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 (+) 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

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 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

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

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

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

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

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

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

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

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

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

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

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

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

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 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

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

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 Example: Evaluate 24 Since 2 is positive, the answer is positive 24 = (2)(2)(2)(2) = +16 or 16

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

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:

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

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

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 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

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

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

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

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

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

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