Introduction Until 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 i 2 = –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. 1 4.3.1: Defining Complex Numbers, i, and i 2
Transcript
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
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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.
