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