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- Imaginary and Complex Numbers

Still Confused?

Try reviewing these fundamentals first.

Still Confused?

Try reviewing these fundamentals first.

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Get Started NowStart now and get better math marks!

Get Started NowStart now and get better math marks!

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Let's find out how to perform some basic operations on complex numbers in polar form! We will briefly introduce the notion of the exponential form of a complex number, then we will focus on multiplication and division on complex numbers in polar form.

Basic concepts: Exponents: Product rule $(a^x)(a^y)=a^{(x+y)}$, Exponents: Division rule ${a^x \over a^y}=a^{(x-y)}$,

Related concepts: Imaginary zeros of polynomials,

Note:

Polar form real part$a=|z|\cos \theta$

imaginary part$b=|z|\sin \theta$

$z=|z|(\cos \theta+i\sin \theta)$

When …

Multiplying:*multiply* the absolute values, and *add* the angles

Dividing:*divide* the absolute values, and *subtract* the angles

Exponential form$z=|z|e^{i \theta}$

Polar form real part$a=|z|\cos \theta$

imaginary part$b=|z|\sin \theta$

$z=|z|(\cos \theta+i\sin \theta)$

When …

Multiplying:

Dividing:

Exponential form$z=|z|e^{i \theta}$

- 1.Multiplying complex numbers in polar forma)$4(\cos(\frac{5\pi}{3})+i \sin(\frac{5\pi}{3})) \cdot 8(\cos(\frac{2\pi}{3})+i \sin(\frac{2\pi}{3}))$b)$(\cos(170^{\circ})+i \sin(170^{\circ}))\cdot 5(\cos(45^{\circ})+i \sin(45^{\circ}))$c)$3(\cos(\pi)+i \sin(\pi))\cdot(\cos(\frac{\pi}{5})+i \sin(\frac{\pi}{5}))\cdot6(\cos(\frac{2\pi}{3})+i \sin(\frac{2\pi}{3}))$
- 2.Dividing complex numbers in polar forma)$20(\cos(\frac{5\pi}{2})+i \sin(\frac{5\pi}{2}))\div 6(\cos(\frac{2\pi}{3})+i \sin(\frac{2\pi}{3}))$b)$3(\cos(\frac{3\pi}{4})+i \sin(\frac{3\pi}{4}))\div 12(\cos(\frac{\pi}{6})+i \sin(\frac{\pi}{6}))$c)$(\cos(262^{\circ})+i \sin(262^{\circ}))\div (\cos(56^{\circ})+i \sin(56^{\circ}))$
- 3.Convert the following complex number to exponential form

$z=3+i$

8.

Imaginary and Complex Numbers

8.1

Introduction to imaginary numbers

8.2

Complex numbers and complex planes

8.3

Adding and subtracting complex numbers

8.4

Complex conjugates

8.5

Multiplying and dividing complex numbers

8.6

Distance and midpoint of complex numbers

8.7

Angle and absolute value of complex numbers

8.8

Polar form of complex numbers

8.9

Operations on complex numbers in polar form

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Get Started Now8.1

Introduction to imaginary numbers

8.2

Complex numbers and complex planes

8.3

Adding and subtracting complex numbers

8.4

Complex conjugates

8.5

Multiplying and dividing complex numbers

8.6

Distance and midpoint of complex numbers

8.7

Angle and absolute value of complex numbers

8.8

Polar form of complex numbers

8.9

Operations on complex numbers in polar form