Menu
Sign up Sign in

AI HL Β· AHL 6

Complex Numbers

0 of 66 skills done

0
AHL 6.3

The complex plane and the modulus

AHL 6.3

The complex plane and the modulus

Target: I can plot a complex number on an Argand diagram and read a complex number from its point. (+3 more)

By the end of this section

Key points

The Argand diagram

  • A complex number needs two coordinates, so it is drawn on a plane: the horizontal axis is the real axis and the vertical axis is the imaginary axis.
  • \(z = x + iy\) is the point \((x, y)\). Plot the real part across and the imaginary part up.
  • The plane is called the complex plane or an Argand diagram. A complex number can also be drawn as a vector from the origin to its point.

Reflections: βˆ’z and z*

  • \(-z = -x - iy\). Both parts change sign, so the point is a reflection of z through the origin.
  • \(z^* = x - iy\). The imaginary part changes sign, so the point is a reflection of z in the real axis.
  • \(-z^* = -x + iy\) is a reflection of z in the imaginary axis.

The modulus

  • The modulus \(|z|\) is the distance from the origin to the point, the length of the vector.
  • For \(z = x + iy\), \(|z| = \sqrt{x^2 + y^2}\), by Pythagoras. For \(z = 2 + 4i\), \(|z| = \sqrt{20} = 4.47\).
  • \(|z| = |z^*| = |-z|\), because reflecting the point does not change its distance from the origin.
  • \(|z|\) is a real number and is never negative. Do not add the two parts: \(|3 + 4i| = 5\), not 7.

Explore: z on the Argand diagram

Drag z (orange) to any point. Read the modulus and the argument, and the three ways of writing z. Predict which quadrant, and what ΞΈ will be, before you read the working.

The point snaps to whole and half numbers. Select it (Tab) and use the arrow keys to move it.

  1. Drag z into the third quadrant. tan⁻¹ of y/x gives a small positive angle. Why is the argument not equal to that angle?
  2. Show z* and βˆ’z. What happens to the modulus? What happens to the argument of z*, and of βˆ’z?
  3. Put z on the negative real axis and switch between the two ranges. Which values of ΞΈ do you get? Why does the second range give Ο€ and not βˆ’Ο€?
  4. Which points have an argument that is a multiple of Ο€/4? What do their coordinates have in common?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Plotting z, βˆ’z and z*

\(z = -3 + 2i\). Represent \(z\), \(-z\) and \(z^*\) on an Argand diagram.

Working
  1. z

    \(z = -3 + 2i \text{ is the point } (-3, 2)\)

    The real part is the x-coordinate and the imaginary part is the y-coordinate.

  2. βˆ’z

    \(-z = 3 - 2i \text{ is the point } (3, -2)\)

    Change the sign of both parts. This is a reflection of z through the origin.

  3. βˆ’4βˆ’3βˆ’2βˆ’11234βˆ’4βˆ’3βˆ’2βˆ’11234ReImz-zz*
    z*

    \(z^* = -3 - 2i \text{ is the point } (-3, -2)\)

    Change the sign of the imaginary part only. This is a reflection of z in the real axis.

Video walkthrough coming
Reading the modulus from a diagram

The diagram shows the point that represents the complex number \(z\). Write down \(z\) and find \(|z|\).

βˆ’12βˆ’10βˆ’8βˆ’6βˆ’4βˆ’224681012βˆ’8βˆ’6βˆ’4βˆ’22468ReImz
Working
  1. Read the coordinates

    \(z = 6 + 8i\)

    The point is 6 across and 8 up.

  2. Length of the vector

    \(|z| = \sqrt{6^2 + 8^2} = \sqrt{100} = 10\)

    \(|z|\) is the distance from the origin: a right-angled triangle with legs 6 and 8.

Video walkthrough coming
The same modulus

\(z = 2 + 5i\). Find \(|z|\), \(|z^*|\) and \(|-z|\), to 3 significant figures. What do you notice?

βˆ’8βˆ’7βˆ’6βˆ’5βˆ’4βˆ’3βˆ’2βˆ’112345678βˆ’6βˆ’5βˆ’4βˆ’3βˆ’2βˆ’1123456ReImzz*-z
Working
  1. |z|

    \(|z| = \sqrt{2^2 + 5^2} = \sqrt{29} = 5.39\)

    Distance from the origin to \((2, 5)\).

  2. |z*|

    \(|z^*| = \sqrt{2^2 + (-5)^2} = \sqrt{29} = 5.39\)

    The sign of a coordinate disappears when you square it.

  3. |βˆ’z|

    \(|-z| = \sqrt{(-2)^2 + (-5)^2} = \sqrt{29} = 5.39\)

    A reflection does not change the distance from the origin.

  4. What you notice

    All three vectors have the same length.

    z, z* and βˆ’z are all the same distance from the origin.

Video walkthrough coming