What is the Number System? Types and Solved Examples

A numeral system, often known as a number system, is a way to represent numbers with symbols. It is an essential tool in programming and mathematics for data representation. It involves block numbers with digits, where the value of each digit is based on where it is located within the system and its base value. Although the binary system of 0s and 1s is what computers utilise most often, number systems are widely used in many different industries for labelling, measuring, and calculations, among other things. Numerous number systems exist, and each is appropriate for a particular set of uses.

What is the Number System?

A writing method for expressing numbers is called a number system. It is the mathematical notation for consistently using digits or other symbols to represent numbers within a specific set. It gives each number a distinct representation and shows the figures’ algebraic and mathematical structures. Arithmetic operations such as addition, subtraction, multiplication, and division can also be performed using it.

Any digit in a number can have its value ascertained by:

• The number
• Where it falls in the number
• The number system’s basis

Types of Number System

The four common types of Number systems are

1. Decimal Number System: Base 10.
2. Binary Number System: Base 2.
3. Octal Number System: Base 8.
4. Hexadecimal Number System: Base 16.

Decimal Number System

The 0 through 9 numbers, or 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, are used in the decimal number system that we use on a daily basis. The base number of the decimal number system is 10, since this number system consists of ten numbers. The base of any integer that is represented without a base is 10. Every value has a power of ten, which indicates that a ten times larger digit is found in the tens place than in the one’s place. When we reach nine, we create the number zero on the right and add one to the left to achieve ten. We keep going forward and so forth.

The digits 7 in the units place, 5 in the tens place, 4 in the hundreds place, and 1 in the thousands place make up the decimal number 1457, whose value can be expressed as follows:

(1×10³) + (4×10²) + (5×10¹) + (7×10⁰)

(1×1000) + (4×100) + (5×10) + (7×1)

1000 + 400 + 50 + 7

1457

Binary Number System

Similar to the widely used decimal or base-10 system, the binary number system is a Base-2 numbering system that adheres to the same mathematical principles. Binary numbers use powers of two (2ⁿ), essentially doubling the value of each consecutive bit as it advances, in place of powers of ten (10ⁿ), such as 1, 10, 100, 1000, and so on.

Computers alter and store all of their data, including words, numbers, images, audio, and video, using the binary number system. Given that there is no such number as “2” in this computer system, 1 + 1 = 10. An algebraic logic system based on the binary system was described in a significant work by British mathematician George Boole in 1854. This system came to be known as Boolean algebra.

Octal Number System

The Octal Number System is a system of numbers consisting of eight base digits, 0 through 7, i.e., 0, 1, 2, 3, 4, 5, 6, and 7. Eight-digit base numbers are referred to as octal numbers. Octal numbers are important and have many applications, such as digital numbering systems and computers.

Digital numbering systems and computers are just two examples of the many uses and significance of octal numbers. Aircraft transponders transmit a four-octal-digit signal known as the “squawk” code in response to a ground radar query. This code is used to distinguish between different aeroplanes on the radar screen.

Hexadecimal Number System

Hexa and deci, which stand for six and ten respectively, are the two words that make up hexadecimal. Integers 0 through 9 and letters A through F are represented in the hexadecimal number system by 16 digits. In other words, the next six digits, from A to F, are symbols, whereas the first nine digits, or numerals, are represented as numbers. The decimal number system, and the hexadecimal number system are quite similar. A symbol is used to symbolise the tenth digit after nine numbers: 10 is A, 11 is B, 12 is C, 13 is D, 14 is E, and 15 is F.

Each byte of plaintext in the Base16 transfer encoding is split into two 4-bit values and two hexadecimal digits using the hexadecimal method. Software engineers and system designers frequently use hexadecimal numbers because they provide a human-friendly representation of binary-coded data. Each hexadecimal digit, sometimes referred to as a nibble (or nybble), represents four bits (binary digits).

Number System Chart

Values of various numbers in the four number systems is shown below:

Decimal	Binary	Octal	Hexa
1	1	1	1
2	10	2	0
3	11	3	2
4	100	4	3
5	101	5	4
6	110	6	5
7	111	7	6
8	1000	10	7
9	1001	11	8
10	1010	12	9
11	1011	13	A
12	1100	14	B
13	1101	15	C
14	1110	16	D
15	1111	17	E
16	10000	20	F
17	10001	21	10
18	10010	22	11
19	10011	23	12
20	10100	24	13

Number System Conversion

Any of the number system categories, including binary, decimal, hexadecimal, etc., can be used to represent numbers. Additionally, it is simple to convert any number that is represented in one number system type to another. Learn how to convert numbers from decimal to binary and vice versa, from hexadecimal to binary and vice versa, and from octal to binary and vice versa using a variety of examples .

Let’s now quickly go over how to convert one number system to another by using a random number, using the various conversion processes that were previously described.

Assume 349 as your number. Thus, 349 appears as follows in several number systems:

• In the binary number system, 349 is 101011101.

• In the decimal numbering system, 349 is 349.

• In the octal number system, 349 is equivalent to 535.

• In the hexadecimal numbering system, 349 is represented by 15D.

Number System Solved Examples

1. Convert (1001001100)₂ to a decimal number.

Solution:

(1001001100)₂

= 1 × 2⁹ + 0 × 2⁸ + 0 × 2⁷ + 1 × 2⁶ + 0 × 2⁵ + 0 × 2⁴ + 1 × 2³ + 1 × 2² + 0 × 2¹ + 0 × 2⁰

= 512 + 64 + 8 + 4

= (588)₁₀

2. Convert 10101₂ into an octal number.

Solution:

Given, 

 10101₂ is the binary number

We can write the given binary number as,

010 101

Now as we know, in the octal number system,

010 → 2

101 → 5

Therefore, the required octal number is (25)₈.

3. Let’s convert 29 to Binary.

Solution:

We repeatedly divide 29 by 2 till we get 0 reminders.

Dividend	Divisor	Reminder
2	29	1
2	14	0
2	7	1
2	3	1
2	1	1
2	0	—

We read the reminder in reverse order. So 29₁₀=10111₂.

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