Modulus & Argument (Edexcel IAL Further Maths)

Argand Diagrams

Spec point — 1.3

What is an Argand diagram?

• An Argand diagram is a 2D Cartesian grid used to visualise complex numbers

• The complex number $x+yi$ is represented by the point with coordinates$(x,y)$

  • The real part is measured along the x -axis

    • called the real axis, written "Re"

  • The imaginary part is measured along the y -axis

    • called the imaginary axis, written "Im"

• Complex numbers can be thought of as points, or as vectors from the origin

Exam Hint If asked to sketch an Argand diagram, it does not need to be to scale (plotted), but should roughly show the correct positions.

Worked Example Sketch, on the same Argand diagram, the complex numbers $5+7i$ and $-4-2i$.

Modulus & Argument

Spec point — 1.1

How do I find the modulus of a complex number?

• The modulus of a complex number is its distance from the origin on an Argand diagram

  • It is written open vertical bar z close vertical bar

  • If $z=x+iy$, then by Pythagoras' theorem

    • $\left| z \right| = \sqrt{x^{2} + y^{2}}$

• A modulus is always positive or zero

  • It cannot be negative

• For example

  • $\vert 3 + 4 \mathrm{i} \vert = \sqrt{3^{2} + 4^{2}} = 5$
  • $\vert 1 - \mathrm{i} \vert = \sqrt{1^{2} + ( - 1 )^{2}} = \sqrt{2}$
  • $\vert - 5 \vert = 5$
  • $\vert 8 \mathrm{i} \vert = 8$
  • $\vert 0 + 0 \mathrm{i} \vert = 0$

What rules does the modulus follow?

• Helpful modulus rules are:

  • $\vert z_{1} z_{2} \vert = \vert z_{1} \vert \vert z_{2} \vert$

  • | z _ 1 / z _ 2 | = (| z _ 1 |)/(| z _ 2 |)

  • $\vert z \vert = \vert z \text{asterisk} \times \vert$

  • $z z \text{asterisk} \times = z \text{asterisk} \times z = \vert z \vert^{2}$

    • Proved using $z=x+yi$ and $z\ast=x-yi$

• Be careful:$\left| z_{1} + z_{2} \right| \text{eq} \left| z_{1} \right| + \vert z_{2} \vert$

  • For example, $z_{1}=3+4i$ and $z_{2}=-3+4i$

    • $\left| z_{1} \right| = \sqrt{3^{2} + 4^{2}} = 5$ and $\vert z_{2} \vert = \sqrt{( - 3 )^{2} + 4^{2}} = 5$
    • so $\vert z_{1} \vert + \vert z_{2} \vert = 10$
    • but $z_{1}+z_{2}=8i$ so open vertical bar z_ 1+z_ 2 close vertical bar=8

How do I find the argument of a complex number?

• The argument of a complex number is the angle in radians that it makes to the positive real axis

  • It is written as $\mathrm{arg}z$
  • The positive direction is anticlockwise

• The range normally used is $-π<\mathrm{arg}z\leqπ$

  • This is called the principal range

  • The sign of the angle depends on the quadrant:

    • The 1^st quadrant is positive acute
    • The 2^nd quadrant is positive obtuse
    • The 3^rd quadrant is negative obtuse
    • The 4^th quadrant is negative acute

• Arguments are found by

  • drawing a sketch
  • forming a right-angled triangle
  • using trigonometry

• The argument of the origin, $\text{arg} ( 0 + 0 \mathrm{i} )$, is undefined

  • No angle can be drawn

Exam Hint Always draw a sketch to see which quadrant the complex number is in.

Worked Example (a) Find the modulus and argument of $z=2+3i$, giving your answers correct to 3 significant figures.

Sketch this on an Argand diagram Form a right-angled triangle

Use $\left| z \right| = \sqrt{x^{2} + y^{2}}$ (or Pythagoras) to find the modulus

$\begin{gathered} \vert z \vert & = \sqrt{2^{2} + 3^{2}} \\ = \sqrt{13} \\ = 3.60555127... \end{gathered}$

$z$ is in the first quadrant so the argument is positive and acute Use trigonometry to find the argument $θ$ in radians

$\begin{gathered} \tan \\theta & = \frac{3}{2} \\ \\theta = \tan ^{- 1} ( \frac{3}{2} ) \\ \\theta = 0.98279372... \end{gathered}$

Round the answers to 3 significant figures

Final answer: $\vert z \vert = 3.61$and$\mathrm{arg}z=0.983$to 3 significant figures

(b) Find the modulus and argument of $w=-1-\sqrt{3}i$, leaving your answers as exact values.

