Rational and Irrational Numbers

A rational number is of the form of p/q, where p and q are integers and the denominator is not equal to zero.

The rational numbers are closed under addition, subtraction and multiplication but not closed for division. Between two rational numbers x and y, there is always a rational number (x+y)/2 such that x< (x+y)/2 < y.

The two ways of locating rational numbers between any two given numbers are:
(i) by locating the mid-value
(ii) by making the denominators same.

A number is called irrational number, if it cannot be written in the form of p/q, where p and q are integers and q is not equal to zero.

The decimal expansion of an irrational number is non-terminating non-recurring.

The negative of an irrational number is irrational. Sum and difference of a rational number and an irrational number is irrational. Product and division of a non-zero rational

number with an irrational number is irrational. The sum, difference, product and quotient

of two irrational numbers need not be an irrational number.

If a and b are two distinct positive rational numbers and ab is not a perfect square, then (ab)1/2 is an irrational number lying between a and b.

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  • Q1

    Is every real number is a rational number ?

    Marks:1
    Answer:

    No, as real numbers are rational and irrational both. For example, 2 is an irrational number, hence it is a real number.

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  • Q2

    Is 1.01001000100001 …… irrational? If so, why?

    Marks:1
    Answer:

    Yes, since it is a decimal number in non-terminating and non-recurring form.

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  • Q3

    State closure property under addition and multiplication for rational numbers and verify with examples that rational numbers are closed under addition, subtraction and multiplication.

    Marks:5
    Answer:



    Thus, we can say that rational numbers are closed under addition, multiplication and subtraction.

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  • Q4

    Find a rational number between

    Marks:1
    Answer:

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  • Q5

    Write an integer which is not a natural number but a whole number.

    Marks:1
    Answer:

    An integer which is not a natural number but a whole number is zero.

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