# Number System Study Notes for GATE & Computer Science Engineering Exams

By BYJU'S Exam Prep

Updated on: September 25th, 2023

Are you preparing for the GATE or Computer Science Engineering exams? Gain a strong foundation in the essential topic of the Number System with our comprehensive study notes. In these study materials, we delve deep into the intricacies of the Number System, equipping you with the knowledge and skills necessary to excel in your exams.

Our Number System Study Notes are meticulously crafted to cater specifically to the needs of GATE and Computer Science Engineering aspirants. Understanding the Number System is crucial for a variety of topics in computer science, such as data representation, digital logic design, and algorithms. With our study notes, you will grasp the core concepts of binary, decimal, octal, and hexadecimal systems, enabling you to tackle complex problems with ease. Whether you are a beginner or seeking to strengthen your existing knowledge, our study notes will empower you to master the intricacies of the Number System and confidently tackle exam questions.

Table of content

## Introduction to Number Systems

In the world of computer science, understanding number systems is fundamental. From binary to hexadecimal, number systems play a crucial role in data representation, digital logic design, and algorithmic operations. Whether you’re preparing for the GATE or Computer Science Engineering exams, this comprehensive guide will equip you with the knowledge and skills to tackle number system-related questions with confidence.

1. The Basics of Number Systems: Dive into the fundamentals of number systems, including the decimal system, binary system, octal system, and hexadecimal system. Learn how to convert between different number systems and gain a solid understanding of their unique properties.

2. Binary System Unraveled: Explore the binary number system in-depth, which lies at the heart of computer systems. Understand binary arithmetic, logical operations, and binary-coded representations. Master the art of bitwise operations and binary manipulation for efficient computing.

3. Octal and Hexadecimal Systems: Delve into the octal and hexadecimal number systems, which offer compact representations of binary numbers. Discover their significance in computer memory, addressing schemes, and digital logic circuits. Learn conversion techniques and how to apply these systems effectively.

4. Number Systems in Data Representation: Learn how number systems are used to represent various data types, such as integers, floating-point numbers, and characters. Understand the principles of signed and unsigned number representations, as well as the impact on arithmetic operations and data storage.

5. Number Systems and Algorithms: Gain insights into how number systems intersect with algorithm design and analysis. Explore topics like number theory, modular arithmetic, and prime numbers, and understand their implications for cryptographic algorithms and data encryption.

• The base or radix of a number system is the number of digits present. The decimal numeral system has a base or radix of 10, where the set of 10 symbols (digits) is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. The weights are in powers of ten.
• In general, a bases-b number (an an-1 … a0 . f1 f2 … fm)b has the value
• (an × bn) + (an-1 ×bn-1) +…+ (a0 × b0) + (f1 × b-1) + (f2 × b-2) +…+ (fm × b-m)

## Weighted Binary System

These binary codes are those which follow the positional weighting principle, In which each position of a number represents a specific weight.

e.g., 8421, 2421, 5421

## Sequential Code

A code is said to be sequential when two subsequent codes, seen as numbers in binary representation, differ by one. The 8421 and excess-3 codes are sequential, whereas the 2421 and 5421 codes are not.

## Non-weighted Codes

Non-weighted codes are codes which are not positionally weighted. That is each position within the binary number isn’t assigned a hard and fast value, Example: Gray Code

## Reflective Code

A code is said to be reflective when a code for 9 may be a complement for the code for 0 and so is for 8 and 1 codes, 7 and 2, 6 and 3, and 5 and 4. Codes 2421, 5421 and excess-3 are reflective, whereas the 8421 code is not.

## BCD (Binary Coded Decimal)

It is a direct assignment of the binary equivalent. Thus encode a decimal number using the common BCD encoding. Each digit is stored during a a4-bit number

BCD encoding for number 127 would be

1 2 7

(0001 0010 0111) → BCD equivalent of 127

whereas the pure binary number would be (01111111)2

BCD

Add (148 + 157) = ?

When the sum of two digits is bigger than or adequate to9, then we’d like to feature then we’d like to feature6 i.e., 0110

### 2421 Code

This is often weighted code, its weights are 2, 4, 2 and 1. A decimal number is represented in 4-bit form and therefore the total 4 bits weight is 2 + 4 + 2 + 1 = 9.

