Use polar form to interpret multiplication, powers and roots as changes of scale and angle.
A complex number can be read as a point. In $z=3+2i$, the $3$ tells us to move three units horizontally and the $2$ tells us to move two units vertically. The point is $(3,2)$; the vector from the origin reaches that same point.
Use the controls to try $(3,2)$, then compare points in different quadrants. What changes when only the real part changes?
Check the logic
Reflecting 3+3i to 3−3i changes which quantity?
Both points have distance 3√2; the principal argument changes from π/4 to −π/4.
Set (a,b) to (3,3), then (3,−3). Compare modulus and argument and explain the symmetry.
$|z|=\sqrt{1.5^2+4^2}\approx4.272$
$\arg(z)=\operatorname{atan2}(4,1.5)\approx1.212$
The modulus is the length of the vector from the origin. The argument is the angle measured from the positive real axis.
Explore the Argand plane
The same point can be described by its distance from the origin and its direction from the positive real axis. These are the modulus and argument. The right triangle connects the distance to the two coordinates:
Its horizontal and vertical components are $x=r\cos\theta$ and $y=r\sin\theta$. Substitute these into $z=x+iy$ to obtain polar form:
For a nonzero number, use the quadrant to choose the principal argument in $-\pi<\theta\le\pi$. Zero has modulus zero but no direction, so its argument is undefined.
The ratio $y/x$ alone cannot distinguish opposite directions. For example, $(2,2)$ and $(-2,-2)$ have the same ratio but principal arguments $\pi/4$ and $-3\pi/4$. The signs of both coordinates resolve the direction.
Checkpoint
Correct. The reference angle is $\pi/3$, and quadrant II gives $\theta=2\pi/3$.
Not quite. Use the signs of the real and imaginary parts to place the argument in the correct quadrant.
The reference angle is $\pi/3$, and quadrant II gives $\theta=2\pi/3$.
Example 1
From coordinates to angle
Write $z=-2+2i$ in polar form using a principal argument.
Multiplying a complex number changes both its distance and its direction. A factor of modulus 2 doubles every distance; a factor of argument $\pi/2$ turns every nonzero point a quarter-turn anticlockwise.
The point (1,1) moves to (−2,2): its distance doubles and its direction turns anticlockwise by π/2. Thus |zw| = 2√2 and arg(zw) = 3π/4. Numbered arrows distinguish the factors and product.
The visual gives two separate rules: multiply the distances and add the angles. If $z_1=r_1\operatorname{cis}\theta_1$ and $z_2=r_2\operatorname{cis}\theta_2$, then
Checkpoint
Correct. Moduli multiply by $2$ and arguments increase by $\pi/3$.
Not quite. The modulus controls scale; the argument controls rotation.
Moduli multiply by $2$ and arguments increase by $\pi/3$.
Example 2
A high power
Find $(1+i)^8$ exactly.
Since $1+i=\sqrt2\operatorname{cis}(\pi/4)$, De Moivre gives
An argument of $2\pi$ points along the positive real axis, so the imaginary component is zero.
A cube root must return to the original number when its distance is cubed and its angle is tripled. Several starting angles can land on the same direction, because a full turn changes no point.
The cube roots of 8 have modulus 2 and angles 0, 2π/3 and 4π/3. Cubing triples each angle, so all three land on the positive real axis. Adjacent roots differ by one third of a full turn.
In general, divide all angles $\phi+2k\pi$ by $n$. For $R>0$, the $n$ roots of $R\operatorname{cis}\phi$ have modulus $R^{1/n}$ and arguments
They are equally spaced around a circle.
Checkpoint
Correct. Four roots divide a full turn into intervals of $2\pi/4=\pi/2$.
Not quite. Root spacing is $2\pi/n$, independent of the original argument.
Four roots divide a full turn into intervals of $2\pi/4=\pi/2$.
Example 3
Roots of $-8i$
Write $-8i=8\operatorname{cis}(-\pi/2)$. Its cube roots have modulus $2$ and arguments
Therefore the roots are $2\operatorname{cis}(-\pi/6)$, $2i$, and $2\operatorname{cis}(7\pi/6)$.
Substituting any one of these into $z^3$ returns $-8i$.
Your score is not saved.
$\sqrt{3^2+(-4)^2}=5$.
$\sqrt{3^2+(-4)^2}=5$.
Multiplication adds arguments and multiplies moduli.
Multiplication adds arguments and multiplies moduli.
$-i$ points down the imaginary axis.
$-i$ points down the imaginary axis.
De Moivre's theorem multiplies the argument by the power.
De Moivre's theorem multiplies the argument by the power.
The roots are equally spaced with $k=0,1,2,3,4$.
The roots are equally spaced with $k=0,1,2,3,4$.
1
5
Want to leave a comment?
Log in