Modulus & Argument (DP IB Analysis & Approaches (AA): HL): Revision Note

Modulus & argument

What is the modulus of a complex number?

• The modulus of a complex number (written $\left|z\right|$ or $r$) is its distance from the origin when plotted on an Argand diagram

  • If $z=x+\mathrm{i}y$, then by Pythagoras

    • $\left|z\right|=\sqrt{x^2+y^2}$

  • A modulus is never negative

• In general, adding two complex numbers does not mean you can add their moduli 

  • $\left|z_1+z_2\right|\ne \left|z_1\right|+\left|z_2\right|$

  • e.g. both $z_1=3+4\mathrm{i}$ and $z_2=-3+4\mathrm{i}$ have a modulus of 5

    • but $z_1+z_2=8\mathrm{i}$ has a modulus of 8

How does the modulus relate to the complex conjugate?

• The modulus is the result of multiplying a complex number by its conjugate

  • $z{z}^{*}=z^{*}z={\left|z\right|}^2$

  • e.g. $z{z}^{*}=\left(x+\mathrm{i}y\right)\left(x-\mathrm{i}y\right)=x^2-{\mathrm{i}}^2{y}^2=x^2+y^2$

What is the argument of a complex number?

• The argument of a complex number (written $\arg z$ or $\theta$) is the angle that it makes to the positive real axis on an Argand diagram

  • measured

    • counter-clockwise

    • in radians

  • where $-\pi <\arg z\le \pi$

• Arguments are

  • positive and acute in the first quadrant

  • positive and obtuse in the second quadrant

  • negative and obtuse in the third quadrant

  • negative and acute in the fourth quadrant

• Arguments can be calculated using right-angled trigonometry

  • A sketch is needed to decide the quadrant

    • positive / negative, or acute / obtuse

  • then the tan ratio can be used

• The argument of zero, $z=0+0\mathrm{i}$, is undefined

  • $\arg 0$ does not exist

    • as no angle can be drawn

What happens to the modulus and argument under multiplication or division?

• When two complex numbers, $z_1$ and $z_2$, are multiplied, $z_1{z}_2$

  • their moduli are also multiplied

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

  • but their arguments are added

    • $\arg \left(z_1{z}_2\right)=\arg z_1+\arg z_2$

• When two complex numbers, $z_1$ and $z_2$, are divided, $\frac{z_1}{z_2}$

  • their moduli are also divided

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

  • but their arguments are subtracted

    • $\arg \left(\frac{z_1}{z_2}\right)=\arg z_1-\arg z_2$