Number System Questions

Number System is one of the most important topics of the Maths subject which the students are required to master. And the best way to master the chapter on the number system is by practicing the questions from the said chapter. But along with the practice students do need the solution of the same so that they can be sure about their progress. Hence Vedantu provides to all the students the practice questions of the number system, as well as solved examples of the same so that before attempting the practise questions students can see the examples and have a mental preparation

An Overview of the Number System

Before going directly into the practice questions of the number system, let us first have a brief understanding of the number system, and revise the concepts of it, so that you find it easy to solve the practice questions.

A method of expressing the numbers on the number line, using numerical symbols, is known as Number System. As you can see there are two terms in particular in this definition to understand in a better manner, which are:

• Number Line: it is a straight line that real numbers at a fixed interval. All the types of numbers are included in the number line, that is to say, natural numbers, rational numbers, integers etc.

• Numerical Symbols: it simply means the mathematical digits, that represents the numbers, which are from 0 to 9.

An Overview of Different Types of Numbers

All the types of numbers are represented in the number line and hence they are all part of the number system, therefore, let us have a quick review of all the types of numbers.

• Natural Number: These are the numbers that we use in our day-to-day life because it is widely used in counting, therefore natural numbers are also called counting numbers. It includes all the positive numbers which are not a fraction, and also, it does not include 0. The range of the natural number is 1 to infinity.

• Whole Numbers: Add zero to the natural number and you have the whole numbers, that is to say, it includes all the natural numbers and also the 0. Therefore, from 0 to infinity all the numbers are whole numbers.

• Integers: It includes all the whole numbers, along with the negative numbers, that is to say, -1, -2, -3. But it does not include the fractions. From the negative infinity number to the positive number in infinity, are all Integers.

• Fractions: The numbers which are written in the form of, where b is always a natural number.

• Rational Numbers: These numbers can also be represented in fractional form, the difference between rational numbers and fractions is that rational numbers can be any integers except for the 0 as the denominator.

• Irrational Numbers: These are the numbers that cannot be represented in a fractional manner such as the root of 2 (\[\sqrt{2}\]) and pie (π)

• Real Numbers: When we combine the whole numbers, integers, and fractions are all real numbers. In a simple manner, all the integers along with the decimals and fractions are real numbers.

• Prime Numbers: The numbers which only have two factors, which are 1 and the number itself are called prime numbers. For example, 37, can only be divided by 1 and by 37 itself.

The Number System is an important chapter of mathematics. A student needs to be strong in the fundamentals of the number system to solve other problems related to Maths. Some students face difficulty in solving sums of the number system. So, here in this article, we have provided some crucial sums relating to the number system. A student can practice these questions, and it would be easy for him/her to understand the chapter. In this article, we have provided various questions based on number systems such as number system questions and answers, number system practice questions, MCQs on number systems and many other important questions.

Number System Questions and Answers

1. Determine whether the numbers are rational or irrational.

• \[ \sqrt{2}\]

• 1.5

• \[\sqrt{100}\]

• 3.14

Ans: A rational number is a number that can be represented in the form of p/q, whereas an irrational number cannot be represented in the form of p/q. So,

\[ \sqrt{2}\] is Irrational.

1.5 is Rational.

\[\sqrt{100}\] is Rational.

3.14 is Irrational.

2. Without Actual Division, a state which of the following is a terminating decimal.

• \[\frac{9}{25}\]

• \[ \frac{37}{78}\]

Ans: In \[\frac{9}{25}\], the prime factors of denominator 25 are 5,5. Thus, it is a terminating decimal. 

In \[ \frac{37}{78}\], the prime factors of denominator 78 are 2, 3, and 13. Thus, it is a non-terminating decimal.

3. Express each of the following as a rational number in the form of p/q, where q ≠ 0.

1. \[\overline{0.6}\]

2. \[ \overline{0.43}\]

Ans:  1. Let x = 0.6666 …..(i)

Multiplying both side of eqn (i) by 10 we get,

10x = 6.6666…..(ii)

Now, subtracting eqn (i) from eqn (ii) we get, 

10x = 6.6666

x = 0.6666

9x = 6.

⇒ x = 6/9 which is equal to ⅔, So the required fraction is ⅔.

