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# Properties of Rational Numbers - Closure

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#### 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.

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Closure Property of Rational Number [00:09:13]
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