10999 form

10999 form

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10999 form

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10999 form

10999 form

10999 form

Example: Convert the complex number 5 ∠ 53° to rectangular form.

Example: Convert the complex number 5 + 2 i to polar form.

Example: Convert the complex number −5 − 2 i to polar form.

Multiplying and Dividing Complex Numbers in Polar Form

  • Multiplication rule: To form the product multiply the magnitudes and add the angles.

10999 form

10999 form

x m · x n = x m + n and x m / x n = x m − n . They suggest that perhaps the angles are some kind of exponents. This guess turns out to be correct. We will prove in the next section that: 10999 form where as usual the base e is the number e = 2.71828… The form r e i θ is called exponential form of a complex number. The next example shows the same complex numbers being multiplied in both forms:

10999 form

10999 form

Proof of Euler’s Identity and Proof that rθ = r e i θ

10999 form

Now that we have Euler’s identity it is easy to prove that rθ = r e i θ . Again we will manipulate the left side until it equals the right side. First, convert the left side to rectangular form: 10999 form Now factor out r and use Euler’s identity on the terms in the brackets. 10999 form The result is that we have converted the left side to equal the right side. This proves that rθ = r e i θ

Examples Using Rectangular, Polar and Exponential Form

Example 2: Calculate i n for n = 1, 2, 3, …, plot the numbers on the complex plane, and spot the pattern.

10999 form

10999 form

Example 3: Evaluate the exponential (3 + 4 i) (6 + 7 i)

10999 form

10999 form

10999 form

10999 form

10999 form

10999 form

10999 form

Example 4: DeMoivre’s Theorem states that the equation x n = C (where C is any complex number and n is any positive integer has n complex roots (solutions) and that they are uniformly distributed in a circle about the origin of the complex plane.

10999 form

10999 form


1099-form.org 1099 Form - What is a 1099 Form? - Form 1099 for 2017

A 1099 Form is used for miscellaneous income tax reporting. Know how to file and report 1099 Forms, and when they are needed.

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