Question

Which of the following statements is correct?
1) There can be a real number which is both rational and irrational
2) The sum of two irrational numbers is always irrational
3) Every integer is a rational number
4) None of these

Answer 1

Hint: Rational numbers are the one which can be expressed in the fraction form or we can say in the p/q form.
Irrational numbers are the one which cannot be expressed in the p/q form.
Using the definition of rational and irrational numbers we will conclude the problem by giving examples.

Complete step-by-step solution:
Let us understand rational, irrational and integer numbers in more detail.
Irrational numbers: any real number that can be expressed as the quotient of two integers. Irrational numbers are non terminating which means that when the number is converted in decimal form it does not terminate but the value after decimal keeps on repeating. Example is the value of pi.
Rational number: Rational number is the one which can be written in p/q form and which can terminate on division by the integer written in the denominator. For example 9/2 fraction has the value 4.5 in decimal form.
Integer number: Numbers which include both the negative and positive numbers but not the fraction and decimal numbers.-1,-2,0,3,4....are examples
Now let's come to the options one by one;
Option 1 says a real number can be both rational and irrational.
No there is no such number which can be both rational and irrational at the same time for example $\sqrt {2,} \sqrt 3 $ are irrational numbers which cannot be rational and 9/2, 8/5 are rational numbers which cannot be irrational at the same time.
Option 2 says the sum of two irrational numbers is always irrational.
No the statement is wrong because when we take two irrational numbers there sum may or may not be irrational such as;
$\sqrt 3 + 6, - \sqrt 3 + 2$ On adding these two numbers we have answer 8 which is an integer and not an irrational number.
Option 3 says every integer is a rational number.
Yes the statement is correct because, every integer is a rational number 4/1, 3/2, 5/1.

Option 3 is correct.

Note: As per the definition of numbers we have real numbers( numbers which do not have imaginary part), whole numbers which do not include negative numbers and include zero, prime numbers which divide by itself and number 1, complex numbers which contain both real and imaginary part.