Section 5.2 The Integers; Order of Operations

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Section 5.2The Integers Order of Operations

• Objectives
• Define the integers.
• Graph integers on a number line.
• Use symbols lt and gt.
• Find the absolute value of an integer.
• Perform operations with integers.
• Use the order of operations agreement.

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Define the Integers

• The set consisting of the natural numbers, 0, and
  the negatives of the natural numbers is called
  the set of integers.
• Notice the term positive integers is another name
  for the natural numbers. The positive integers
  can be written in two ways
• Use a sign. For example, 4 is positive
  four.
• Do not write any sign. For example, 4 is also
  positive four.

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The Number Line

• The number line is a graph we use to visualize
  the set of integers, as well as sets of other
  numbers.
• Notice, zero is neither positive nor negative.

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The Number LineGraphing Integers on a Number Line

• Example Graph
• -3
• 4
• 0
• Solution Place a dot at the correct location for
  each integer.

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Use the Symbols lt and gt

• Looking at the graph, -4 and -1 are graphed
  below.
• Observe that -4 is to the left of -1 on the
  number line. This means that -4 is less than -1.
• Also observe that -1 is to the right of -4 on the
  number line. This means that -1 is greater then
  -4.

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Use the Symbols lt and gt

• The symbols lt and gt are called inequality
  symbols.
• These symbols always point to the lesser of the
  two real numbers when the inequality statement is
  true.

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Use the Symbols lt and gtUsing the Symbols lt and gt

• Example Insert either lt or gt in the shaded area
  between integers to make each statement true
• -4 3
• -1 -5
• -5 -2
• 0 -3
• Solution The solution is illustrated by the
  number line.

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Use the Symbols lt and gtExample Continued

• -4 lt 3 (negative 4 is less than 3) because -4 is
  to the left of 3 on the number line.
• -1 gt -5 (negative 1 is greater than negative 5)
  because -1 is to the right of -5 on the number
  line.
• -5 lt -2 ( negative 5 is less than -2) because -5
  is to the left of -2 on the number line.
• 0 gt -3 (zero is greater than negative 3) because
  0 is to the right of -3 on the number line.

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Use the Symbols lt and gt

• The symbols lt and gt may be combined with an equal
  sign, as shown in the following table

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Absolute Value

• The absolute value of an integer a, denoted by
  a, is the distance from 0 to a on the number
  line.
• Because absolute value describes a distance, it
  is never negative.
• Example Find the absolute value
• -3 b. 5 c. 0
• Solution

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Absolute ValueExample Continued

1. -3 3 because -3 is 3 units away from 0.
2. 5 5 because 5 is 5 units away from 0.
3. 0 0 because 0 is 0 units away from itself.

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Addition of Integers
Examples

• Rule
• If the integers have the same sign,
• Add their absolute values.
• The sign of the sum is the
• same sign of the two numbers.
• If the integers have different signs,
• subtract absolute values
• Subtract the smaller absolute
• value from the larger absolute
• value.
• The sign of the sum is the same as the sign of
  the number with the larger absolute value.

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Addition of IntegersExample

• A good analogy for adding integers is
  temperatures above and below zero on the
  thermometer. Think of a thermometer as a number
  line standing straight up. For example,

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Addition of IntegersAdditive Inverses

• Additive inverses have the same absolute value,
  but lie on opposite sides of zero on the number
  line.
• When we add additive inverses, the sum is equal
  to zero.
• Example
• 18 (-18) 0
• (-7) 7 0
• In general, the sum of any integer and its
  additive inverse is 0
• a (-a) 0

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Subtraction of Integers

• For all integers a and b,
• a b a (-b).
• In words, to subtract b from a, add the additive
  inverse of b to a. The result of subtraction is
  called the difference.
• Example Subtract
• a. 17 (-11) b. -18 (-5) c. -18 - 5

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Subtraction of IntegersExample Continued

• Solution

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Multiplication of Integers

• The result of multiplying two or more numbered is
  called the product of the numbers.
• Think of multiplication as repeated addition or
  subtraction that starts at 0. For example,

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Multiplication of Integers Rules

• Rule
• The product of two integers with different
• signs is found by multiplying their absolute
• values. The product is negative.
• The product of two integers with the same
• signs is found by multiplying their absolute
• values. The product is positive.
• The product of 0 and any integer is 0
• .
• If no number is 0, a product with an odd
• number of negative factors is found by
• multiplying absolute values. The product is
• negative.
• If no number is 0, a product with an even
• number of negative factors is found by
• multiplying absolute values. The product is
• positive.

Examples

• 7(-5) -35

• (-6)(-11) 66

• -17(0) 0

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Exponential Notation

• Because exponents indicate repeated
  multiplication, rules for multiplying can be used
  to evaluate exponential expressions.
• Example Evaluate a. (-6)2 b. -62 c.
  (-5)3 d. (-2)4
• Solution

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Division of Integers

• The result of dividing the integer a by the
  nonzero integer b is called the quotient of
  numbers.
• We write this quotient as or a / b.
• Example

means that 4(-3) -12.
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Division of IntegersRules
Examples

• Rule
• The quotient of two integers with different signs
  is found by dividing their absolute values. The
  quotient is negative.
• The quotient of two integers with the same sign
  is found by dividing their absolute values. The
  quotient is positive.
• Zero divided by any nonzero integer is zero.
• Division by 0 is undefined.

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Order of Operations

• Perform all operations within grouping symbols.
• Evaluate all exponential expressions.
• Do all the multiplications and divisions in the
  order in which they occur, working left to right.
• Finally, do all additions and subtractions in the
  order in which they occur, working left to right.
• We also use the acronym PEMDAS, parenthesis,
  exponents, multiplication and division, addition
  and subtraction, for the order of operations.

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Order of OperationsUsing PEMDAS

• Example Simplify 62 24 22 3 1.
• Solution There are no grouping symbols. Thus, we
  begin by evaluating exponential expressions.
• 62 24 22 3 1 36 24 4 3 1
• 36 6 3 1
• 36 18 1
• 18 1
• 19

Then we divide since occurs before
multiplication.
Next, multiply.
Finally, we add and subtract to obtain the final
answer.