Unit 6

SIGNED NUMBERS

2

ABSOLUTE VALUE The absolute value of a number is the distance

from the number 0. The symbol for absolute value is The number is placed between the bars |16| The absolute value of –16 and 16 are the same

because each is 16 units from 0 Written with the absolute value symbol:

16 = –16 = 16

3

ADDITION OF SIGNED NUMBERS

Procedure for adding two or more numbers with the same signs Add the absolute values of the

numbers If all the numbers are positive, the

sum is positive If all the numbers are negative, prefix

a negative sign to the sum

4

ADDITION OF SIGNED NUMBERSEXAMPLES

9 + 5.8 + 12

4 1/2 + 6 1/3 + 8 2/5

(–7) + (–10) + (–5)

(–3 1/3) + (–5 2/9) + (–4 1/2)

= 26.8 Ans

= 19 7/30 Ans

= –22 Ans

= –13 1/18 Ans

5

Procedure for adding a positive and a negative number:

• Subtract the smaller absolute value from the larger absolute value

• The answer has the sign of the number having the larger absolute value

ADDITION OF SIGNED NUMBERS

–10 + 14 = 4 Ans

–64.3 + 42.6 = –21.7 Ans

6

ADDITION OF SIGNED NUMBERS

Procedure for adding combinations of two or more positive and negative numbers: Add all the positive numbers Add all the negative numbers Add their sums, following the

procedure for adding signed numbers

7

SUBTRACTION OF SIGNED NUMBERS

Procedure for subtracting signed numbers: Change the sign of the number

subtracted (subtrahend) to the opposite sign

Follow the procedure for addition of signed numbers

8

EXAMPLES

6 – (–15) = 6 + 15 = 21 Ans

–17.3 +(– 9.5) = –17.3 –9.5 = –26.8 Ans

–76.98 – (–89.74) = –76.98 + 89.74 = 12.76 Ans

–1 2/3 +(– 4 5/6) = –1 2/3 –4 5/6 = –6 1/2 Ans

9

MULTIPLICATION OF SIGNED NUMBERS

Procedure for multiplying two or more signed numbers Multiply the absolute values of the

numbers If all numbers are positive, the product is

positive Count the number of negative signs

An odd number of negative signs, gives a negative product

An even number of negative signs gives a positive product

10

EXAMPLES

Multiply each of the following:

(–5)(–3)

(17)(–4)(0.5)

(–3)(–2)(–1)(–3.2)

(2.5)(5.7)(6.24)(1.376)(–1.93)

= 15 Ans

= –34 Ans

= 19.2 Ans

= –236.1430656 Ans

11

DIVISION OF SIGNED NUMBERS

Procedure for dividing signed numbers Divide the absolute values of the

numbers Determine the sign of the quotient

If both numbers have the same sign (both negative or both positive), the quotient is positive

If the two numbers have unlike signs (one positive and one negative), the quotient is negative

12

DIVISION OF SIGNED NUMBERS

Divide each of the following:

24.2 –4

(–4 2/3) (–2 1/2)

= –6.05 Ans

= 1 13/15 Ans

8

30 = 0 Ans

13

POWERS OF SIGNED NUMBERS

Determining values with positive exponents Apply the procedure for multiplying signed

numbers to raising signed numbers to powers

A positive number raised to any power is positive A negative number raised to an even power is

positive A negative number raised to an odd power is

negative

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POWERS OF SIGNED NUMBERS

Evaluate:

42 = (4)(4)

(–3)3 = (–3)(–3)(–3)

–24 = – (2)(2)(2)(2)

(–2)4

= 16 Ans

= –27 Ans

= –16 Ans

= (–2)(–2)(–2)(–2) = 16 Ans

15

POWERS OF SIGNED NUMBERS

Determining values with negative exponents Invert the number (write its

reciprocal) Change the negative exponent to a

positive exponent Ans.25or41

21

22

2

Ans0.0625or161

41

42

2

16

ROOTS OF SIGNED NUMBERS

A root of a number is a quantity that is taken two or more times as an equal factor of the number Roots are expressed with radical signs An index is the number of times a root

is to be taken as an equal factor The square root of a negative number

has no solution in the real number system

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ROOTS OF SIGNED NUMBERS

Determine the indicated roots for the following problems:

