## Chapter 2: Real Numbers

### Abstract

In this chapter, we will learn about the concept of real numbers, which includes rational and irrational numbers. We will also study the properties of real numbers and their representation on the number line.

### Multiple Choice Questions

- Which of the following is an irrational number?

a) √2

b) √3

c) √5

d) √6

Answer: a) √2 - Which of the following is a rational number?

a) √2

b) √3

c) √5

d) √6

Answer: None of the above - Which of the following is an example of a terminating decimal?

a) 0.333…

b) 0.666…

c) 0.71828…

d) 0.999…

Answer: c) 0.71828… - Which of the following is an example of a repeating decimal?

a) 0.333…

b) 0.666…

c) 0.71828…

d) 0.999…

Answer: a) 0.333… - Which of the following is an example of a whole number?

a) 3.14

b) 4.56

c) 5

d) 6.78

Answer: c) 5

### Very Short Questions and Answers

- What is a real number?

Answer: A real number is a number that can be represented on the number line. - What are rational numbers?

Answer: Rational numbers are numbers that can be expressed as the ratio of two integers. - What are irrational numbers?

Answer: Irrational numbers are numbers that cannot be expressed as the ratio of two integers. - What is a terminating decimal?

Answer: A terminating decimal is a decimal that ends with a sequence of zeros. - What is a repeating decimal?

Answer: A repeating decimal is a decimal that has a sequence of digits that repeats indefinitely.

### Short Questions and Answers

- Convert the following decimal to a fraction: 0.5

Answer: 0.5 = 5/10 = 1/2 - Convert the following fraction to a decimal: 3/4

Answer: 3/4 = 0.75 - Find the sum of the following two numbers: 3.14 and 2.718

Answer: 3.14 + 2.718 = 5.858 - Find the difference between the following two numbers: 5 and 2.718

Answer: 5 – 2.718 = 2.282 - Find the product of the following two numbers: 0.333… and 0.71828…

Answer: 0.333… × 0.71828… = 0.23999…

### Long Questions and Answers

- Prove that the sum of two rational numbers is a rational number.

Answer: Let a and b be two rational numbers. Then, a = p/q and b = r/s, where p, q, r, and s are integers and q and s are not equal to zero. The sum of a and b is given by: a + b = (p/q) + (r/s) = (p + r)/(q + s) Since p, q, r, and s are integers, the sum (p + r) is also an integer. Similarly, q and s are not equal to zero, so q + s is also not equal to zero. Therefore, the sum (p + r)/(q + s) is a rational number. - Prove that the product of two rational numbers is a rational number.

Answer: Let a and b be two rational numbers. Then, a = p/q and b = r/s, where p, q, r, and s are integers and q and s are not equal to zero. The product of a and b is given by: a × b = (p/q) × (r/s) = (pr)/(qs) Since p, q, r, and s are integers, the product pr is also an integer. Similarly, q and s are not equal to zero, so qs is also not equal to zero. Therefore, the product (pr)/(qs) is a rational number.