Table of Contents

## What are Integers?

Integers are a fundamental concept in mathematics and represent a set of whole numbers, including their negative counterparts and zero. The set of integers is denoted by the symbol ℤ. Integers can be positive, negative, or zero, and they do not include fractional or decimal parts.

## Definition of Integers

Integers, denoted by the symbol ℤ, encompass the set of whole numbers, including positive numbers, negative numbers, and zero. The set is expressed as ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}, where the ellipsis indicates an infinite extension in both positive and negative directions.

Here are some key points about integers:

**Positive Integers:**These are whole numbers greater than zero. Examples include 1, 2, 3, 100, and so on.**Negative Integers:**These are whole numbers less than zero. Examples include -1, -2, -3, -100, and so on.**Zero:**Zero is also considered an integer and is neither positive nor negative. It serves as the origin on the number line.**Notation:**The set of integers is often denoted as ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}. The ellipsis (…) indicates that the set extends infinitely in both directions.**Addition and Subtraction:**Integers can be added and subtracted from one another. The result may be positive, negative, or zero, depending on the values involved.**Number Line:**Integers can be represented on a number line, with positive integers to the right of zero and negative integers to the left.**Multiplication and Division:**When multiplying or dividing integers, the result follows specific rules. The product of two positive integers is positive, the product of two negative integers is positive, and the product of a positive and a negative integer is negative.**Absolute Value:**The absolute value of an integer is its distance from zero on the number line, disregarding its sign. For example, the absolute value of -5 is 5.

### Properties of Integers:

**Closure under Addition and Multiplication:**- The sum or product of any two integers is also an integer.

**Identity Elements:**- Zero serves as the additive identity, and one is the multiplicative identity for integers.

**Associative and Commutative Properties:**- Integers follow the associative and commutative properties under addition and multiplication.

### Rules Governing Integers

**Addition and Subtraction:**- Adding integers of the same sign yields a sum with the same sign.
- Subtracting an integer is equivalent to adding its additive inverse.

**Multiplication and Division:**- The product of two integers with the same sign is positive.
- The product of two integers with different signs is negative.
- Division involves similar rules, and the quotient inherits the sign of the dividend.

**Absolute Value:**- The absolute value of an integer is its distance from zero on the number line, irrespective of its sign.

## Rules of Integers in Subtraction

Subtraction Example | Result |
---|---|

Positive – Positive | 12 – 3 = 9 |

Negative – Positive | (-15) – 5 = -20 |

Positive – Negative | 8 – (-2) = 10 |

Negative – Negative | (-7) – (-4) = -3 |

Subtracting Zero | 10 – 0 = 10 |

Subtracting from Zero | 0 – 6 = -6 |

Subtracting from Itself | 5 – 5 = 0 |

Subtracting a Negative | 7 – (-3) = 10 |

Subtracting Powers of 10 | 300 – 10^{2} = 200 |

## Rules of Integers in Multiplication

Multiplication Example | Result |
---|---|

Positive × Positive | 3 × 5 = 15 |

Negative × Negative | (-4) × (-2) = 8 |

Positive × Negative | 6 × (-3) = -18 |

Multiplicative Identity | (-7) × 1 = -7 |

Multiplicative Zero | 9 × 0 = 0 |

Associative Property | (2 × 3) × 4 = 2 × (3 × 4) |

Commutative Property | 5 × (-2) = (-2) × 5 |

Multiplying by Powers of 10 | 4 × 10^{3} = 4000 |

## Rules of Integers in Division

Division Example | Result |
---|---|

Positive ÷ Positive | 12 ÷ 3 = 4 |

Positive ÷ Negative | 15 ÷ (-5) = -3 |

Negative ÷ Positive | (-8) ÷ 2 = -4 |

Negative ÷ Negative | (-12) ÷ (-3) = 4 |

Division by Zero | 10 ÷ 0 is undefined |

Even ÷ Even | 14 ÷ 2 = 7 (Even) |

Odd ÷ Odd | 9 ÷ 3 = 3 (Odd) |

Even ÷ Odd | 8 ÷ 3 = 2 (Even) |

Odd ÷ Even | 7 ÷ 2 = 3 (Odd) |

Fractional Quotient | 7 ÷ 2 = 3.5 |

Dividing by Powers of 10 | 900 ÷ 10^{2} = 9 |

## Integers Examples

**1.Example: 3+(−5)**

**Solution:** 3+(−5)=−2

Explanation: When adding a positive integer (3) to a negative integer (-5), we move left on the number line, resulting in a negative integer (-2).

**2. Example: 8−(−2)**

**Solution:** 8 – (-2) = 10

Explanation: Subtracting a negative integer is equivalent to adding its positive counterpart.

Thus, 8 – (-2) is the same as 8 + 2, resulting in 10.

**3. Example: 6(−4)×6**

**Solution:** (−4)×6 =−24

Explanation: The product of a negative integer and a positive integer is negative. In this case, (−4)×6 results in -24.