Modulus & Argument (Edexcel International AS Further Maths: Further Pure 1): Revision Note

Argand Diagrams

What is an Argand diagram?

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

• The complex number $x+y\mathrm{i}$ 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

Modulus & Argument

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 $\left|z\right|$

  • If $z=x+\mathrm{i}y$, 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

  • $|3+4\mathrm{i}|=\sqrt{3^2+4^2}=5$

  • $|1-\mathrm{i}|=\sqrt{1^2+{\left(-1\right)}^2}=\sqrt{2}$

  • $|-5|=5$

  • $\left|8\mathrm{i}\right|=8$

  • $|0+0\mathrm{i}|=0$

What rules does the modulus follow?

• Helpful modulus rules are:

  • $\left|z_1{z}_2\right|=\left|z_1\right|\left|z_2\right|$

  • $\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}$

  • $\left|z\right|=|z*|$

  • $zz*=z*z=|z{|}^2$

    • Proved using $z=x+y\mathrm{i}$ and $z*=x-y\mathrm{i}$

• Be careful:$\left|z_1+z_2\right|\ne \left|z_1\right|+\left|z_2\right|$

  • For example, $z_1=3+4\mathrm{i}$ and $z_2=-3+4\mathrm{i}$ 

    • $\left|z_1\right|=\sqrt{3^2+4^2}=5$ and $\left|z_2\right|=\sqrt{{\left(-3\right)}^2+4^2}=5$

    • so $\left|z_1\right|+\left|z_2\right|=10$

    • but $z_1+z_2=8\mathrm{i}$ so $\left|z_1+z_2\right|=8$

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 $\arg z$

  • The positive direction is anticlockwise

• The range normally used is $-\pi <\arg z\le \pi$

  • 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, $\arg \left(0+0\mathrm{i}\right)$, is undefined

  • No angle can be drawn

Modulus-Argument Form

What is modulus-argument form?

• All complex numbers can be written in the form:

  • $z=r\left(\cos \theta +\mathrm{isin}\theta \right)$

  • where $r=\left|z\right|$ and $\theta =\arg z$

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

• Negative arguments should be shown clearly without simplifying

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

    • Simplifying them gives the Cartesian form

    •  $z=2\left(\frac{1}{2}+\mathrm{i}\left(-\frac{\sqrt{3}}{2}\right)\right)=1-\sqrt{3} \mathrm{i}$

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

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

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

  • The argument is $-\theta$