Rounding Whole Numbers

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Chapter 2
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Definitions
The set of integers contain , -3
,-2, -1 ,0, 1, 2, 3...
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Using a Number Line
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Ordering Numbers

• a gtb if a lies to the right of b on the
  number line.
• a lt b if a lies to the left of b on the
  number line.

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Definitions
A POSITIVE number is greater than zero. A
NEGATIVE number is less than zero. Zero is
neither positive nor negative .
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Ordering IntegersPut a lt or gt sign
between each number.

• -4 6
• -25 -46
• 0 -10
• 24 -2
• -1001 -999

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Definition
Two numbers that are the same distance away from
zero on a number line, but on opposite sides of
zero, are opposites or negatives.
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Definition
If a is any number, the opposite of a is denoted
by -a (read the opposite of a)
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ExampleFind the opposite of each number

• -4
• 3
• 9
• -5
• a
• x

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The Double Negative Rule
If a is any number,
-(-a) a
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ExampleSimplify each of the following.

• -(-8)
• -(-4)
• -(-7)
• -(-s)
• -(-u)
• -(-(-3))

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Definition
The absolute value of any number is the distance
between the number and zero on a number line.
a (read the absolute value of a)
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ExampleFind each of the following

• 4
• 3
• -9
• -5
• -17
• --23

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Bar Graphs
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Adding Integers
We can use a number line to determine how to add
integers.
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Adding Integers

• If you are adding two numbers with the same sign,
  then both are positive, or both are negative.
• For example 4 3 or -4 (-3)

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Adding Integers

• If you are adding two numbers with the different
  signs, then one is positive, and one is
  negative.
• For example 4 (-3) or -4 3

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Adding Integerswith the same sign.

• To add two integers with the same sign, add their
  absolute values and attach their common sign to
  the sum. If both integers are positive, their sum
  is positive. If both integers are negative their
  sum is negative.

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Examples

• -4 (-5)
• -2 (-13)
• -7 (-9)
• -6 (-2) (-8)

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Adding Integerswith different signs.

• To add two integers with different signs,
  subtract their absolute values, the smaller from
  the larger. Then attach to that result the sign
  of the integer with the larger absolute value.

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Examples

• -4 5
• 2 (-13)
• 7 (-9)
• -6 2

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More complicated examples.

• -2 6 (-5)
• (-9 12) 6 (-8)
• 9 (-3) 5 (-4)
• -5 9 15 (-24)

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Definition
If a is any number, the opposite of a is denoted
by -a (read the opposite of
a) Sometimes we call -a the additive inverse
of a because a (-a) 0.
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Application Examples
Use the following graphs, determine how much
profit the company made or lost for the entire
year.
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Bar Graphs
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Subtracting Integers
We can use a number line to determine how to
subtract integers.
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Rule for Subtracting
If a and b are any numbers, then
a - b a (-b) Subtraction is
the same as adding the opposite of the number to
be subtracted. So given a subtraction problem,
you can always change it to an addition problem.
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Examples

• -4 - (-5)
• -2 - (-13)
• -7 - (-9)
• -6 - (-2) - (-8)

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More complicated examples.

• -2 - 6 - (-5)
• (-9 12) - 6 - (-8)
• 9 - (-3) 5 - (-4)
• -5 - 9 15 - (-24)

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Finding the change from a to b
If a and b are any numbers, the change in
going from a to b is given by
b - a
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Example
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Example
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Multiplying Integers
Recall that multiplication indicates repeated
addition. 3(5) 5 5 5
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Multiplying Integers

• If you multiply two positive numbers, the answer
  is positive, multiply as usual.
• For example 4(3) 12

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Multiplying Integers

• If you are multiplying two numbers with the
  different signs the answer is negative.
• For example 3(-4) (-4) (-4) (-4) -12

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Multiplying Integerswith different signs.

• To multiply two numbers with different signs,
  multiply their absolute values. Then make the
  answer negative.
• Example 5(-8)

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Examples
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Consider the following problems
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Multiplying Two Negative Numbers

• If you multiply two negative numbers, multiply
  their absolute values, the answer is positive.
• For example -4(-3) 12

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Examples
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Multiplying Two Numbers

• To multiply any two numbers, multiply their
  absolute values.
• The product of two numbers with the same sign is
  positive.
• The product of two numbers with different signs
  is negative.

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More complicated examples.

• -2 (6) (-5)
• (-9) (12) 6 (-8)
• 9 (-3) 5 (-4)
• -5 ( 9) (15)(-24)

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Powers of Integers

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Opposites

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Important Example

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Examples
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Dividing Two Numbers

• To divide any two numbers, divide their absolute
  values.
• The quotient of two numbers with the same sign is
  positive.
• The quotient of two numbers with different signs
  is negative.

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Examples
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Examples
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Order of Operations AgreementPEMDAS

• Parentheses first (or any grouping symbol).
• Exponents
• Multiplication or
• Division
• Addition or
• Subtraction
• Similar operations are done from left to right.

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Simplify the following.
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Simplify the following.
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When working on a problem with a fraction bar, do
the operations in the numerator first, followed
by the operations in the denominator, then do the
division last.
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Simplify the following.
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Break into groups of two and work the following
problems on page 120.Problems 47 - 65 odd.
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Solving Equations

• Use the addition property of equality
• Use the subtraction property of equality
• Use the multiplication property of equality
• Use the division property of equality

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Let a,b, and c be any numbers.

• Addition Property of Equality if a b,
  then a c b c
• Subtraction Property of Equality if a b,
  then a - c b - c

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Let a,b, and c be any numbers, where c is not
equal to 0.

• Multiplication Property of Equality if
  a b, then a(c) b(c)
• Division Property of Equality if
  a b, then a / c b / c

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Solving Equations

• To solve an equation, do the following
• Simplify each side of the equation.
• Isolate the variable by addition or subtraction.
• Solve the equation by multiplication or division.
• Check the solution.

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Example
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Example
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Examples
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More Examples