Question

is r is a rational number and s is a rational number then r + s and r -s are​

Answer 1

Answer

• (r + s) is a rational number
• (r - s) is a rational number

Explanation:

A number that can be expressed as ratio of two integers i.e. in the form x/y. where the denominator is a non-zero integer as division by zero is not defined. i.e. y ≠ 0.

It is given that r and s are rational numbers. So,

• r can be written as a/b
• s can be written as c/d

Where,

• a, b, c and d are integers.
• b ≠ 0, d ≠ 0

For r + s,

r + s = a/b + c/d

= (da + cb)/bd

( bd, da and cb are integers because product of two integers is an integer. So, (da + cb) is an integer as sum of two integers is an integer. )

• The numerator (da + cb) is an integer
• The denominator bd is an integer

r + s can be expressed as ratio of two integers. So, r + s is a rational number. It shows that sum of two rational number is a rational number.

So, we can say rational number are closed under addition

For r - s,

(r - s) = a/b - c/d = (da - cb)/bd

• The numerator (da - cb) is an integer
• The denominator bd is an integer

( bd, da and cb are integers because the product of two integers is an integer

da and cb are integers and difference of two integers is an integer. So (da - cb) is also an integer )

r - s can be expressed as ratio of two integers. So, (r - s) is a rational number. It shows that the difference of two rational numbers is also a rational number.

So, we say rational numbers are closed under subtraction.

More:

a/b × c/d = ac/bd

ac and bd are rational as product of two integers is an integer.

So, Rational numbers are closed under multiplication

Rational numbers are not always closed under division because division by zero is not defined so a number like q/0 is not defined.

But if we exclude 0 ( as the denominator) rational numbers are closed under division