Hence, the 2421 code represents the decimal numbers which are from 0 to 9.

### Excess-3 Code

Excess-3 code is a non-weighted code usually used to represent decimal numbers. The code receives its name from the fact that each binary code is the correspondence of 8421 code plus 0011 (3).

e.g.,

### Gray Code

This is variable, weighted code and is cyclic in nature. This means that it is arranged in order so that every transition from one value to the next value involves only a one-bit Change.

### Binary to Gray Code Conversion

1. Write down the number in binary codes.
2. The Most Significant Bit (MSB) of the grey code will be the same as the MSB of binary code.
3. Perform XOR operation on MSB and next bit to the MSB in the binary number.
4. Repeat step 3 till all bits of binary number have been XO Red, the resultant code is the grey code equivalent to binary code.

### Gray Code to Binary Conversion

2. Copy this bit as the MSB of the binary number.
3. Now, perform the Ex-OR operation of this bit with the next bit of the binary number.
4. Repeat step 3 till all bits of grey coded number have been used in the XOR operation. The resultant number is the binary equivalent of the grey number.

## Complements

Complements are used in the digital computer system for simplifying the subtraction operation and for logical manipulation.

>>types of complements for each base r system as,

1. The r′ s complement.
2. The (r′ – 1)′s complement.

### The r’s Complement

Here given no is a positive number N with base r with an integer part of n digits. The r’s complement of N is given as rn – N for N ≠ 0 and 0 for N = 0.

e.g., 10’s complement of (25.639)10 is (102 – 25.639)

100 – 25.639 = 74.361, here the number of digits in the integer part is 2

means n = 2

### The (r – 1)′s Complement

Given, a positive number N with base r with an integer part of n digits and a fraction part of m digits, the (r – 1)’s complement of N is derived as rn – r–m – N.

• 9’s complement of (52520)10 is (105 – 1 – 52520)
= 99999 – 52520 = 47479
Because the number of integer parts is 5, so rn = 105 and no fractional part is
Present’ so r–m = 10–0 = 1
• 9’s complement of (0.3267)10 is (100 – 10–4 – 0.3267)
= 1 – 0.0001 – 0.3267
= 0.9999 – 0.3267 = 0.6732
No integer part, so 10n = 100 = 1
• 1’s complement of (101100)2 is (26 – 20)10 – (101100)2
= (64 – 1)10 – (101100)2
= (63)10 = (101100)2
= 111111 – 101100 = 010011

Key Points

• 520 → Here, n = 3, but (052) → here n = 2.
• In the latter example, 0 is of no significance.

### Representation of Integers

The are three possible ways to represent a number

1. Signed magnitude method
2. 1’s complement method
3. 2’s complement method

## Signed Magnitude Method

The number is divided into two parts, one is the sign bit and the other part for magnitude, In the example we are using the 5-bit register to represent –6 and +6.

Range of Number For the n bit register, MSB will be a sign bit and (n – 1) bits will be the magnitude.

Key Points

• The drawback of the signed magnitude method is that 0 will be having 2 different representations one will be 10000 i.e., –0 and the other one will be 00000 + 0.

## 1’s Complement Method

Positive numbers are represented in the same way as in the sign-magnitude method. If the number is negative, then it is represented by the help of 1’s complement method. For this, we need to represent the number with a positive sign and then take its 1’s complement of this number.

e.g., Suppose we are using the 5-bit register. The representation of –6 will be as below.

Key Points

• The only drawback of 1’s complement method is that there are two different representations for zero, one is –0 and the other is +0.

## 2’s Complement Method

Positive numbers are represented in the same way as in sign magnitude. For representing a negative number, we take 2’s complement of the corresponding positive number.

### Properties of 2’s Complement

This representation usually uses binary arithmetic operations on signed integers, resulting in the current 2’s complement result.

• Positive Numbers Positive 2’s complement numbers are represented as the simple binary.
• Negative Numbers Negative 2’s complement numbers are represented as the binary number that when added to a positive number of the same magnitude equals zero.

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