Number System

What is Number System?

In digital electronics, a number system or system of numeration, allows the information or values to be represented in the form of digits. Basically, the number system helps in representing the data within a digital computer.

In order to understand any number system, base or radix is very much important as base corresponds to the total number of individual digits any number system is having.

Introduction

In general, the numerical value of any physical quantity can be represented in either analog or digital form. In the case of analog form, the physical quantity is represented as a continuous range of values between two extreme values. While in the case of digital representation, the quantity is discretely represented in steps.

Previous researches have summarized that digital techniques and systems offer simplicity in terms of design consideration, easier data storage, high accuracy, etc. We are aware of the fact that the real world is analog in nature, implying various physical quantities like temperature, pressure, force, velocity, position, etc. are analog in nature. Due to the advantages associated with digital systems, it becomes necessary to convert analog quantities into digital ones.

The key characteristics that define a number system are:

1. The total number of separate digits in a number system.
2. The place value of different digits in a number system.
3. The highest total numbers that can be written using the given number of digits.

It is to be noted here that the total number of dependent digits that a number system holds is defined as the radix or base of the number system. Also, the place values of various digits in the integer part of a number system are given as r⁰, r¹, r^2, and so on while for fractional part, r^-1, r^-2, and so on where r corresponds to the radix of the number system.

The highest total numbers that we can write of an n digits number system are rⁿ.

Here we will discuss the major categorization of number systems,

Decimal Number System

A number system that includes 10 different digits or symbols within it is known as a decimal or radix-10 number system. The digits include in the decimal number system are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. All the numbers higher than value 9 are represented by the combination of these 9 digits.

The place value of the mixed decimal number, considering the decimal point is given as 10⁰, 10¹, 10²—- for integer and 10^-1, 10^-2—— for a fraction.

The magnitude of a given decimal number is obtained by the addition of the product of various digits with their respective place values.

Example: (1267.89)₁₀

Here, 1267 is the integer part and 89 is in fractional form.

1267 = 7 * 10⁰ + 6 * 10¹ + 2 * 10² + 1 * 10³

1267 = 7 + 60 + 200 + 1000 = 1267

0.89 = 8 * 10^-1 + 9 * 10^-2

0.89 = 0.8 + 0.09 = 0.89

This shows that the respective place values are a function of the radix of the subsequent number system and the position of the digits.

Binary Number System

The number system that includes two binary values 0 and 1 only in it is known as a binary number system. As it consists of merely 2 digits thus, it has a base of 2 thus called the radix-2 number system. However, in this also, if one needs to write higher-order numbers then similar to decimal number system, a combination of the two binary numbers is used.

Like decimal number system, the place values of binary digits are given as 2⁰, 2¹, 2² —- and 2^-1, 2^-2, 2^-3 —–

Example: (101011)₂

101011 = 1 * 2⁰ + 1*2¹ + 0*2² + 1*2³ + 0*2⁴ + 1*2⁵

101011 = 1 + 2 + 0 + 8 + 0 + 32

101011 = (43)₁₀

This number system is advantageous as any information can be easily represented in form of 0s and 1s, thereby offering ease in hardware implementation. This also simplifies the circuit that performs various arithmetic operations like addition, subtraction, multiplication, division, etc. as data is present in the form of 0s and 1s.

Octal Number System

The number system that includes 8 digits from 0 to 7 forms an octal number system. The various independent digits included in the octal number system are 0, 1, 2, 3, 4, 5, 6, 7. It consists of total of 8 digits thus also known as radix 8.

In order to represent higher digit values, the combination of various octal numbers is used similar to the binary and decimal number systems. The place values of various digits of octal numbers are 8⁰, 8¹, 8² —- and 8^-1, 8^-2, and so on.

Example: (317)₈

317 = 7*8⁰ + 1*8¹ + 3*8²

317 = 3 + 8 + 192

(317)₈ = (203)₁₀

Hexadecimal Number System

The hexadecimal number system is constituted of 16 individual digits. These include 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Here the decimal equivalent of A is 10, B is 11, C is 12, D is 13, E is 14, F is 15. Similar to the above-discussed number systems, here also, the place values are written in reference to the consideration of radix 16.

So, for the hexadecimal number system, the place value representation is such that 16⁰, 16¹, 16² —- and 16^-1, 16^-2, 16^-3, and so on. Through this number system, one can condensely write large binary numbers within the system.

Example: (A8)₁₆

A represents 10 whose binary equivalent is 1010 and the binary equivalent of 8 is 1000.

(A8)₁₆ = (10101000)₂

(10101000)₂ = 0*2⁰ + 0*2¹ + 0*2² + 1*2³ + 0*2⁴ + 1*2⁵ + 0*2⁶ + 1*2⁷

(10101000)₂ = 8 + 32 + 128

(10101000)₂ = (168)₁₀

The hexadecimal number system helps in writing the information in a complex manner by reducing large strings of binary numbers.