Title: Section 5.2 The Integers; Order of Operations
1Section 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.
2Define 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.
3The 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.
4The Number LineGraphing Integers on a Number Line
- Example Graph
- -3
- 4
- 0
- Solution Place a dot at the correct location for
each integer.
5Use 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.
6Use 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.
7Use 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.
8Use 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.
9Use the Symbols lt and gt
- The symbols lt and gt may be combined with an equal
sign, as shown in the following table
10Absolute 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
11Absolute ValueExample Continued
- -3 3 because -3 is 3 units away from 0.
- 5 5 because 5 is 5 units away from 0.
- 0 0 because 0 is 0 units away from itself.
12Addition 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.
13Addition 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,
14Addition 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
15Subtraction 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
16Subtraction of IntegersExample Continued
17Multiplication 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,
18Multiplication 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
19Exponential 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
20Division 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.
21Division 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.
22Order 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.
23Order 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.