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Section 5.2 The Integers; Order of Operations

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Section 5.2 The Integers; Order of Operations Objectives Define the integers. Graph integers on a number line. Use symbols . Find the absolute value of an integer. – PowerPoint PPT presentation

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Title: Section 5.2 The Integers; Order of Operations


1
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.

2
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.

3
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.

4
The Number LineGraphing Integers on a Number Line
  • Example Graph
  • -3
  • 4
  • 0
  • Solution Place a dot at the correct location for
    each integer.

5
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.

6
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.

7
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.

8
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.

9
Use the Symbols lt and gt
  • The symbols lt and gt may be combined with an equal
    sign, as shown in the following table

10
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

11
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.

12
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.

13
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,

14
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

15
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

16
Subtraction of IntegersExample Continued
  • Solution

17
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,

18
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

19
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

20
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.
21
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.

22
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.

23
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.
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