Question

What numbers added to $0.4$ will produce an irrational number?

Answer 1

Hint: Here, we have to add such a number to $0.4$ that it will produce an irrational number. Here, we have a rational number $0.4$ so we will add an irrational number to it so that it becomes an irrational number. In irrational numbers the decimal expansion is non-terminating and non- repeating means that after decimal the value never ends.

Complete step by step answer:
In Mathematics, numbers are classified into Real numbers and imaginary numbers. Real numbers can be defined as the numbers that exist in reality and are used in mathematical operations whereas imaginary numbers are the numbers that do not exist in reality.Further real numbers are again classified into rational numbers and irrational numbers.

Rational numbers are the numbers that can be expressed in the form of fraction or the number which can be written in the form of $\dfrac{p}{q}$ where $(q \ne 0)$. Rational numbers are terminating and repeating means after decimal there are finite numbers or the numbers are repeated again. For example- $0.2,\,0.33$

Irrational numbers are the numbers that cannot be expressed in the form of fraction or the numbers that cannot be expressed in the form of $\dfrac{p}{q}$ where $(q \ne 0)$.Here, we have a rational number $0.4$ so we will add an irrational number to it so that it becomes an irrational number as when we add a rational number to an irrational number the result we get is an irrational number.

Now, adding $\pi $ or $3.14287 \ldots $ to $0.4$. We get,
$ \Rightarrow 0.4 + 3.14287 \ldots = 3.54287 \ldots $
Which is an irrational number.
Now adding $\sqrt 2 $ to $0.4$. We get,
$ \therefore 0.4 + 1.4142135 \ldots = 1.81421 \ldots $
Which is also an irrational number.

Hence, it is clear that if we add any irrational number to $0.4$ we get an irrational number.

Note: For a number to be rational, only the denominator should not be equal to zero.However, $0$ can be written in the place of a numerator. We can write it as $\dfrac{0}{1}$ which is a rational number. Note that the square root of natural numbers other than perfect squares are irrational numbers. If $p$ is a prime number then its square root i.e., $\sqrt p $ is an irrational number. The sum or difference of rational and irrational numbers is always an irrational number whereas the sum or difference of two irrational numbers is not always an irrational number it can be a rational number.