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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

PRACTICE PROBLEMS Perform the indicated operations: 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)

PRACTICE PROBLEMS (Cont) –79.54 – 65.39 –98.6 – (–45.3) 6 3/4 – (–7 1/3) (4 5/6 + 3 1/3) – (–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)

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

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)

Practice Problems

PROBLEM ANSWER KEY 1. –11 2. 73 3. –5 13/20 4. 1.49 5. 2.05 6. –2 7/60 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 1/3 16. –285.12 17. 17 25/36 18. 268.272 19. 1.45 20. –41 21. 9 1/5 22. 0 23. –125 24. –1/125 or –0.008 25. 0.3136 26. 4 27. 1/4 or 0.25 28. –3 29. No solution 30. 2 31. –2/3 32. –30 33. –5 34. 45 35. 0

PROBLEM ANSWER KEY A B V