Worksheet: Argand Diagram

In this worksheet, we will practice identifying complex numbers plotted on an Argand diagram and discovering their geometric properties.

Q1:

Find the value of 𝑍 given 𝑍 on the Argand diagram below.

  • A𝑍=−3+5𝑖
  • B𝑍=3−5𝑖
  • C𝑍=3+5𝑖
  • D𝑍=−3−5𝑖

Q2:

What is the real part of the complex number shown?

Q3:

What is the imaginary part of the complex number shown?

Q4:

If the number 𝑍=8+𝑖 is represented on Argand diagram by the point 𝐴, determine the Cartesian coordinates of that point.

  • A(−8,1)
  • B(−8,−1)
  • C(8,−1)
  • D(8,1)

Q5:

In which quadrant of the Argand diagram does the complex number 3−2𝑖 lie?

  • Afourth
  • Bsecond
  • Cthird
  • Dfirst

Q6:

Let us consider a complex number, 𝑧, with nonzero real and imaginary parts.

If the real and imaginary parts of 𝑧 are of the same sign, in which quadrant(s) of the Argand diagram could 𝑧 appear?

  • A2nd or 4th
  • B3rd or 4th
  • C1st or 2nd
  • D1st or 3rd
  • E1st or 4th

If the real and imaginary parts of 𝑧 are of opposite signs, in which quadrant(s) of the Argand diagram could 𝑧 appear?

  • A1st or 2nd
  • B2nd or 4th
  • C3rd or 4th
  • D1st or 3rd
  • E1st or 4th

Q7:

The Argand diagram shows the complex numbers 𝑊, 𝑋, 𝑌, and 𝑍. Which of these numbers is 2−5𝑖?

  • A𝑍
  • B𝑌
  • C𝑊
  • D𝑋

Q8:

What complex number lies at the midpoint of 𝑧 and 𝑧 on the given complex plane?

  • A8+10𝑖
  • B4+5𝑖
  • C2+2𝑖
  • D5+4𝑖
  • E4+4𝑖

Q9:

In what quadrant does 𝑧 lie?

  • Athe first quadrant
  • Bthe third quadrant
  • Cthe second quadrant
  • Dthe fourth quadrant

Q10:

Find the value of 𝑍 given 𝑍 on the Argand diagram below.

  • A𝑍=3−3𝑖
  • B𝑍=−3−3𝑖
  • C𝑍=−3+3𝑖
  • D𝑍=3+3𝑖

Q11:

Describe the geometric transformation that maps every complex number 𝑧 to its conjugate 𝑧 in the complex plane.

  • AReflection in the line ReIm(𝑧)=(𝑧)
  • BReflection in the line ReIm(𝑧)=−(𝑧)
  • CRotation by 180∘ about the origin
  • DReflection in the real axis
  • EReflection in the imaginary axis

Q12:

Given that points 𝐴 and 𝐵 represent the complex numbers 𝑍 and 𝑍 on an Argand diagram, then 𝐵 is the image of 𝐴 under which transformation?

  • Areflection in the 𝑦-axis
  • Breflection in the origin point
  • Creflection in the 𝑥-axis

Q13:

Seven complex numbers 𝑧, 𝑧, 𝑧, 𝑧, 𝑧, 𝑧, and 𝑧 are represented on the Argand diagram.

Which of the complex numbers is −3+2𝑖?

  • A𝑧
  • B𝑧
  • C𝑧
  • D𝑧
  • E𝑧

What complex number is represented by 𝑧?

  • A−4+𝑖
  • B1−4𝑖
  • C4+𝑖
  • D−4−𝑖
  • E−1−4𝑖

Which complex number has equal real and imaginary parts?

  • A𝑧
  • B𝑧
  • C𝑧
  • D𝑧
  • E𝑧

Which two complex numbers are a conjugate pair? What is their geometric relationship?

  • A𝑧 and 𝑧 are a conjugate pair; they are related by a rotation about the origin of 𝜋 radians.
  • B𝑧 and 𝑧 are a conjugate pair; they are related by reflection in the real axis (𝑥-axis).
  • CNone of the complex numbers are a conjugate pair.
  • D𝑧 and 𝑧 are a conjugate pair; they are related by reflection in the imaginary axis (𝑦-axis).
  • E𝑧 and 𝑧 are a conjugate pair; they are related by reflection in the real axis (𝑥-axis).

