IntroductionUntil now, you have been told that you cannot take the square of –1 because there is no number that when squared will result in a negative number. In other words,

the square root of –1, or , is not a real number. French mathematician René Descartes suggested the imaginary unit i be defined so that i2 = –1. The imaginary unit enables us to solve problems that we would not otherwise be able to solve. Problems involving electricity often use the imaginary unit.

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4.3.1: Defining Complex Numbers, i, and i2

Key Concepts• All rational and irrational numbers are real numbers. • The imaginary unit i is used to represent the non-

real

value, . • An imaginary number is any number of the form bi,

where b is a real number, i = , and b ≠ 0. • Real numbers and imaginary numbers can be

combined to create a complex number system.• A complex number contains two parts: a real part and

an imaginary part.

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4.3.1: Defining Complex Numbers, i, and i2

Key Concepts, continued• All complex numbers are of the form a + bi, where a

and b are real numbers and i is the imaginary unit. • In the general form of a complex number, a is the real

part of the complex number, and bi is the imaginary part of the complex number. Note that if a = 0, the complex number a + bi is wholly imaginary and contains no real part: 0 + bi = bi.

• If b = 0, the complex number a + bi is wholly real and contains no imaginary part: a + (0)i = a.

• Expressions containing imaginary numbers can also be simplified.

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4.3.1: Defining Complex Numbers, i, and i2

Key Concepts, continued• To simplify in for n > 4, divide n by 4, and use the

properties of exponents to rewrite the exponent.

• The exponent can be rewritten using the quotient:

n ÷ 4 = m remainder r, or , where r is the

remainder when dividing n by 4, and m is a whole number.

• Then n = 4m + r, and r will be 0, 1, 2, or 3. Use the properties of exponents to rewrite in.

in = i 4m + r = i

4m • i r

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4.3.1: Defining Complex Numbers, i, and i2

Key Concepts, continued• i

4 = 1, so i to a multiple of 4 will also be 1:

i 4m = (i

4)m = (1)m = 1. • The expression i

r will determine the value of in. • Use i

0, i 1, i

2, and i 3 to find in.

If r = 0, then: If r = 1, then:

in = i 4m + r in = i

4m + r

in = i 4m • ir in = i

4m • i r

in = 1 • i 0 in = 1 • i

1

in = 1 • 1 = 1 in = i, or

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4.3.1: Defining Complex Numbers, i, and i2

Key Concepts, continuedIf r = 2, then: If r = 3, then:

in = i 4m + r in = i

4m + r

in = i 4m • ir in = i

4m • i r

in = 1 • i 2 in = 1 • i

3

in = 1 • –1 = –1 in = i • –1 = –i, or• Only the value of the remainder when n is divided by

4 is needed to simplify in.

i 0 = 1, i 1 = i, i

2 = –1, and i 3 = –i

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4.3.1: Defining Complex Numbers, i, and i2

Key Concepts, continued• Properties of exponents, along with replacing i with its

equivalent value of , can be used to simplify the expression in. Start with n = 0.

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4.3.1: Defining Complex Numbers, i, and i2

Key Concepts, continued• When simplifying i

3, use the property i 3 = i

2 • i 1, and

the determined values of i 2 and i.

• When simplifying i 4, use the property i

4 = i 2 • i

2, and the determined value of i

2.

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4.3.1: Defining Complex Numbers, i, and i2

Common Errors/Misconceptions• incorrectly identifying the real and imaginary parts of a

complex number • assuming that a real number isn’t part of the complex

number system • incorrectly dividing a power of i by 4 • incorrectly simplifying the expressions i

2, i 3, and i

4, including stating i

2 = 1

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4.3.1: Defining Complex Numbers, i, and i2

Guided Practice

Example 3Rewrite the radical using the imaginary unit i.

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4.3.1: Defining Complex Numbers, i, and i2

Guided Practice: Example 3, continued

1. Rewrite the value under the radical as the product of –1 and a positive value.

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4.3.1: Defining Complex Numbers, i, and i2

Guided Practice: Example 3, continued

2. Rewrite the radical as i.

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4.3.1: Defining Complex Numbers, i, and i2

Guided Practice: Example 3, continued

3. If possible, rewrite the positive value as the product of a square number and another whole number. 32 = 16 • 2, and 16 is a square number.

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4.3.1: Defining Complex Numbers, i, and i2

Guided Practice: Example 3, continued

4. Simplify the radical by finding the square root of the square number. The simplified expression will be in the form whole number • i • radical.

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4.3.1: Defining Complex Numbers, i, and i2

✔

Guided Practice: Example 3, continued

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4.3.1: Defining Complex Numbers, i, and i2

Guided Practice

Example 4Simplify i 57.

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4.3.1: Defining Complex Numbers, i, and i2

Guided Practice: Example 4, continued

1. Find the remainder of the power of i when divided by 4. 14 • 4 = 56; therefore, 57 ÷ 4 = 14 remainder 1.

The remainder is 1.

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4.3.1: Defining Complex Numbers, i, and i2

Guided Practice: Example 4, continued

2. Use the remainder to simplify the power of i. i 57 = 1 • i 1 = i

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4.3.1: Defining Complex Numbers, i, and i2

✔

Guided Practice: Example 4, continued

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4.3.1: Defining Complex Numbers, i, and i2