1. 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 NUMBERS EXAMPLES
= Ans
4 1/ / /5
= 19 7/30 Ans
(–7) + (–10) + (–5)
= –22 Ans
(–3 1/3) + (–5 2/9) + (–4 1/2)
= –13 1/18 Ans

5. 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
– = 4 Ans
– = –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) = = 21 Ans
–17.3 +(– 9.5) = –17.3 –9.5 = –26.8 Ans
–76.98 – (–89.74) = – = 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
= – 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
= –6.05 Ans
(–4 2/3)  (–2 1/2)
= 1 13/15 Ans
= 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)
= 16 Ans
(–3)3 = (–3)(–3)(–3)
= –27 Ans
–24 = – (2)(2)(2)(2)
= –16 Ans
(–2)4
= (–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

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:

18. COMBINED OPERATIONS
The same order of operations applies to terms with exponents as in arithmetic
Find the value of (–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, 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) = × 109 Ans

22. PRACTICE PROBLEMS Perform the indicated operations: 7 + (–18)
(–25) + 98
(–2 1/4) + (–3 2/5)
(–5.76)
– (–8.97)
–7 2/ /5 + (–3 1/2) + 2 ¼
98 – (–67)

23. PRACTICE PROBLEMS (Cont)
–79.54 – 65.39
–98.6 – (–45.3)
6 3/4 – (–7 1/3)
(4 5/ /3) – (–1 1/2 – 3 2/3)
(–98.7 – (–54.3)) – (3.59 – 4.76)
8.4(–6.9)
(–4)(–97)
(1 1/3)(–2 1/2)
(–3)(–5.4)(3.2)(–5.5)
(–3 1/2)(2 1/3)(–2 1/6)

24. PRACTICE PROBLEMS (Cont)
(7.2)(–4.6)(–8.1)
– 7.25  –5
16.4  –0.4
(–4 3/5)  (–1/2)
0  (–4 3/5)
(–5) 3
(–5) –3
(.56) 2
(–1/2) –2
(–1/2) 2

25. PRACTICE PROBLEMS (Cont)
4(–3)  (–2)(–5)
4 + (–6)(–3)  (–2)
(–4)(2)(–6) + (–8 + 2)  2
7 + 6(–2 + 7) + (–7) + (–5)(8 – 2)

26. Practice Problems

27. PROBLEM ANSWER KEY 1. –11 2. 73 3. –5 13/20 4. 1.49 5. 2.05 6. –2 7/60
1. – –5 13/20
–2 7/60
– –53.3
/ / –43.23
– –3 1/3
– /
– /5
– –1/125 or –0.008
/4 or 0.25
– No solution
–2/ – –5

28. PROBLEM ANSWER KEY A B V