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PRESENTATION 3 Signed Numbers

PRESENTATION 3 Signed Numbers. 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 (+)

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PRESENTATION 3 Signed Numbers

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  1. PRESENTATION 3Signed 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 9 + 5.6 + 2.1 • All the numbers have the same sign so add and assign a positive sign to the answer 9 + 5.6 + 2.1 = +16.7 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) + 7 + 3 + (–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 –10from 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

  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 1.46 × 105 • In scientific notation, 0.00003 is written as 3 × 10–5 • The number –3.8 × 10-4 is written as a whole number as –0.00038

  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

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