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2-1 (b) Modulus, amplitude and principal amplitude

2-1 (b) Modulus, amplitude and principal amplitude

Point 2 of 8 — Lesson 1: Complex Numbers in Trigonometric and Exponential Form and De Moivres TheoremQuestion bank: 0 questions (0 with pictures)

02 2-1 (b) Modulus, amplitude and principal amplitude

Definitions

  • Modulus ∣z∣=r: the distance of the point z(x,y) from the origin. r=∣z∣=x2+y2
  • Amplitude (argument) θ of z: the angle from the positive real axis to the vector of z, with tan⁡θ=yx.
  • Principal amplitude: the value of the amplitude lying in the interval ]−π,π].

Finding the principal amplitude from the quadrant of z

  • First find the quadrant in which the point (x,y) lies, then use α=tan⁡−1∣yx∣ as the acute reference angle.
  • First quadrant (x>0, y>0): θ=α.
  • Second quadrant (x<0, y>0): θ=π−α.
  • Third quadrant (x<0, y<0): θ=−π+α.
  • Fourth quadrant (x>0, y<0): θ=−α.
  • On the axes: positive real axis θ=0; negative real axis θ=π; positive imaginary axis θ=π2; negative imaginary axis θ=−π2.
  • Radians convert to degrees with degrees=radians×180∘π.
✍️ Worked examples

Example 1: Find the modulus and the principal amplitude of z1=−3+i and z2=−1−i.

Solution:

  • For z1: x=−3, y=1, so z1 lies in the second quadrant.
  • r=3+1=2.
  • Reference angle α=tan⁡−113=π6.
  • θ=π−π6=5π6.
  • For z2: x=−1, y=−1, so z2 lies in the third quadrant.
  • r=1+1=2.
  • Reference angle α=tan⁡−1(1)=π4.
  • θ=−π+π4=−3π4.

Properties of the modulus and amplitude

  • ∣z∣≥0 for every complex number z.
  • The amplitude has infinitely many values: θ+2πn where n∈Z.
  • ∣z∣=∣zˉ∣=∣−z∣.
  • z zˉ=∣z∣2.
  • If the principal amplitude of z is θ, then that of zˉ is −θ.

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