap.

Ideas to try
  1. Slide to 90∘90^\circ. The amber point is 4iz=−8+20i4iz=-8+20i, and the red dashed segment from 3w3w is the vector 1+5i1+5i.
  2. Slide back to 0∘0^\circ: the same segment no longer equals 1+5i1+5i, which is why the rotation matters.
Open the full interactive problem →

Worked example

(1+i)/(1−i)(1+i)/(1-i) becomes (1+i)2/2=i(1+i)^2/2=i. The numerator has modulus 2\sqrt2 and argument π/4\pi/4; the denominator has the same modulus and argument −π/4-\pi/4, so the quotient has modulus 11 and argument π/2\pi/2.

An inverse tangent value alone may select the wrong quadrant.

Check your understanding

Find the modulus and principal argument of −1+i-1+i using the interval (−π,π](-\pi,\pi].

Mastery learning objectives

At the end of this chapter, I should be able to:

General

Recognise that the real numbers are a subset of the complex numbers.

Recall i=−1i=\sqrt{-1} (the principal square root), so i2=−1i^2=-1.

Simplify powers of i, including i2=−1i^2=-1, i3=−ii^3=-i, i4=1i^4=1, and the repeating cycle.

Represent complex numbers in an Argand diagram.

Describe geometrical effects of conjugation, negation, addition, subtraction and multiplication by i.

Using a graphing calculator

Perform basic operations on two or more complex numbers.

Obtain the square roots of a complex number.

Find the conjugate of a complex number.

Verify the roots of an equation.

Determine the modulus and argument of a complex number.

Using an analytical method

Perform basic operations on two or more complex numbers.

Find unknowns by comparing real and imaginary parts of a complex equation.

Obtain the square roots of a complex number.

Find the conjugate of a complex number.

Perform basic operations involving a conjugate.

Apply conjugate properties to simplify complex expressions and equations.

Express a complex number as a+bi by realising the denominator.

Solve simultaneous equations involving two or more complex numbers.

Solve for the complex roots of a quadratic equation.

Apply the general theory of polynomial equations to find roots. In particular:

  • State the other root of a real-coefficient polynomial given a nonreal root.
  • Find all linear or quadratic factors of a polynomial given root(s).

Determine the modulus and argument of a complex number.

Represent a complex number in an Argand diagram.

Choose a practice question

Common question typeLecture examplesDiscussion questionsReview problems (follow up)
Equality of complex numbers1,3,4,13,141,21,6,7
Solve an equation involving a complex number63,5,69(b),10,16,17(a)
Solve simultaneous complex equations8,943,9(a),11
Solve polynomial equations10,11,127,84,5,8,12,13
Sketch on an Argand diagram and/or describe geometrical effects2,16914
Determine modulus and argument1510,1115,17(b)

Practice questions

Practice section 1 · Questions 1–6

Explain why the new quadratic roots are not real · Relate real numbers to complex numbers · +4 more

Practice section 2 · Questions 7–12

Find paired real unknowns by comparing components · Practise comparing components with real parameters · +4 more

Practice section 3 · Questions 13–18

Recall rationalisation before complex division · Divide exactly and identify real and imaginary parts · +4 more

Practice section 4 · Questions 19–24

Recover a quartic and deduce reciprocal roots · Reuse a cube after multiplying by i · +4 more

Practice section 5 · Questions 25–30

Sketch complex sums and prove a square · Apply the nine modulus and argument rules · +4 more

Practice section 6 · Questions 31–36

Real coefficients and a known complex root · A quadratic with complex coefficients · +4 more

Practice section 7 · Questions 37–42

Real parts, imaginary parts, moduli and arguments · Use both conjugate constraints to determine z · +4 more

Practice section 8 · Questions 43–48

Reuse a known square after multiplying by i · Reuse a square root before undoing a scale and translation · +4 more

Practice section 9 · Questions 49–54

Use a purely imaginary root to factor and transform a quartic · Use rotation to find area and real quadratic factors · +4 more

More practice

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