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# Performing Arithmetic with Complex Numbers

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## The Imaginary Unit

Is there any number that, when multiplied by itself, equals -1? Among the real numbers, the answer is no. However, the system of numbers can be expanded to include the imaginary unit i, defined as
i2=-1.
From this definition, it follows that which allows the square root of any negative number to be found. What results is called an imaginary number. Once i replaces the negative sign, the square root of the remaining positive number can be evaluated as usual.
Condition: a>0

## Complex Numbers

The set of complex numbers, represented by the symbol is formed by all numbers that can be written in the form z=a+bi, where a and b are real numbers, and i is the imaginary unit. Here, a is called the real part and b is called the imaginary part of the complex number.

If b0, the number is an imaginary number. Conversely, if b=0, the number is real. Both real numbers and imaginary numbers are subsets of the complex number set.

## Performing Operations with Complex Numbers

Complex numbers can be added, subtracted and multiplied just like real numbers.

### Method

Adding and subtracting complex numbers is done by combining like terms. Imaginary numbers and real numbers are not like terms.

### Multiplication

When multiplying two complex numbers, the Distributive Property can be used. Meaning, each term of the first number is multiplied by each term of the second number.

## Simplify the complex expressions

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Simplify the following expressions.
Show Solution expand_more

Let's simplify the expressions, one at a time.

### Example

The square root of a negative number can be found by first writing the number as a product, then using
6i

### (8i)2

Here, we can use the Power of a Product Property and that i2=-1, to simplify the expression.
(8i)2
64i2
64(-1)
-64

### (2+4i)+(1−i)

For this expression, we'll combine the real and the imaginary parts.
(2+4i)+(1i)
2+4i+1i
2+1+4ii
3+3i

### (1−i)(2+4i)

Here, we can use the Distributive Property to multiply two complex numbers. We'll also use the fact that i2=-1.
(1i)(2+4i)
12+14ii2i4i
2+4i2i4i2
2+4i2i4(-1)
2+4i2i+4
2+4+4i2i
6+2i