#### notes

**1) Whole numbers -**

Operation |
Numbers |
Remarks |

Addition |
0 + 7 = 7 + 0 = 7 ,2 + 3 = 3 + 2 = 5 For any two whole numbers a and b, a + b = b + a |
Addition is commutative . |

Subtraction | 3 - 2 = 1 but 2 - 3 = -1 For any teo whole numbers a and b , a-b ≠ b - a |
Subtraction is not commutative. |

Multiplication | 2 × 3 = 6 and 3 × 2 = 6 For any two whole numbers a and b, a × b = b × a |
Multiplication is commutative . |

Division | 4 ÷ 2 = 2 but 2 ÷ 4 = 0.5 For any two whole numbers a and b, a ÷ b ≠ b ÷ a |
Division is not commutative . |

**2 ) Integers :**

Operation |
Numbers |
Remarks |

Addition |
-2 + 3 = 1 and 3 + (-2) = 1 |
Addiion is commutative. |

Subtraction | (-2)-(1) = -3 but (1) - (-2) = 3 | Subtraction is not commutative. |

Multiplication | -2 * (-7) = 14 and (-7) * (-2) = 14 | Multiplication is commutative. |

Division | -2 ÷ 3 = - 1.666... but 3 ÷ -2 = -1.5 | Division is not commutative . |

**3) Rational numbers : ****a. Addition** **: **

example- `-2/3 + 5/7 = 1/21` and `5/7+(-2/3)=1/21`

We say that two rational numbers can be added in any order. We say that addition is commutative for rational numbers. That is, for any two rational numbers a and b,

a + b = b + a.

**b. Subtraction :**

Example - `2/3 - 5/4 =-7/12` and `5/4 - 2/3 = 7/12`

Note that subtraction is not commutative for integers and integers are also rational numbers. So, subtraction will not be commutative for rational numbers too.

**c. Multiplication ****:**

Example - `-7/3 xx 6/5 = -42/15 = 6/5 xx -7/3`

We say that multiplication is commutative for rational numbers. In general, a × b = b × a for any two rational numbers a and b.

**d. Division :**

Example - `-5/4 ÷3/7 = -35/12`, `3/7 ÷ (-5/4) = (12/-35)`

The expressions on both sides are not equal. So division is not commutative for rational numbers.