PRESENTATION 3 Signed Numbers SIGNED NUMBERS In algebra

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 o 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) o 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 o Multiply the absolute values of the numbers § If all numbers are positive, the product is positive o 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 o Divide the absolute values of the numbers o 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 o 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 4 2 • Since 2 is positive, the answer is positive 24 = (2)(2) = +16 or 16

POWERS OF SIGNED NUMBERS • Example: 3 (– 4) • Since a negative number is raised to an odd power, the answer is negative (– 4)3 = (– 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: • • – 2 (– 5) 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 (+) Even Negative (–) No Solution Odd Positive (+) Odd Negative (–)

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 + 3 (– 2) (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 × • • • 3 10 )(3. 2 × 9 10 ) Multiply the decimals 5. 7 × 3. 2 = 18. 24 Multiply the powers of 10 s using the rules for exponents (103)(109) = 1012 Combine both parts 18. 24 × 1012