Complex numbers as points: modulus, amplitude and trigonometric form
الأسبوع 4 · الأعداد المركبة على مستوى أرجاند والمرافق ومقياس العدد المركب وسعته والسعة الأساسية والصورة المثلثية · 3 دروس
1
Complex numbers as points: the Argand plane, the conjugate and polar coordinates
Pure Mathematics Sec 3 Term 1, Unit 2, Lesson 1
GoalPlot complex numbers as points, find conjugates and −z, and move between x + yi and polar coordinates.
New words
Argand plane · the plane where z = x + yi is the point (x, y)conjugate · x − yi, the mirror image of x + yi in the real axispolar coordinates · (r, θ): the distance from the origin and the angle from the positive x-axis
Explanation
A complex number z = x + yi (with i² = −1, as in Secondary 1) has a real part x and an imaginary part y. Because it is a pair of real numbers, it can be drawn as the point (x, y). The plane used for this is the Argand plane (in Arabic مستوى أرجاند): the horizontal axis holds real parts, the vertical axis holds imaginary parts. The conjugate x − yi is the mirror image of z in the horizontal axis. The number −z = −x − yi is the point opposite z through the origin. Adding a real number slides the point sideways: if z = 4 + 3i then 1 + z = 5 + 3i. All numbers with real part 3 lie on the vertical line x = 3, and all numbers with imaginary part −1 on the horizontal line y = −1. A point can also be described by polar coordinates (r, θ): r is the distance from the origin and θ the angle from the positive x-axis, turned anticlockwise. We measure θ in radians: π radians = 180°, so π/6 = 30°, π/4 = 45°, π/3 = 60° and π/2 = 90°. The links are x = r cos θ, y = r sin θ, r = √(x² + y²) and tan θ = y/x.
Worked examples
Represent z = 4 + 3i, its conjugate, −z and the conjugate of −z. The points are z = (4, 3), conjugate (4, −3), −z = (−4, −3) and the conjugate of −z, −4 + 3i = (−4, 3). The four points are the corners of a rectangle centred on the origin; the pairs are mirror images in the two axes.
Plot 3 + 2i, 3 − i and 3. All have real part 3, so they lie on the vertical line x = 3. Any number 3 + yi, for any real y, lies on that line.
Change between forms. The point (−2, 2) has r = √(4 + 4) = 2√2 (about 2.83). It lies on the line y = −x in the second quadrant, 45° from the negative x-axis, so θ = π − π/4 = 3π/4. Going back: (r, θ) = (6, π/6) gives x = 6 cos 30° = 3√3 and y = 6 sin 30° = 3, the point (3√3, 3).
Class activity
Mirror game. Plot z = 2 + 3i on squared paper. Without calculating, plot its conjugate, −z and the conjugate of −z, and say which pairs are mirror images in the vertical axis.
Practice
1
Write the point of z = −3 + 5i, of its conjugate and of −z.
For z = 2 − i find 1 + z and the conjugate of 1 + z, as points.
1 + z = 3 − i = (3, −1). Its conjugate 3 + i = (3, 1).
3
Convert the polar coordinates (r, θ) = (8, π/3) to x + yi.
x = 8 cos 60° = 4 and y = 8 sin 60° = 4√3, so the number is 4 + 4√3 i.
4
Describe the set of numbers z with imaginary part 4, and the set with real part −2.
Imaginary part 4: the horizontal line y = 4. Real part −2: the vertical line x = −2.
5
Plot z = 2 + 2i, its conjugate and the conjugate of −z. Which two of the three points are mirror images in the vertical axis?
z = (2, 2); conjugate (2, −2); the conjugate of −z is −2 + 2i = (−2, 2). The points (2, 2) and (−2, 2), that is z and the conjugate of −z, are mirror images in the vertical axis.
