Let me see and write what I will get by adding, subtracting, multiplying and dividing (divisor is non-zero) two rational numbers.

As rational numbers are number which can be expressed in form of ,

Where p and q are integers and q≠0

Let’s take 2 rational number and

On adding 2 rational numbers.

Where result is in form of

As multiplying and adding integers gives an integer

(ys) can’t be 0 because y,s≠0

∴ Adding 2 rational number gives a rational number

On subtracting 2 rational numbers.

Where result is in form of

As multiplying and subtracting integers gives an integer

(ys) can’t be 0 because y,s≠0

∴ Subtracting 2 rational number gives a rational number

On multiplying 2 rational numbers.

Where result is in form of

As multiplying an integers gives an integer

(ys) can’t be 0 because y,s≠0

∴ Multiplying 2 rational number gives a rational number

On Dividing 2 rational numbers.

Where result is in form of

As multiplying an integer gives an integer

(yr) can be 0 because r can be 0

∴ Dividing 2 rational number doesn’t gives a rational number