Sketch this on an Argand diagram Form a right-angled triangle

Use $\left| z \right| = \sqrt{x^{2} + y^{2}}$ (or Pythagoras) to find the modulus

$\begin{gathered} \vert z \vert & = \sqrt{( - 1 )^{2} + ( - \sqrt{3} )^{2}} \\ = \sqrt{4} \\ = 2 \end{gathered}$

$z$ is in the third quadrant so the argument is negative and obtuse Use trigonometry to first find $α$ in radians

$\begin{gathered} \tan \\alpha & = \frac{ \sqrt{3} }{1} \\ \\alpha = \tan ^{- 1} ( \sqrt{3} ) \\ \\alpha = \frac{\pi }{3} \end{gathered}$

Then find $θ$ by subtracting $α$ from 180° ($π$ radians)

$\begin{gathered} \\theta = \pi - \\alpha \\ = \pi - \frac{\pi }{3} \\ = \frac{2 \pi }{3} \end{gathered}$

Remember that the argument here must be a negative angle

Final answer: $\vert z \vert = 2$and$\mathrm{arg}z=-\frac{2π}{3}$

These answers must be exact

Modulus-Argument Form

Spec point — 1.1

What is modulus-argument form?

• All complex numbers can be written in the form:

  • $z = r ( \cos \\theta + \text{isin} \\theta )$

  • where $r = \vert z \vert$ and $θ=\mathrm{arg}z$

    • This is called the **modulus-argument (or polar) **form

• Negative arguments should be shown clearly** without simplifying**

  • $z = 2 ( \cos ( - \frac{\pi }{3} ) + \text{isin} ( - \frac{\pi }{3} ) )$

    • Simplifying them gives the Cartesian form
    • $z = 2 ( 1 \frac{1}{2} + \mathrm{i} ( - \frac{ \sqrt{3} }{2} ) ) = 1 - \sqrt{3} \mathrm{i}$

• Be careful: $z = r ( \cos \\theta - \text{isin} \\theta )$ is not in modulus-argument form (due to the minus sign)

  • Rewrite it as $z = r ( \cos ( - \\theta ) + \text{isin} ( - \\theta ) )$

    • This uses the symmetries$\cos ( - \\theta ) \equiv \cos ( \\theta )$ and $\sin ( - \\theta ) \equiv - \sin \\theta$

  • The argument is $-θ$

Worked Example Write $z=-4+4i$ in the exact form $r(\mathrm{cos}θ+i\mathrm{sin}θ)$ where $r>0$ and $-π<θ\leqπ$.

Draw a sketch to find the modulus, $r$, and argument, $θ$, of $z$ Form a right-angled triangle

Use $\left| z \right| = \sqrt{x^{2} + y^{2}}$ (or Pythagoras) to find the modulus

$\begin{gathered} \vert z \vert & = \sqrt{( - 4 )^{2} + ( 4 )^{2}} \\ = \sqrt{32} \\ = 4 \sqrt{2} \end{gathered}$

$z$ is in the second quadrant so the argument is positive and obtuse Use trigonometry to first find $α$ in radians

$\begin{gathered} \tan \\alpha & = \frac{4}{4} \\ \\alpha = \tan ^{- 1} ( 1 ) \\ \\alpha = \frac{\pi }{4} \end{gathered}$

Then find $θ$ by subtracting $α$ from 180° ($π$ radians)

$\begin{gathered} \\theta = \pi - \\alpha \\ = \pi - \frac{\pi }{4} \\ = \frac{3 \pi }{4} \end{gathered}$

Write the final answer in the form $r(\mathrm{cos}θ+i\mathrm{sin}θ)$

$z = 4 \sqrt{2} ( \cos ( \frac{3 \pi }{4} ) + \mathrm{i} \sin ( \frac{3 \pi }{4} ) )$

Leave the modulus and argument exact