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Understanding Signed Numbers in Mathematics

Learn about absolute value, addition, subtraction, multiplication, division, powers, roots, scientific and engineering notation in signed numbers.

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Understanding Signed Numbers in Mathematics

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  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 NUMBERSEXAMPLES 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

  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 –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 = –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 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 × 102Ans • The problem below uses engineering notation when multiplying two numbers • (3.08 × 103) × (6.2 × 106) = (3.1)(6.2) × (103)( 106) = 19.22 × 109Ans

  22. 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)

  23. 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)

  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 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

  28. PROBLEM ANSWER KEY • A • B • V

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