36

3 64

Ans6(6)(6)

Ans44)4)(4)((3

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COMBINED OPERATIONS

The same order of operations applies to terms with exponents as in arithmetic

Find the value of 36 + (–3)[6 + (2)3(5)]: 36 + (–3)[6 + (2)3(5)] Powers or exponents

first = 36 + (–3)[6 + (8)(5)] Multiplication within

the brackets = 36 + (–3)[6 + 40] Evaluate the brackets = 36 + (–3)(46) Multiply = 36 + (–138) Add

= –102 Ans

19

SCIENTIFIC NOTATION

In scientific notation, a number is written as a whole number or decimal between 1 and 10 multiplied by 10 with a suitable exponent In scientific notation, 1,750,000 is

written as 1.75 × 106

In scientific notation, 0.00065 is written as 6.5 × 10–4

9.8 × 103 in scientific notation is written as 9,800 as a whole number

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ENGINEERING NOTATION

Engineering notation is similar to scientific notation, but the exponents of 10 are written in multiples of three 32,500 is written as 32.5 × 103 in

engineering notation 832,000,000 is written as 832 × 106 in

engineering notation -22,100,000 is written as -22 × 106 in

engineering notation

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SCIENTIFIC AND ENGINEERING NOTATION

The problem below uses scientific notation when multiplying two numbers (1.2 × 103)(5 × 10–1) = (1.2)(5) × (103)(10–

1) = 6 × 102 Ans The problem below uses engineering

notation when multiplying two numbers (3.08 × 103) × (6.2 × 106) = (3.1)(6.2) ×

(103)( 106) = 19.22 × 109 Ans

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PRACTICE PROBLEMS Perform the indicated operations:

1. 7 + (–18)2. (–25) + 983. (–2 1/4) + (–3 2/5)4. 7.25 + (–5.76)5. –4.38 + (–8.97) + 15.46. –7 2/3 + 6 4/5 + (–3 1/2) + 2 ¼7. 98 – (–67)

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PRACTICE PROBLEMS (Cont)8. –79.54 – 65.399. –98.6 – (–45.3)10. 6 3/4 – (–7 1/3)11. (4 5/6 + 3 1/3) – (–1 1/2 – 3 2/3)12. (–98.7 – (–54.3)) – (3.59 – 4.76)13. 8.4(–6.9)14. (–4)(–97)15. (1 1/3)(–2 1/2)16. (–3)(–5.4)(3.2)(–5.5)17. (–3 1/2)(2 1/3)(–2 1/6)

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PRACTICE PROBLEMS (Cont)18. (7.2)(–4.6)(–8.1)19. – 7.25 –520. 16.4 –0.421. (–4 3/5) (–1/2)22. 0 (–4 3/5)23. (–5) 3

24. (–5) –3

25. (.56) 2

26. (–1/2) –2

27. (–1/2) 2

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PRACTICE PROBLEMS (Cont)

32. 4(–3) (–2)(–5)33. 4 + (–6)(–3) (–2)34. (–4)(2)(–6) + (–8 + 2) 235. 7 + 6(–2 + 7) + (–7) + (–5)(8 – 2)

3

3

3

27

8.31

8.3036.2927.28

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Practice Problems

195

134

1023.1104.2

1075.3103.2.37

xx

xx

7

19

1023.1

1075.3.36

x

x

239

135

1043.11096.1

1045.5103.8.38

xx

xx

27

1. –11 2. 73 3. –5 13/20 4. 1.49 5. 2.05 6. –2 7/60 7. 165 8. –144.93 9. –53.3 10. 14 1/12 11. 13 1/3 12. –43.23 13. –57.96 14. 388 15. –3 1/316. –285.12 17. 17 25/36 18. 268.272 19. 1.45 20. –41 21. 9 1/522. 0 23. –125 24. –1/125 or –0.00825. 0.3136 26. 4 27. 1/4 or 0.2528. –3 29. No solution 30. 231. –2/3 32. –30 33. –534. 45 35. 0

PROBLEM ANSWER KEY

28

PROBLEM ANSWER KEY

36. A

37. B

38. V

121004.3 x

29201092.23orx

231061.1

x