The expression
has no real answer. The symbol
i is created to represent
and is called an
imaginary value. Since
,
i2 = −1. Any expression that is a product of a real number with
i is called a
pure imaginary number.
Example 1: Simplify each of the following.
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This last expression is commonly written as
so that the
i is not mistakenly written under the radical.
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(6 i)(4 i) = 24 i2 = 24(−1) = −24
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For this last example, all imaginary values had to be put into their “ i-form” before any simplifying could be done. Note that
That is, the product rule for radicals does not hold (in general) with imaginary numbers.
When
i is raised to powers, it has a repeating pattern.
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When i is raised to any whole number power, the result is always 1, i, −1 or −i. If the exponent on i is divided by 4, the remainder will indicate which of the four values is the result.
Example 2: Simplify each of the following.
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i34
Since 34 divided by 4 has a remainder of 2, i34 = i2 = −1.
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i95
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Since 95 divided by 4 has a remainder of 3, i95 = i3 = − i.
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i108
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Since 108 divided by 4 has a zero remainder, i108 = i0 = 1.
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i53
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Since 53 divided by 4 has a remainder of 1, i53 = i1 = i.
Complex numbers and complex conjugates. A complex number is any expression that is a sum of a pure imaginary number and a real number. A complex number is usually expressed in a form called the a + bi form, or standard form, where a and b are real numbers. The expressions a + bi and a − bi are called complex conjugates. Complex conjugates are used to rationalize the denominator when dividing with complex numbers.
Arithmetic with complex numbers is done in a similar manner as arithmetic with polynomials. The following are definitions for arithmetic with two complex numbers call ( a + bi) and ( c + di).
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Combining like terms and factoring out the i,
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Using the distributive property,
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Rationalizing the denominator,
Example 3: Find the sum, difference, product, and quotient of (4 + 3
i) and (5 −4
i).
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Quotient: Rationalize the denominator.
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Example 4: Simplify
.
Since 6
i is 0 + 6
i, its complex conjugate is 0 − 6
i. Therefore,
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Linear Sentences in One Variable
Radicals and Complex Numbers