unit 6 signed numbers. 2 absolute value the absolute value of a number is the distance from the...
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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
14
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
17
ROOTS OF SIGNED NUMBERS
Determine the indicated roots for the following problems:
36
3 64
Ans6(6)(6)
Ans44)4)(4)((3
18
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
20
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
21
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
22
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)
23
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)
24
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
25
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
26
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