#### notes

**1) Whole number : **

Operation |
Numbers |
Remarks |

Addition | 3 + (5 + 2) = 10 and (3 + 5) + 2 = 10 For any three whole numbers a, b and c a + (b + c) = (a + b) + c |
Addition is Associative . |

Subtraction | 3 - (2 - 1) = 2 but (3 - 2 )- 1 = 0 For any three whole numbers a, b and c a - (b - c) ≠ (a - b)- c |
Subtraction is not Associative . |

Multiplication | - 7 × (2 × 5) = (7 × 2) × 5 = 70 For any three whole numbers a, b and c a × (b × c) = (a × b) × c |
Multiplication is Associative |

Division | 3 ÷ (5 ÷ 10) = 6 but (3 ÷ 5) ÷ 10 = 0.06 For any three whole numbers a, b and c a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c |
Division is not Associative . |

**2 ) Integers :**

Operation |
Numbers |
Remarks |

Addition | (-3) + (5 + 1) = 3 and (-3 + 5) + 1 = 3 For any three integers a, b and c a + (b + c) = (a + b) + c |
Addition is Associative . |

Subtraction | - 5 – (7 – 3) = 1 but (5 – 7) – 3 = -5 For any three integers a, b and c a - (b - c) ≠ (a - b) - c |
Subtraction is not Associative . |

Multiplication |
5 × [(–7) × (– 8) = 280 and |
Multiplication is Associative . |

Division | [(–10) ÷ 2] ÷ (–5) =25 but (–10) ÷ [2 ÷ (– 5)] = 1 For any three integers a, b and c a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c |
Division is not Associative . |

**3) Rational numbers - **

(a) Addition :

we have `-2/3 + [3/5 + (-5/6)] = -2/3 + (-7/30) = -27/30 = -9/10`

`[-2/3 + 3/5] + (-5/6) = -1/15 + (-5/6) = -27/30 = -9/10`

we say that addition is associative for rational numbers. That is, for any three rational numbers a, b and c,

a + (b + c) = (a + b) + c.

**(b) Subtraction** **:**

We have `-2/3 - [-4/5 -1/2] = 19/30 ,

[2/3 - (-4/5)] - 1/2 = 26/30`

Subtraction is not associative for rational numbers.

That is, for any three rational numbers a, b and c,

a - (b - c) ≠ (a - b) - c.

**(c) Multiplication : **

We have `-7/3 xx (5/4 xx 2/9) = (-7/3 xx 5/4) xx 2/9 = -35/54`

We observe that multiplication is associative for rational numbers. That is for any three rational numbers a, b and c,

a × (b × c) = (a × b) × c.

**(d) Division :**

Let us see if `1/2 ÷ [-1/3 ÷ 2/5] = [1/2 ÷ (-1/3)] ÷ 2/5 `

We have, LHS = `1/2 ÷ [-1/3 ÷ 2/5] = -30 / 10`

RHS =` [1/2 ÷ (-1/3)] ÷ 2/5 = -15/4`

We say that division is not associative for rational numbers.

That is, for any three rational numbers a, b and c,

a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c.