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CHAPTER XVI

IMAGINARY NUMBERS

130. As any even power of a number is positive, any expression involving an indicated even root of a negative number is called an imaginary expression.

To distinguish them from imaginary numbers, all other numbers are called real numbers.

The algebraic sum of a real number and an imaginary number is called a complex number.

In this chapter, only indicated square roots of negative numbers are considered, as any indicated even root of a negative number may be made to assume a form involving only a square root of a negative number.

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Since Vab Va× √b, when a and b are positive, it is assumed that this is also true when either a or b is negative. That is, - 9

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So that any imaginary square root may be factored into a real number and the imaginary factor √- 1.

1 is called the imaginary unit and for brevity is sometimes expressed by i.

Since the square of the square root of a number is the number itself, (√− 1)2 = −1. This is the important fact underlying operations with imaginary numbers.

The successive powers of V-1 are,

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(√− 1)3 = (√−1)2 (√ = 1) = (− 1)(√ − 1) =

1

(V-1) 4 = (√− 1)2(√= 1)2 = (− 1)(− 1) = + 1 (√1)(√1) = (+ 1)(√ − 1) = +√−1

(√ - 1)5

and so on.

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2

Remembering that (1)2 = 1 and (- 1)1 = + 1, the value of any power of - 1 can easily be found.

[graphic]

CARL FRIEDRICH GAUSS
(After a Lithograph From a Painting by Jensen)

Born at Braunschweig, 1777. Died at Goettingen, 1855.

Gauss was a physicist and astronomer and the greatest mathematician of modern times. He is responsible for the acceptance and understanding of imaginary quantities and complex numbers.

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Thus, (1)17 = (√− 1) 16 (√ − 1)' = (+ 1)(√− 1) + √−1.

Since every fourth power of √ = 1 yields a factor + 1.

Similarly (√ 1)15 = (√ — 1)12(√1)3 =

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(+ 1)(√ = 1)2 (√ − 1) = (+ 1)( − 1)(√− 1)

= - √ - 1. Hence any exponent of √- 1 may be reduced by any multiple of 4. As, (√1)17 = (√ − 1)1 and (√ − 1)15 = (√ − 1)3.

OPERATIONS WITH IMAGINARIES

131. 1. Add 3 + 61 and 5-√9.

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3 +61 +593 + 6√ − 1 + 5 −√+9√−1 = 3 +61 +53√1 = 8+ 3√1

2. Multiply √ 3 + 2√− 2 by 3√ — 3 — 5√√ — 2.

Factoring out the imaginary unit,

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(√3√ = 1 + 2√2√−1)(3√3√-1-5√√√√-1) = [(√3 + 2√2)√=1][(3√3 - 5√2)√= 1] =

- 1(√3 +2√2)(3√3-5√2) =

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-1(9+66-5√6-20)=1(√6 – 11) = 11 −√6

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4. Divide 6- √-8 by √5+ √ − 6.

√6-√-8 √6-√8-1

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EXERCISE 92

Simplify:

22

1. (√− 1). 2. (√− 1)13. 3. (√−1)12. 4. (√−1)22 5. (1)19.

Add:

6. √ 36 + 3681 - √144.

7. 427 + √12 - 2√3.

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8. √2+3√3√32 √12. 9. √x2 + √+ 16x2 - 3√ - x2.

a1

10. √-at-√ = 4a2b2 + √-ba.

Multiply.

11. √5 by √ — 7.

13. √16 by - √-8.

14. 3 + √− 2 by 2— √− 3.

12. √ 25 by √9.

15. a√x + b√y by a√x.— b√ — y.

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17. x by 1.

18.√x2 by √

y2.

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19. √5-6 by V-6 - √ - 8. 20. 837 by 3 - 8-7.

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4. Multiply 35 - 4√6 by 6√5 +8√6.

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11. Find the value of (= 1 + √ = 3)2 + (-1-√3).

12. Find the value of

2

2

(1 + √ − 3)(1 + √ − 3)(1 + √ − 3).

13. Find the value of

7+ 3-5 7-3√-5

+

7-3-57 +3√ - 5

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