Real Number System

Rational numbers are numbers that can be expressed as fractions or ratios of two integers (where the denominator is not zero). Examples include 1/2, -3/4, 5/1 (which equals 5). When you work with fractions, you're working with rational numbers.

Pi is an irrational number because it cannot be expressed as a simple fraction of two integers. Its decimal representation (3.14159...) continues infinitely without repeating or terminating. Rational numbers can be written as fractions, while odd and even are properties of integers only, not applicable to pi.

A perfect number equals the sum of its proper divisors (all positive factors except the number itself). For 6: 1+2+3=6; for 28: 1+2+4+7+14=28; for 496: 1+2+4+8+16+31+62+124+248=496.

Real numbers include rational numbers, irrational numbers, and whole numbers, but NOT imaginary numbers. Imaginary numbers (involving i, where i²=-1) are a separate category from real numbers.

A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding the number itself). For example, $6 = 1 + 2 + 3$. The next examples are 28 and 496.

The statement is false because real numbers include rational numbers (fractions, terminating/repeating decimals) and irrational numbers (non-repeating decimals), but imaginary numbers (like the square root of -1) are NOT part of the real number system.

Both e (Euler's number, approx 2.718) and pi (π, approx 3.14159) are fundamental mathematical constants classified as irrational numbers - they cannot be expressed as fractions of integers and have non-terminating, non-repeating decimal expansions. Irrational numbers are a subset of real numbers that cannot be written as p/q where p and q are integers. Options A, B, and D are incorrect because e and pi are not whole numbers, not imaginary (they have real values), and not rational (they cannot be expressed as fractions).

Value of 2/3 = 0.66 Value of 4/5 = 0.8 Value of - 7/10 = - 0.7 Value of 16/20 = 0.8 Value of 3/4 = 0.75 Valvue of - 3/4 = - 0.75 Value of 3/10 = 0.3 Therefore, 3/4 lies between the given two rational numbers.

Correct, as this option is less than the given number.

• 17771/ 23333 > - 17772/ 23333

Option (1) is incorrect, as there are infinite rational numbers between 1/20 and 11/20. Option (2) is incorrect, as there are infinite rational numbers between 1/41 and 31/41. Option (3) is correct, as there are 11 whole numbers between 20 and 32, i.e. 21 to 31. Option (4) is correct, as there can be infinite rational numbers between any two rational numbers. Hence, 5^th option is the answer.

This answer is correct.

Both (1) & (2) are irrationalas shown above

$\sqrt 2 \times \sqrt 3 = \sqrt 6$, which is an irrational number.

Consider $\sqrt p$ and $\sqrt q$. If p and q are prime numbers, then $\sqrt p \times \sqrt q$ will be irrational. ^{_$\therefore$}  Product of $\sqrt 5$ and $\sqrt 7$ is an irrational number.