# Positional Number System

A number is a method used for representing an arithmetic value, measure, or count, of a physical quantity. A number system is defined as a method of naming and representing numbers. The concept of number system helps in defining the rules associated with the numbers and different operations on numbers.

A number system is determined with the help of its radix or base. The radix or base of a number system is nothing but the total number of symbols used in the number system for representing the different numbers. For example, in the decimal number system, there are 10 symbols, i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Thus, the base or radix of the decimal number system is 10.

The number systems are broadly classified into two types, namely −

• Positional Number System

• Non-Positional Number System

In this article, we will discuss the positional number system in detail. So let’s begin with the basic introduction of the positional number system.

## What is the Positional Number System?

Positional number system is the type of number system in which the weight or value of the digit (or symbol) depends upon its position in the number. The positional number system is also known as weighted number system. This is because, in the positional number system, there is a weight associated with the position in the number.

Therefore, in the positional number system, each digit of the number is weighted according to its position of occurrence in the number. When we travel toward left along the number, the weights increase by a constant factor that is equivalent to the base of the number system. Also, in the positional number system, a radix point (.) is used to differentiate the positions corresponding to integral weights from the positions corresponding to the fractional weights.

## Types of Positional Number Systems

There are four very popular positional number systems, which are:

• Decimal Number System

• Binary Number System

• Octal Number System

Let us discuss each of these number systems in detail.

## Decimal Number System

The decimal number system is the most familiar number system for us. It is called decimal number system because its base or radix is ten (10). This means, it has 10 unique symbols, i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to represent different numbers of the system. Also, there is no symbol for its base, i.e. for ten.

The decimal number system is a positional or weighted number system. Therefore, in this number system, the value of a symbol depends on its position (or location) with respect to the decimal point or radix point.

In the decimal number system, any number, whether it is an integer or fraction or mixed number, of any magnitude can be represented by using these ten symbols only. Each of these symbols is called a digit.

In a decimal number, the most left digit has the greatest positional weight out of all the digits present in the number, this is called the MSD (Most Significant Digit). On the other hand, the right most digit has the least positional weight out of all the digits present in the number, it is called LSD (Least Significant Digit).

In the decimal number system, the digits present on the left of the decimal point form the integral part of the decimal number, while the digits present on the right of the decimal point form the fractional part of the number. The digits in the integral part of the decimal number have weights that are positive powers of 10 (base), whereas, the digits in the fractional part of the decimal number have weights that are negative powers of 10.

A decimal number is generally expressed as,

$$\mathrm{d_nd_{n-1}d_{n-2}...d_1d_0\cdot d_{-1}d_{-2}...d_{-m}}$$

The value of this number is evaluated as,

$$\mathrm{(d_n\times 10^n)+(d_{n-1}\times 10^{n-1})+(d_{n-2}\times 10^{n-2})+...+(d_1\times 10^1)+(d_0\times 10^0)+(d_{-1}\times 10^{-1})+(d_{-2}\times 10^{-2})+...+(d_{-m}\times 10^{-m})}$$

For example, consider a decimal number 512.26, then

$$\mathrm{5\times 10^2+1\times 10^1+2\times 10^0+2\times 10^{-1}+6\times 10^{-2}}$$

Therefore,

$$\mathrm{500 + 10 + 2 + 0.2 + 0.06}$$

Consider another decimal number with same digits, 125.62, then it can be expended as,

$$\mathrm{1\times 10^2+2\times 10^1+5\times 10^0+6\times 10^{-1}+2\times 10^{-2}}$$

Or,

$$\mathrm{100 + 20 + 5 + 0.6 + 0.02}$$

Hence, from these two examples, it is clear that the same digit has different value, when placed in different position. This also proves that the decimal number system is a positional number system.

## Binary Number System

The binary number system is also a positional number system or a weighted number system. The base or radix of the binary number system is 2, which means it has two unique symbols, i.e. 0 and 1 to represent numbers. Similar to decimal number system, the base of the binary number system itself cannot be a symbol.

In the binary number system, each digit is called a bit (binary digit). Thus, a binary number is nothing but a string of binary 0s and 1s. There is a binary point (radix point) that separates the integral and fractional parts of the binary number. Each binary digit or bit in a binary number carries a weight according to its position with respect to the binary point.

In the case of binary number system, the weight of each position is expressed in terms of power of 2, i.e. $2^n$ , where n = …, -3, -2, -1, 0, 1, 2, 3,…. The positive power of 2 represents the weight of a bit on the left side of the binary point, while the negative power of 2 represents the weight of a bit on the right side of the binary point.

A binary number with $(n^{+1})$ bits in its integral part and –m bits in its fractional part is expressed as,

$$\mathrm{d_nd_{n-1}d_{n-2}...d_1d_0\cdot d_{-1}d_{-2}...d_{-m}}$$

The decimal equivalent of this binary number is given by,

$$\mathrm{(d_{n}\times 2^n)+(d_{n-1}\times 2^{n-1})+(d_{n-2}\times 2^{n-2})+...+(d_1\times 2^1)+(d_0\times 2^0)+(d_{-1} \times 2^{-1})+(d_{-2}\times 2^{-2})+...+(d_{-m}\times 2^{-m})}$$

The binary number system is mainly used in digital systems because digitals systems use two state switching devices like diodes, transistors, etc. Where, the binary 1 is used to represent the ON state of the device, and the binary 0 is used to represent the OFF state of the device.

## Octal Number System

Octal Number System is again a type of positional number system. That means, the value of a digit in an octal number depends upon its position in the number. The base or radix of the octal number system is 8, thus, the octal number system has eight unique symbols, i.e. 0, 1, 2, 3, 4, 5, 6, and 7. The octal number system was widely used in early minicomputers.

The generalized form of an octal number having "n+1" digits in its integral part and "–m" digits in its fraction part is,

$$\mathrm{d_nd_{n-1}d_{n-2}...d_1d_0\cdot d_{-1}d_{-2}...d_{-n}}$$

The decimal equivalent of this octal number is,

$$\mathrm{(d_n\times 8^n)+(d_{n-1}\times 8^{n-1})+(d_{n-2}\times 8^{n-2})+...+(d_1\times 8^1)+(d_0\times 8^0)+(d_{-1}\times 8^{-1})+(d_{-2}\times 8^{-2})+...+(d_{-m}\times 8^{-m})}$$