Key Terms
|
absolute value
|
A number's distance from 0 on the number line.
|
|
integers
|
Counting numbers, their opposites, and zero$$... –3, –2, –1, 0, 1, 2, 3 ...$$
|
|
negative number
|
A number less than zero.
|
|
opposites
|
The number that is the same distance from zero on the number line, but on the opposite side of zero.
|
Key Concepts
3.1 - Introduction to Integers
-
Opposite Notation
-
−a means the opposite of the number a
-
The notation −a is read the opposite of a.
-
Absolute Value Notation
-
The absolute value of a number n is written as |n|.
-
|n| ≥ 0 for all numbers.
3.2 - Add Integers
-
Addition of Positive and Negative Integers
|
5 + 3
|
−5 + (−3)
|
|
both positive, sum positive
|
both negative, sum negative
|
|
When the signs are the same, the counters would be all the same color, so add them.
|
|
−5 + 3
|
5 + (−3)
|
|
different signs, more negatives
|
different signs, more positives
|
|
sum negative
|
sum positive
|
|
When the signs are different, some counters would make neutral pairs; subtract to see how many are left.
|
Table 3.110
|
5 – 3 = 2
|
–5 – (–3) = –2
|
|
2 positives
|
2 negatives
|
|
When there would be enough counters of the color to take away, subtract.
|
|
–5 – 3 =
|
5 – (–3) = 8
|
|
5 negatives, want to subtract 3 positives
|
5 positives, want to subtract 3 negatives
|
|
When there would not be enough of the counters to take away, add neutral pairs.
|
-
Subtraction Property
-
a − b = a + (−b)
-
a − (−b) = a + b
-
Solve Application Problems
-
Identify what you are asked to find.
-
Write a phrase that gives the information to find it.
-
Translate the phrase to an expression.
-
Simplify the expression.
-
Answer the question with a complete sentence.
3.4 - Multiply and Divide Integers
-
Multiplication of Signed Numbers
-
To determine the sign of the product of two signed numbers:
|
Same Signs
|
Product
|
|
Two positives
|
Positive
|
|
Two negatives
|
Positive
|
|
Different Signs
|
Product
|
|
Positive • negative
|
Negative
|
|
Negative • positive
|
Negative
|
-
Division of Signed Numbers
-
To determine the sign of the quotient of two signed numbers:
|
Same Signs
|
Quotient
|
|
Two positives
|
Positive
|
|
Two negatives
|
Positive
|
|
Different Signs
|
Quotient
|
|
Positive & negative
|
Negative
|
|
Negative & positive
|
Negative
|
-
Multiplication by −1
-
Multiplying a number by −1 gives its opposite: −1a = − a
-
Division by −1
-
Dividing a number by −1 gives its opposite: a ÷ (−1) = −a
3.5 - Solve Equations Using Integers; The Division Property of Equality
-
How to determine whether a number is a solution to an equation.
-
Substitute the number for the variable in the equation.
-
Simplify the expressions on both sides of the equation.
-
Determine whether the resulting equation is true.
-
If it is true, the number is a solution.
-
If it is not true, the number is not a solution.
-
Properties of Equalities
|
Subtraction Property of Equality
|
Addition Property of Equality
|
Division Property of Equality
|
|
For any numbers a, b, c, if a = b then a − c = b − c.
|
For any numbers a, b, c, if a = b then a + c = b + c.
|
For any numbers a, b, c, and c ≠ 0 If a = b, then \(\dfrac{a}{c} = \dfrac{b}{c}\).
|
Contributors and Attributions