#### notes

**1. Whole numbers - **

Operation |
Numbers |
Remarks |

Addition | 0 + 5 = 5, a whole number 4 + 7 = 11 In general, a + b is a whole number for any two whole numbers a and b. |
Whole numbers are closed under addition . |

Subtraction | 5 – 7 = – 2, which is not a whole number. |
Whole numbers are not closed under subtraction. |

Multiplication | 0 × 3 = 0, a whole number 3 × 7 = 21 . Is it a whole number? . In general, if a and b are any two whole numbers, their product ab is a whole number. |
Whole numbers are closed under multiplication. |

Division | 5 ÷ 8 = `5/8` which is not a whole number. | Whole numbers are not closed under division. |

**2 . Integers : **

Operation |
Numbers |
Remarks |

Addition | – 6 + 5 = – 1, an integer Is – 7 + (–5) an integer? Is 8 + 5 an integer? In general, a + b is an integer for any two integers a and b. |
Integers are closed under addition. |

Subtraction | 7 – 5 = 2, an integer Is 5 – 7 an integer? – 6 – 8 = – 14, an integer – 6 – (– 8) = 2, an integer Is 8 – (– 6) an integer? In general, for any two integers a and b, a – b is again an integer. Check if b – a is also an integer. |
Integers are closed under subtraction. |

Multiplication | 5 × 8 = 40, an integer Is – 5 × 8 an integer? – 5 × (– 8) = 40, an integer In general, for any two integers a and b, a × b is also an integer. |
Integers are closed under multiplication. |

Division | 5 ÷ 8 = `5/8` which is not an integer. | Integers are not closed under division. |

**3. Rational numbers :**

A number which can be written in the form `p/q`, where p and q are integers and q ≠ 0 is called a rational number.

For example, `-2/3,6/7,9/-5` are all rational numbers.**1)** How to add two rational numbers. lets check it.

for example - `3/8 + (-5)/7 =( 21 + (-40))/56= -19/56` - it is a rational number.

We say that rational numbers are closed under **addition**. That is, for any two rational numbers a and b, a + b is also a rational number.

**2)** Will the difference of two rational numbers be again a rational number? We have

for example - `(-5)/7 - 2/3 =(-5 xx 3 - 2xx 7)/21 = (-29)/21` -> It is rational number.

We say that rational numbers are closed under **subtraction**. That is, for any two rational numbers a and b, a – b is also a rational number.

**3)** Let us now see the product of two rational numbers.

for example- `-2/3 xx 4/5 = -8/15; 3/7 xx 2/5 = 6/35`

(both the products are rational numbers)

We say that rational numbers are closed under **multiplication**. That is, for any two rational numbers a and b, a × b is also a rational number.

**4)** We note that `-5/3 ÷2/5 = -25/6 ` -> (it is rational number)

We find that for any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under **division**.