Question

Can we show that \[\;0.01001000100001......\] is an irrational number using a contradiction method?

Answer 1

Hint: Here in this question, we have to show that a given decimal number is an irrational number using a contradiction method. To show this try to express the given number in the form of a simple fraction or If we try to express an irrational number in decimal form then check whether its decimal number is neither terminating nor recurring.

Complete step-by-step answer:
Irrational numbers are the real numbers or any number which cannot be represented as a simple fraction is termed as irrational number. It cannot be expressed in the form of a ratio, such as \[\dfrac{p}{q}\], where p and q are integers, but \[q \ne 0\] are known as irrational numbers. If we try to express an irrational number in decimal form then it is neither terminating nor recurring.
Examples: \[\sqrt 2 \], \[\sqrt 3 \], the value of \[\pi = 3.14159265358979 \ldots \] and so on… are the examples of the irrational number.
Whereas any number which can be represented in the form of \[\dfrac{p}{q}\], where p and q are integers, but \[q \ne 0\] are known as rational numbers.
Therefore, Irrational numbers are a contradiction of rational numbers.
Consider the given question
We have to show that \[\;0.01001000100001......\] is an irrational number using a contradiction method.
On observing the given number \[\;0.01001000100001......\] the decimals are neither terminating nor recurring. Hence it is a contradiction of rational numbers. Therefore it is an irrational number.

Note: The above question need not be proven by contradiction method because all the irrational numbers are considered as real numbers, which should not be rational numbers. It means that irrational numbers cannot be expressed as the ratio of two numbers. The irrational numbers can be expressed in the form of non-terminating fractions and in different ways. For example, the square roots which are not perfect squares will always result in an irrational number.