The Main Idea

Complex numbers combine a real part and an imaginary part. The complex plane is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Plotting a complex number [latex]a+bi[/latex] works just like plotting a point [latex](x,y)[/latex] in the rectangular plane, except the vertical axis is labeled “Imaginary” instead of “y.”

Quick Tips: Plotting Complex Numbers

1. Identify the Parts

   • Real part = [latex]a[/latex] (x-coordinate).

   • Imaginary part = [latex]b[/latex] (y-coordinate).

2. Plot as a Point

   • Plot [latex]a+bi[/latex] at [latex](a,b)[/latex] on the complex plane.

The Main Idea

Complex numbers can be expressed not only in rectangular form [latex]a+bi[/latex], but also in polar form, which uses a magnitude (distance from the origin) and an angle (direction from the positive real axis). This form highlights the geometric meaning of complex numbers and is especially useful for multiplication, division, and powers.

The polar form of a complex number is:

[latex]z = r(\cos\theta + i\sin\theta)[/latex]

where:

• [latex]r = \sqrt{a^{2}+b^{2}}[/latex] is the magnitude (modulus).

• [latex]\theta = \tan^{-1}\left(\dfrac{b}{a}\right)[/latex] is the argument (angle), adjusted for quadrant.

Quick Tips: Converting to Polar Form

1. Find the Magnitude

   • [latex]r=\sqrt{a^{2}+b^{2}}[/latex].

2. Find the Angle

   • [latex]\theta = \tan^{-1}\left(\dfrac{b}{a}\right)[/latex].

   • Adjust [latex]\theta[/latex] for the correct quadrant.

3. Write in Polar Form

The Main Idea

Polar form expresses a complex number as [latex]z = r(\cos\theta + i\sin\theta)[/latex]. To return to rectangular form [latex]a+bi[/latex], we simply evaluate the cosine and sine at the given angle and multiply by [latex]r[/latex]. This lets us move seamlessly between the two representations: geometric (polar) and algebraic (rectangular).

Quick Tips: Polar → Rectangular

1. Recall the Relationship

   • [latex]z = r(\cos\theta + i\sin\theta)[/latex].

   • So [latex]a = r\cos\theta[/latex], [latex]b = r\sin\theta[/latex].

2. Use Calculators When Needed

   • If [latex]\theta[/latex] is not a special angle, compute [latex]\cos\theta[/latex] and [latex]\sin\theta[/latex] with a calculator.

3. Check Your Quadrant

   • Make sure the signs of [latex]a[/latex] and [latex]b[/latex] match the quadrant of [latex]\theta[/latex].

The Main Idea

Polar form makes multiplying and dividing complex numbers much easier than working in rectangular form. Instead of expanding and simplifying, we use the magnitudes and angles directly. Multiplication means multiplying magnitudes and adding angles, while division means dividing magnitudes and subtracting angles. This is a direct application of trigonometric identities and Euler’s formula.

Formulas:

• Product: [latex]z_{1}z_{2} = r_{1}r_{2}\big(\cos(\theta_{1}+\theta_{2}) + i\sin(\theta_{1}+\theta_{2})\big)[/latex]

• Quotient: [latex]\dfrac{z_{1}}{z_{2}} = \dfrac{r_{1}}{r_{2}}\big(\cos(\theta_{1}-\theta_{2}) + i\sin(\theta_{1}-\theta_{2})\big)[/latex]

Quick Tips: Multiplying and Dividing in Polar Form

1. Multiplication Rule

   • Multiply the magnitudes: [latex]r_{1}r_{2}[/latex].

   • Add the angles: [latex]\theta_{1}+\theta_{2}[/latex].

2. Division Rule

   • Divide the magnitudes: [latex]\dfrac{r_{1}}{r_{2}}[/latex].

   • Subtract the angles: [latex]\theta_{1}-\theta_{2}[/latex].

3. Keep Results in Polar Form

   • These operations are simplest in polar form.

   • If rectangular form is needed, convert at the end using [latex]x=r\cos\theta, y=r\sin\theta[/latex].

The Main Idea

Polar form makes it especially easy to compute powers and roots of complex numbers using De Moivre’s Theorem. Instead of expanding binomials, we work directly with magnitudes and angles.

• De Moivre’s Theorem (Powers):
  [latex]z^{n} = \big(r(\cos\theta + i\sin\theta)\big)^{n} = r^{n}\big(\cos(n\theta) + i\sin(n\theta)\big)[/latex]

• nth Roots of a Complex Number:
  [latex]z_{k} = r^{\tfrac{1}{n}}\Big(\cos\Big(\dfrac{\theta+2k\pi}{n}\Big) + i\sin\Big(\dfrac{\theta+2k\pi}{n}\Big)\Big), \quad k=0,1,\dots,n-1[/latex]

This guarantees that powers give a single answer, while roots give n distinct solutions, evenly spaced around a circle in the complex plane.

Quick Tips: Powers and Roots

1. Powers with De Moivre’s Theorem

   • Raise the magnitude to the nth power.

   • Multiply the angle by n.

2. Work an Example (Power): Find [latex](1+i)^{4}[/latex].

   • Convert: [latex]r=\sqrt{1^{2}+1^{2}}=\sqrt{2}, \ \theta=45^\circ[/latex].

   • Apply De Moivre: [latex]z^{4} = (\sqrt{2})^{4}\big(\cos(4\cdot 45^\circ)+i\sin(4\cdot 45^\circ)\big)[/latex].

   • [latex]= 4(\cos 180^\circ + i\sin 180^\circ) = 4(-1+0i)=-4[/latex].

3. nth Roots

   • Take the nth root of the magnitude: [latex]r^{1/n}[/latex].

   • Divide the angle by n, then add [latex]\dfrac{2k\pi}{n}[/latex] for each root.

4. Work an Example (Roots): Find the cube roots of [latex]8(\cos 0^\circ+i\sin 0^\circ)[/latex].

   • Magnitude: [latex]8^{1/3}=2[/latex].

   • Angles: [latex]\dfrac{0^\circ+360^\circ k}{3}, \ k=0,1,2[/latex].

   • Roots:

     • [latex]2(\cos 0^\circ+i\sin 0^\circ)=2[/latex].

     • [latex]2(\cos 120^\circ+i\sin 120^\circ) = -1+\sqrt{3}i[/latex].

     • [latex]2(\cos 240^\circ+i\sin 240^\circ) = -1-\sqrt{3}i[/latex].

5. Remember the Geometry

   • Powers “stretch” the vector and rotate it.

   • Roots divide the angle into equal arcs, producing evenly spaced solutions around a circle.