Introduction:
In mathematics, real numbers are the numbers that can be found on a number line. They include both rational and irrational numbers. Rational numbers can be expressed as a fraction, while irrational numbers cannot be expressed as a fraction. In this explanation, we will discuss why every real number is either irrational or rational.

Rational numbers:
Rational numbers are those numbers that can be expressed as a quotient or fraction of two integers. Any number that can be written in the form of p/q, where p and q are integers and q is not equal to zero, is a rational number. For example, 1/2, -3/4, and 5 are all rational numbers. Rational numbers can be positive, negative, or zero.

Irrational numbers:
Irrational numbers are those numbers that cannot be expressed as a fraction or quotient of two integers. They cannot be represented as terminating or repeating decimals. Irrational numbers are non-repeating and non-terminating decimals. For example, √2, π (pi), and e (Euler's number) are all irrational numbers.

Proof that every real number is either irrational or rational:
To prove that every real number is either irrational or rational, we need to consider two cases:

Case 1: Rational Numbers
Every rational number is a real number, so we can say that every rational number is a subset of real numbers. Rational numbers can be expressed as fractions, which means they can be written in the form p/q, where p and q are integers and q is not equal to zero. Since rational numbers are a subset of real numbers, we can conclude that every rational number is a real number.

Case 2: Irrational Numbers
Irrational numbers are those numbers that cannot be expressed as fractions. They are non-repeating and non-terminating decimals. Since irrational numbers cannot be expressed as fractions, we can conclude that they are not rational numbers. However, irrational numbers are still real numbers because they can be found on the number line.

Conclusion:
In conclusion, every real number is either rational or irrational. Rational numbers can be expressed as fractions, while irrational numbers cannot be expressed as fractions. Both rational and irrational numbers are subsets of real numbers. Therefore, any number that can be found on the number line is either rational or irrational.

It's true because real number consists of Rational and irrational number