Question

State whether the statement is true or false. The sum of two irrational numbers is irrational.

Answer 1

Hint: Numbers which cannot be expressed in the form of a fraction are known as irrational numbers. From which we get, that non terminating non-recurring decimals and surds to be irrational numbers. Now to check whether the given statement is true or false just consider any two irrational numbers, add them and check whether it is irrational or not.

Complete step-by-step answer:
Irrational numbers are real numbers which cannot be written in the form of a fraction, that is in the form of $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q$ is not equal to zero.
Non Terminating non-recurring decimals and surds are irrational numbers.
Now to check the given statement let’s use an example.
Let the two irrational numbers be $5\sqrt 5 $ and $\sqrt 3 $
Now when we add these two irrational numbers, we get $5\sqrt 5 + \sqrt 3 $
This is also an irrational number.
Same way, consider the irrational numbers $2 + \sqrt 7 $ and $2 - \sqrt 7 $
Now when we add these two numbers we get,
$2 + \sqrt 7 + 2 - \sqrt 7 = 2 + 2 = 4$
It is a rational number.
Hence the sum of two irrational numbers may be rational or irrational
From this it is clear that the given statement is false.

Note: The statement can be proved false by using two non-terminating non-recurring decimals instead of surds
For example, let’s consider the two irrational numbers to be $5.213475921.........$ and $6.321475895........$
Now when we add these two numbers, we get
 $5.213475921......... + 6.321475895........ = 11.534951816......$
We can see that the resulting number is also an irrational number
Same way, consider two irrational numbers $5.213475921.........$ and $ - 5.213475921.........$
When we add these numbers we get,
$5.213475921......... - 5.213475921......... = 0$
It is a rational number.
Hence the statement can be proved false even in this way