# 3.S: Integers (Summary)

### 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. |

#### 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 \(\frac{a}{c} = \frac{b}{c}\). |