2. Let x = 0.43434343….(i)

Multiplying both sides of eqn (i) by 100 we get,

100x = 43.43434343…..(ii)

Now, subtracting eqn (i) from eqn (ii) we get,

100x = 43.43434343

x = 0.43434343

99x = 43

⇒ x = \[\frac{43}{99}\], Hence the fraction is \[\frac{43}{99}\].

4. Find 4 rational numbers between 1 and 2.

Ans: To find 4 rational numbers between 1 and 2, we need to divide and multiply both the numbers by (4 + 1) which is 5. So we get,

\[1 \times \frac{5}{5} = \frac{5}{5}\] and \[ 2 \times \frac{5}{5}\] = \[\frac{10}{5}\], Therefore the rational numbers are:

\[\frac{5}{5}\], \[\frac{6}{5}\], \[\frac{7}{5}\], \[\frac{8}{5}\], \[\frac{9}{5}\], \[\frac{10}{5}\].

5. Compare the following numbers.

(i) 0 and \[-\frac{9}{5}\].

(ii) \[-\frac{17}{20}\] and \[-\frac{13}{20}\].

(iii) \[\frac{40}{29}\] and \[\frac{141}{29}\].

Ans: (i) We know that a negative number is always less than 0. Therefore,

0 > - \[\frac{9}{5}\].

(ii) Here the denominator is the same and we know that -17 < -13. Therefore, 

\[\frac{-17}{20}\] < \[\frac{-13}{20}\].

(iii) Here the denominator is the same and we know that 40 < 141. Therefore,

\[\frac{40}{29}\] < \[\frac{141}{29}\].

6. Write the following in decimal numbers and state what expansion it is.

(i) \[\frac{40}{100}\]  (ii) \[\frac{9}{10}\]  (iii) \[\frac{9}{37}\] (iv) \[\frac{103}{5}\]

Ans: (i) \[\frac{40}{100}\] is 0.40, and it is terminating.

(ii) \[\frac{9}{10}\] is 0.9, and it is ending.

(iii) \[\frac{9}{37}\] is 0.243243… it is non-terminating.

(iv) \[\frac{103}{5}\] is 20.6, and it is terminating.

7. Insert one rational number between 3/5 and 7/9.

Ans:  If a and b are two rational numbers, then one rational number between these two will be \[\frac{a + b}{2}\]. Hence the required rational number will be 

\[\frac{1}{2} (\frac{3}{5} + \frac{7}{9}) = \frac{1}{2} (\frac{27 + 35}{45}) = \frac{1}{2} \times \frac{62}{45} = \frac{31}{45}\]

So, the rational number is \[\frac{31}{45}\].

Questions on Number System Conversion

Here, we have provided some number system math questions which are based on number system conversion. 

1. Convert each of the following into a decimal number.

(i) \[\frac{4}{15}\]

(ii) \[2\frac{5}{12}\]

(iii) \[\frac{9}{27}\]

(iv) \[5\frac{31}{55}\]

2. Convert the following into a rational number.

(i) \[0.\overline{227}\]

(ii) \[0.\overline{2104}\]

Number System Practice Questions

As we know, practice makes everyone perfect, so for the better understanding of students, we have provided some number system important questions for practice.

1. Show the number √5 on the number line.

• \[\sqrt{2}\]

• 1.5

• \[ \sqrt{100}\]

• 3.14

Ans: A rational number is a number which can be represented in the form of p/q, whereas an irrational number cannot be represented in the form of p/q. So,

\[\sqrt{2}\] is Irrational.

1.5 is Rational.

\[\sqrt{100}\] is Rational.

3.14 is Irrational.

2. Insert three rational numbers between 4 and 5.

• \[\frac{9}{25}\]

• \[ \frac{37}{78} \]

Ans: In \[ \frac{9}{25}\], the prime factors of denominator 25 are 5,5. Thus, it is a terminating decimal. 

In \[\frac{37}{78} \], the prime factors of denominator 78 are 2, 3, and 13. Thus, it is a non-terminating decimal.

3. Represent the following rational numbers in decimal form

(i) \[\frac{18}{42}\]    (ii) \[-\frac{11}{13}\]

4. Rationalise the denominator of

\[ \frac{1}{3-\sqrt{5}} \]

5. Simplify the following expression (24 - 32)a. ( 5 + 23)