1. PRESENTATION 3 Signed Numbers

2. SIGNED NUMBERS
In algebra, plus and minus signs are used to indicate both operation and direction from a reference point or zero
Positive and negative numbers are called signed numbers
A positive number is indicated with either no sign or a plus sign (+)
A negative number is indicated with a minus sign (–)
Example: A Celsius temperature reading of 20 degrees above zero is written as +20ºC or 20ºC; a temperature reading 20 degrees below zero is written as –20ºC

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
Example: Add
All the numbers have the same sign so add and assign a positive sign to the answer
= or 16.7

5. ADDITION OF SIGNED NUMBERS
Example: Add (–6.053) + (–0.072) +
(–15.763) + (–0.009)
All the numbers have the same sign (–) so add and assign a negative sign to the answer
(–6.053) + (–0.072) + (–15.763) + (–0.009)
= –21.897

6. 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

7. ADDITION OF SIGNED NUMBERS
Example: Add +10 and (–4)
Different signs, so subtract and assign the sign of the larger absolute value
10 – 4 = 6
Prefix the positive sign to the difference
(+10) + (– 4) = +6 or 6

8. ADDITION OF SIGNED NUMBERS
Example: Add (–10) and +4
Different signs, so subtract and assign the sign of larger absolute value
10 – 4 = 6
Prefix the negative sign to the difference
(–10) + (+ 4) = –6

9. 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

10. ADDITION OF SIGNED NUMBERS
Example: Add (–12) (–5) + 20
Add all the positive numbers and all the negative numbers
30 + (–17)
Add the sums using the procedure for adding signed numbers
30 + (–17) = +13 or 13

11. 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

12. SUBTRACTION OF SIGNED NUMBERS
Example: Subtract 8 from 5
Change the sign of the subtrahend to the opposite sign
8 to –8
Add the signed numbers
5 + (–8) = –3

13. SUBTRACTION OF SIGNED NUMBERS
Example: Subtract –10 from 4
Change the sign of the subtrahend to the opposite sign
–10 to 10
Add the signed numbers
4 + (10) = 14

14. 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

15. MULTIPLICATION OF SIGNED NUMBERS
Example: Multiply 3(–5)
Multiply the absolute values
Since there is an odd number of negative signs (1), the product is negative
3(–5) = –15

16. MULTIPLICATION OF SIGNED NUMBERS
Example: Multiply (–3)(–1)(–2)(– 3)(–2)(–1)
Multiply the absolute values
Since there is an even number of negative signs (6), the product is positive
(–3)(–1)(–2)(–3)(–2)(–1) = +36 or 36

17. 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

18. DIVISION OF SIGNED NUMBERS
Example: Divide –20 ÷ (–4)
Divide the absolute values
Since there is an even number of negative signs (2), the quotient is positive
–20 ÷ (–4) = +5 or 5

19. DIVISION OF SIGNED NUMBERS
Example: Divide 24 ÷ (–8)
Divide the absolute values
Since there is an odd number of negative signs (1), the quotient is negative
24 ÷ (–8) = –3

20. 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

21. POWERS OF SIGNED NUMBERS
Example: Evaluate 24
Since 2 is positive, the answer is positive
24 = (2)(2)(2)(2) = +16 or 16

22. POWERS OF SIGNED NUMBERS
Example: (–4)3
Since a negative number is raised to an odd power, the answer is negative
(–4)3 = (–4)(–4)(–4) = –64

23. NEGATIVE EXPONENTS
Two numbers whose product is 1 are multiplicative inverses or reciprocals of each other
For example:
A number with a negative exponent is equal to the reciprocal of the number with a positive exponent:

24. POWERS OF SIGNED NUMBERS
Determining values with negative exponents
Invert the number (write its reciprocal)
Change the negative exponent to a positive exponent

25. POWERS OF SIGNED NUMBERS
Example: (–5)–2
Write the reciprocal of (–5)–2 and change the negative exponent –2 to a positive exponent +2
Simplify

26. 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

27. ROOTS OF SIGNED NUMBERS
The expression is a radical
The 3 is the index and 64 is the radicand
Use the following chart to determine the sign of a root based on the index and radicand
Index
Radicand
Root
Even
Positive (+)
Negative (–)
No Solution
Odd

28. ROOTS OF SIGNED NUMBERS
Example: Determine the indicated roots for the following problems:

29. COMBINED OPERATIONS
The same order of operations applies to terms with exponents as in arithmetic
Parentheses
Powers and roots
Multiply and divide from left to right
Add and subtract from left to right

30. COMBINED OPERATIONS Example: Evaluate 50 + (–2)[6 + (–2)3(4)]
50 + (–2)[6 + (–2)3(4)] Powers or exponents first
= 50 + (–2)[6 + (–8)(4)] Multiplication in []
= 50 + (–2)[6 + –32] Evaluate the brackets
= 50 + (–2)(–26) Multiply
= 50 + (52) Add
50 + (–2)[6 + (–2)3(4)] = 102

31. 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

32. SCIENTIFIC NOTATION Examples:
In scientific notation, 146,000 is written as × 105
In scientific notation, is written as 3 × 10–5
The number –3.8 × is written as a whole number as –

33. SCIENTIFIC AND ENGINEERING NOTATION
Example: Multiply (5.7 × 103)(3.2 × 109)
Multiply the decimals
5.7 × 3.2 = 18.24
Multiply the powers of 10s using the rules for exponents
(103)(109) = 1012
Combine both parts
18.24 × 1012