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{{ printedBook.courseTrack.name }} {{ printedBook.name }} 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 $i_{2}=-1.$ From this definition, it follows that $-1 =i,$ 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.

The set of complex numbers, represented by the symbol $C,$ 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 $b =0,$ 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.

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

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

Simplify the following expressions. $-36 (8i)_{2}(2+4i)+(1−i)(1−i)(2+4i)$

Show Solution

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

$-36 $

$(2+4i)+(1−i)$

RemoveParRemove parentheses

$2+4i+1−i$

CommutativePropAddCommutative Property of Addition

$2+1+4i−i$

SimpTermsSimplify terms

$3+3i$

$(1−i)(2+4i)$

MultParMultiply parentheses

$1⋅2+1⋅4i−i⋅2−i⋅4i$

MultiplyMultiply

$2+4i−2i−4i_{2}$

IDefPow$i_{2}=-1$

$2+4i−2i−4(-1)$

MultiplyMultiply

$2+4i−2i+4$

CommutativePropAddCommutative Property of Addition

$2+4+4i−2i$

SimpTermsSimplify terms

$6+2i$

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