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The Imaginary Unit i Solutions

The imaginary unit i is a special number in math that is unique because it does not have a real number value .
The imaginary unit i is defined as: $i^{2} = - 1$
The number i is the principal square root of –1, or: $i = \sqrt{- 1}$
Imaginary numbers can be simplified using the following guidelines:

Rules for the Imaginary Unit

$i^{0} = 1$
$i = \sqrt{- 1}$
$i^{2} = - 1$
$i^{3} = - i$	$i^{3} = i \cdot i^{2} = i \cdot - 1 = - i$
$i^{4} = 1$	$i^{4} = {(i^{2})}^{2} = {(- 1)}^{2} = 1$

When evaluating an expression using the number i, use the exponent rules to rewrite the base as $i^{4}$ raised to a power because $i^{4} = 1$ will be easier to work with.
If there is a remainder , it determines if the answer will be i, –1, or –i.

Note

While you should have your Formula Sheet at all times, using it for this particular lesson will be very important. This lesson uses the exponent rules, algebraic properties, and the rules for the imaginary unit.

Example 1

Evaluate.

$i^{9} i^{8} \cdot i {(i^{4})}^{2} \cdot i {(1)}^{2} \cdot i$

$i$

$i^{10} i^{8} \cdot i^{2} {(i^{4})}^{2} \cdot i^{2}$

${(1)}^{2} \cdot (- 1) - 1$

$i^{11} i^{8} \cdot i^{3}$

${(i^{4})}^{2} \cdot i^{3} {(1)}^{2} \cdot (- i) - i$

$i^{12}$

${(i^{4})}^{3} {(1)}^{3} = 1$

Note

Notice that a remainder determines the value. Dividing the exponent by 4 and then determining the remainder may be helpful when evaluating expressions with i.

Example 2

Evaluate.

$i^{39}$
$39 \div 4 = 9 R 3 {(i^{4})}^{9} \cdot i^{3} - i$

$39 \div 4 = 9 R 3 {(i^{4})}^{9} \cdot i^{3} - i$

$i^{85}$

$85 \div 4 = 21 R 1 {(i^{4})}^{21} \cdot i i$

$i^{46}$

$i^{46} 46 \div 4 = 11 R 2 {(i^{4})}^{11} \cdot i^{2} - 1$