3.S: Integers (Summary)
- Page ID
- 6041
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| 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. | |
3.3 - Subtract Integers
- Subtraction of Integers
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 \(\dfrac{a}{c} = \dfrac{b}{c}\). |