Transcript absolute value

Unit 6
SIGNED NUMBERS
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
2
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
3
ADDITION OF SIGNED NUMBERS
EXAMPLES
9 + 5.8 + 12 = 26.8 Ans
4 1/2 + 6 1/3 + 8 2/5 = 19 7/30 Ans
(–7) + (–10) + (–5) = –22 Ans
(–3 1/3) + (–5 2/9) + (–4 1/2) = –13 1/18 Ans
4
ADDITION OF SIGNED NUMBERS
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
–10 + 14 = 4 Ans
–64.3 + 42.6 = –21.7 Ans
5
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
6
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
7
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
8
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
9
EXAMPLES

Multiply each of the following:
(–5)(–3) = 15 Ans
(17)(–4)(0.5) = –34 Ans
(–3)(–2)(–1)(–3.2) = 19.2 Ans
(2.5)(5.7)(6.24)(1.376)(–1.93) = –236.1430656 Ans
10
DIVISION OF SIGNED NUMBERS

Procedure for dividing signed numbers
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
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
11
DIVISION OF SIGNED NUMBERS

Divide each of the following:
24.2  –4 = –6.05 Ans
(–4 2/3)  (–2 1/2) = 1 13/15 Ans
 3
0     = 0 Ans
 8
12
POWERS OF SIGNED NUMBERS

Determining values with positive
exponents

Apply the procedure for multiplying signed
numbers to raising signed numbers to
powers
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

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
13
negative
POWERS OF SIGNED NUMBERS

Evaluate:
42 = (4)(4) = 16 Ans
(–3)3 = (–3)(–3)(–3) = –27 Ans
–24 = – (2)(2)(2)(2) = –16 Ans
(–2)4 = (–2)(–2)(–2)(–2) = 16 Ans
14
POWERS OF SIGNED NUMBERS

Determining values with negative
exponents


Invert the number (write its reciprocal)
Change the negative exponent to a positive
exponent
2
2
 4
1
1
 2  or .25 Ans
2
4
2
1
1
 2 
or 0.0625 Ans
4
16
15
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
16
ROOTS OF SIGNED NUMBERS

Determine the indicated roots for the
following problems:
36  (6)(6)  6 Ans
3
 64  3 (4)(4)(4)   4 Ans
17
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)]:
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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
18
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
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

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
19
ENGINEERING NOTATION

Engineering notation is similar to
scientific notation, but the exponents of
10 are written in multiples of three
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

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
20
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
21
PRACTICE PROBLEMS

Perform the indicated operations:
1.
2.
3.
4.
5.
6.
7.
7 + (–18)
(–25) + 98
(–2 1/4) + (–3 2/5)
7.25 + (–5.76)
–4.38 + (–8.97) + 15.4
–7 2/3 + 6 4/5 + (–3 1/2) + 2 ¼
98 – (–67)
22
PRACTICE PROBLEMS (Cont)
8. –79.54 – 65.39
9. –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)
23
PRACTICE PROBLEMS (Cont)
18. (7.2)(–4.6)(–8.1)
19. – 7.25  –5
20. 16.4  –0.4
21. (–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
24
PRACTICE PROBLEMS (Cont)
28.
29.
30.
31.
3
3
3
 27
 36
8
8
27
32. 4(–3)  (–2)(–5)
33. 4 + (–6)(–3)  (–2)
34. (–4)(2)(–6) + (–8 + 2)  2
35. 7 + 6(–2 + 7) + (–7) + (–5)(8 – 2)
25
Practice Problems
19 

 3.75 x10 

36. 
7

 1.23 x10 


4 
13 

 2.3 x10  3.75 x10 


37. 
 5 
19 

2
.
4
x
10
1
.
23
x
10






 5 
13 

 8.3 x10  5.45 x10 


38. 
9 
23 

1
.
96
x
10
1
.
43
x
10






26
PROBLEM ANSWER KEY
1.
4.
7.
10.
13.
16.
19.
22.
25.
28.
31.
34.
–11
1.49
165
14 1/12
–57.96
–285.12
1.45
0
0.3136
–3
–2/3
45
2.
5.
8.
11.
14.
17.
20.
23.
26.
29.
32.
35.
73
2.05
–144.93
13 1/3
388
17 25/36
–41
–125
4
No solution
–30
0
3.
6.
9.
12.
15.
18.
21.
24.
27.
30.
33.
–5 13/20
–2 7/60
–53.3
–43.23
–3 1/3
268.272
9 1/5
–1/125 or –0.008
1/4 or 0.25
2
–5
27
PROBLEM ANSWER KEY
36.
37.
38.
12
3A.04 x10
3
2B.92 x10 or 2920
V
1.61x10
23
28