Q14:

Using the Argand diagram shown, find the value of 𝑧+𝑧.

Q15:

Four complex numbers 𝑧, 𝑧, 𝑧, and 𝑧 are shown on the Argand diagram.

Find the image of the points 𝑧, 𝑧, 𝑧, and 𝑧 under a transformation that maps 𝑧 to 𝑖𝑧.

  • A𝑧=−3𝑧=−2+3𝑖𝑧=2−𝑖𝑧=−𝑖
  • B𝑧=3𝑧=2−3𝑖𝑧=−2+𝑖𝑧=𝑖
  • C𝑧=3𝑖𝑧=−3+2𝑖𝑧=1−2𝑖𝑧=1
  • D𝑧=−3𝑖𝑧=3−2𝑖𝑧=1−2𝑖𝑧=−1
  • E𝑧=−3𝑧=−2−3𝑖𝑧=2+𝑖𝑧=𝑖

By plotting these points on an Argand diagram, or otherwise, give a geometric interpretation of the transformation.

  • AThe transformation represents a reflection in the 𝑦-axis.
  • BThe transformation represents a counterclockwise rotation by an angle of 𝜋2 radians about the origin.
  • CThe transformation represents a counterclockwise rotation by an angle of 𝜋 radians about the origin.
  • DThe transformation represents a clockwise rotation by an angle of 𝜋2 radians about the origin.
  • EThe transformation represents a reflection in the 𝑥-axis.

Q16:

Three complex numbers 𝑧, 𝑧, and 𝑧 are represented on the Argand diagram.

Find the image of points 𝑧, 𝑧, and 𝑧 under the transformation that maps 𝑧 to 2𝑧.

  • A𝑧=4𝑧=−𝑖𝑧=−6+2𝑖
  • B𝑧=4𝑧=−2𝑖𝑧=−6+4𝑖
  • C𝑧=2𝑧=−2𝑖𝑧=−3+4𝑖
  • D𝑧=1𝑧=−𝑖2𝑧=−32+𝑖
  • E𝑧=4𝑖𝑧=2𝑧=−4−6𝑖

By plotting these points on an Argand diagram, or otherwise, give a geometric interpretation of the transformation.

  • AThe transformation represents a dilation parallel to the 𝑦-axis with scale factor two centered at 𝑦=0.
  • BThe transformation represents a dilation with scale factor two centered at the origin.
  • CThe transformation represents a dilation parallel to the 𝑥-axis with scale factor two centered at 𝑥=0.
  • DThe transformation represents a counterclockwise rotation by 𝜋2 radians about the origin followed by a dilation with scale factor two centered at the origin.
  • EThe transformation represents a dilation with scale factor 12 centered at the origin.

Q17:

In which quadrant of the Argand plane does the complex number 7+9𝑖3−4𝑖 lie?

  • Athe fourth quadrant
  • Bthe first quadrant
  • Cthe third quadrant
  • Dthe second quadrant

Q18:

Find all the solutions to 𝑧=1.

  • A𝑧=𝑖,−𝑖,√32+12𝑖,−√32+12𝑖,√32−12𝑖,−√32−12𝑖
  • B𝑧=1,−1,𝑖,−𝑖,√22+√22𝑖,−√22−√22𝑖
  • C𝑧=𝑖,−𝑖,√32+𝑖,−√32+𝑖,√32−𝑖,−√32−𝑖
  • D𝑧=1,−1,1+√32𝑖,1−√32𝑖,−1+√32𝑖,−1−√32𝑖
  • E𝑧=1,−1,12+√32𝑖,12−√32𝑖,−12+√32𝑖,−12−√32𝑖

By plotting the solutions on an Argand diagram, or otherwise, describe the geometric properties of the solutions of 𝑧=1.

  • AThe solutions are evenly spaced around the unit circle centered at the origin.
  • BThe solutions are evenly spaced around a unit semicircle centered at the origin in the half plane Im(𝑧)≥0.
  • CThe solutions are evenly spaced around the unit square centered at the origin.
  • DThe solutions form a regular hexagon inscribed within the unit circle centered at the origin with one vertex at 𝑧=𝑖.
  • EThe solutions form a regular hexagon centered at the origin with one vertex at 𝑧=√32+𝑖.

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