Number Systems

The number system is the system of naming or representing numbers. Number is a mathematical value that helps to count or measure objects and it helps in performing various mathematical calculations.

Definition

A number system is defined as a system of writing to express numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner. It provides a unique representation of every number and represents the arithmetic and algebraic structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction, multiplication and division.

The value of any digit in a number can be determined by:

• The digit
• Its position in the number
• The base of the number system

Types of Number Systems

There are various types of number systems in mathematics. The four most common number system types 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 (Base 10 Number System)

The decimal number system has a base of $10$ because it uses ten digits from $0$ to $9$. In the decimal number system, the positions successive to the left of the decimal point indicates units, tens, hundreds, thousands and so on. This system is expressed in decimal numbers. Every position shows a particular power of the base ($10$).

Example of Decimal Number System:

The decimal number $1234$ consists of the digit $4$ in the units position, $3$ in the tens place, $2$ in the hundreds position, and $1$ in the thousands place whose value can be written as:

$(1\times 10^{3})+(2\times 10^{2})+(3\times 10^{1})+(4\times 10^{0})$

$(1\times 1000)+(2\times 100)+(3\times 10)+(4\times 1)$

$(1000+200+30+4)$

$1234$

Binary Number System (Base 2 Number System)

The base $2$ number system is also known as the Binary number system wherein, only two binary digits exist, i.e., $0$ and $1$. The figures described under this system are known as binary numbers which are the combination of $0$ and $1$. For example, $1110$ is a binary number.

Example of Binary Number System:

Write $(14)_{10}$ as a binary number.

Solution:

$2$	$14$
$2$	$7$	$0$	$\uparrow$
$2$	$3$	$1$	$\uparrow$
	$1$	$1$	$\uparrow$
	$\rightarrow$	$\rightarrow$

Steps:

Divide the number $14$ by $2$ , quotient is $7$ and remainder is $0$

Divide the quotient $7$ by $2$ , quotient is $3$ and remainder is $1$

Divide the quotient $3$ by $2$ , quotient is $1$ and remainder is $1$

Write the numbers as mentioned by the direction of the arrows above - $1110$

$(14)_{10}=(1110)_{2}$

Octal Number System (Base 8 Number System)

In the octal number system, the base is $8$ and it uses numbers from $0$ to $7$ to represent numbers. Octal numbers are commonly used in computer applications.

For example, $(14110)_{8}$ is a octal number which is an equivalent of $(2158)_{10}$

Hexadecimal Number System (Base 16 Number System)

In the hexadecimal system, numbers are written or represented with base $16$. In the hexadecimal system, the numbers are first represented just like in the decimal system, i.e. from $0$ to $9$. Then, the numbers are represented using the alphabet from $A$ to $F$.

For example, $(F2)_{16}$ is a hexadecimal number which is an equivalent of $(242)_{10}$