A set of symbols and rules represent numbers on the number line in a number system. These symbols are digits, ranging from 0 to 9. Number systems are necessary for various calculations, from simple arithmetical to complex mathematical calculations.

Number Line

A number line is expressed as a horizontal straight line, with numbers placed at equal intervals along its length. The ends of a number line are not defined, extending infinitely in both directions.

On a number line, numbers on the left are smaller in value than those on the right. A number line contains all types of numbers, natural, integers, and rational numbers.

Types of Number Systems

Several number systems exist, but the four most common ones are,

Decimal Number System

The decimal number system is of ten digits, 0, 1, 2 to 9; since it uses ten digits, it is called a base -10 system. The decimal system is the most widely used number system in the world.

The decimal system is known by another name,  the base 10 number system, where each digit has a specific place value, extending from right to left. The successive positions are defined as units, tens, hundreds, thousands, etc. Each location has a power of the base, 10.

The counting follows the series logic of counting digits in sets of 10. In the first set, we have digits from 1 to 9; then, 1 is added before 0 to get 10. For subsequent numbers, we count the second set of tens by adding 1 before 1, 2, and more to get 11, 12 and the like. For calculating the third set of tens, we add 2 before all the numbers to get 20, 21, and 22 to 29. The logic continues for each subsequent set of tens.

To illustrate, let us take the number 4321. The illustration will be an excellent example for class 9 NCERT maths chapter1 solutions.

Placement of digits is, in-unit place 1, tens' place 2, in hundreds' place 3 and thousands' 4.

Mathematically, we can write,

(4x10⁰ 3) + (3 x 10⁰ 2) + (2 x 10⁰ 1) + (1 x 10⁰ 0)

= (4 x 1000) + (3 x 100) + (2 x 10) + (1 x 1)

= 4000 + 300 + 200 + 1

= 4321

Binary Number System

In the base -2 number system, there are only two digits, 0 and 1, and it is known as the binary system. This system finds applications in computers for storing and processing data.

The digits 0 and 1 are termed binary digits or bits as abbreviations. The digit 2 as a subscript identifies a base-2 number.

The binary system does not use other digits, such as 2, 3, 4 etc. A binary number example is 10001₂.

Octal Number System

The Octal number system is called the base-8 number system and uses the eight digits, 0 to 7. In the octal system, each place is a power of eight, whereas, in the decimal system, each place is a power of ten.

Octal is used as a computer language, though it is not a very popular language.

Hexadecimal Number System

The base-16 number system uses a combination of numbers and alphabets and is called the hexadecimal number system. The numbers from 0 to 9 and alphabets A to F  represent the numbers.

The base -16 is primarily used in computers and other digital systems to store large numbers or high volumes of data. For example, a computer usually expresses colours in hexagonal language. To understand the system, we start counting from 1 and after 9, and the next digit is A and not 10. The series continues till F, and then we have 10, 11 to 19 and again 1A, 1B, 1C, etc. In this system, 10 does not represent the digit 10 but a value 1-0, indicating 1 group of 16 numbers from 0 to 9 and A to F without a single leftover number.

Number System:  Converting Decimal to Binary

Numbers are convertible from one base system to another by using specific procedures. For example, there are many methods of converting decimal to binary. We examine the most common division by 2, quotient and remainder method.

1.  Divide the decimal number by 2 and note the remainder.

2.  Next, divide the quotient by 2 and once more note the remainder.

3.  Repeat steps till the quotient is 0.

4.  Next, write the last remainder first, followed by the other reminders in the reverse order.

Example:

We are converting the decimal number 15 to binary.

15₁₀

Therefore, 15₁₀ = 1111₂

In the same way, we can convert from decimal to other systems like Octal, Hexadecimal;  first, we divide the given number by the base of the required number; and then, note down the quotient/remainder; by repeating the process of division of the quotient by the base till the resultant quotient is less than the base.

• What is the smallest whole number?

The smallest whole number is 0. The reason is that whole numbers do not have negative values, and no positive number is below 0.

• While converting a decimal system to binary, does the base of the number change from 2 to 10 – Yes or no?

The answer is no.

• How will the binary values of 0, 1, 2 and 3 appear in the decimal to binary conversion table?

The binary value of 0 is 0; 1 is 1; 2 is 10; 3 is 11.

• What differentiates real numbers and integers?

All Integers are real numbers, but the reverse is not true. Integers are whole numbers with negatives and do not include fractions or decimals.

The class 9 maths chapter 1 is crucial for students to build a foundation of the number system for future study of mathematics. The MSVgo app enables students to speed up problem-solving skills and comfortably tackle NCERT class 9 maths chapter 1 problems. Video solutions to textbook questions help students in quick revision and preparation for exams. Join MSVgo app  let your child be a MSVgo champ!

Numbers can be expressed in all of the divisions of the number system, including binary, decimal, hex, etc. Every number that is expressed in any of the forms of the number system can also easily be translated to another.

In this chapter, we learned about the basics of number systems. We learned about the properties Of irrational numbers, types of number systems, and utilizing this base knowledge we can understand the rationalisation laws of radicals. These concepts can be utilized to solve mathematical problems.

1. What are the 4 types of number systems? 

The four most prevalent kinds of number system are: 

• Decimal Machine Number (Base- 10) 
• System of Binary Numbers (Base- 2) 
• System of Octal Numbers (Base-8) 
• Hexadecimal System of Numbers (Base- 16) 

2. What are the different types of number systems? 

The four most prevalent kinds of number system are: 

• Decimal Machine Number (Base- 10) 
• System of Binary Numbers (Base- 2) 
• System of Octal Numbers (Base-8) 
• Hexadecimal System of Numbers (Base- 16) 

3. What is the highest base number system? 

System of Decimal Number [Base-10] 

With the least value being 0 and the greatest value being nine. On the left, the digit or column has the greatest value, while on the right, the digit has the least value.

4. What is number system with example? 

A system for representing numbers of a certain type (that is, expressing or writing them). Example: There are several systems for the representation of numbers for counting. These include: the usual system of “base ten” or “decimal”: 

5. What is the real number system in math?

All rational numbers, such as integer-5 and fraction 4/3, and all irrational numbers, such as √2, are included in the real numbers (1.41421356…, the square root of 2, an irrational algebraic number). Transcendental numbers, such as π, are included within the irrationals.