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# 10999 form

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# 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.

*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: 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:

### Proof of Euler’s Identity and Proof that *r* ∠ *θ* = *r e* *i* *θ*

Now that we have Euler’s identity it is easy to prove that *r* ∠ *θ* = *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: Now factor out *r* and use Euler’s identity on the terms in the brackets. The result is that we have converted the left side to equal the right side. This proves that *r* ∠ *θ* = *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.

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

**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.

# 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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