Trackpad Ver. 2.0 Class 7

CHAPTER 1

NUMBER SYSTEM

Hello, Shweta! Welcome to class VII.

Thank you, Techie! We missed you during the break. Now we are fresh and eager to begin the new session. What are we going to discuss today?

My dear Shweta, throughout your primary classes, your teachers have made you count from 1 to 100, 101 to 200, and so on. This system of representing numbers is known as the decimal number system. However, this is not the only system available to us. There are many more.

That does sound interesting, Techie, but we are confused. Please tell me in detail.

Maintain that curiosity, kid! You’ll find out everything in this chapter. Let’s get started!

DATA REPRESENTATION

The smallest piece of data that can be recognised and used by the computer is known as the bit or binary digit. A bit is a single binary value i.e., 1 or 0. A computer is an electronic device that has two states: On and Off. These two states of the computer are represented by two digits: 1 and 0. Here, 1 represents the electronic state On, and 0 represents the electronic state Off.

NUMBER SYSTEMS

A number system is a way to express quantities used for counting, comparing amounts, performing calculations and representing values. A computer represents all kinds of data and information like text, numbers, audio and video in binary form. The total number of digits used in a number system is called its base or radix. There are four types of number systems: Binary number system, Decimal number system, Octal number system and Hexadecimal number system.

DECIMAL NUMBER SYSTEM

A number system made up of 10 digits from 0 to 9, is known as decimal number system. The base of the decimal number system is 10. It is also known as the base-10 system.

In the decimal number system, the position of the first digit towards the left side of the decimal point is 0. The position of the second digit towards the left side of the decimal point is 1. Similarly, the position of the first digit towards the right side of the decimal point is –1. The position of the second digit towards the right side of the decimal point is –2, and so on.

In simple words, the successive position to the left of the decimal point represents units, tens, hundreds, thousands, and so on.

The value of a number is determined by multiplying the digits with the weight of their position and adding the results. This method is known as the expansion method. The right-most digit of the number has the lowest weight. This digit is called Least Significant Digit (LSD). The left-most digit of a number has the highest weight. This digit is called the Most Significant Digit (MSD).

The digit 7 in the number 724 is the most significant digit and 4 is the least significant digit.

The table shows the positional weight of decimal number (724)10.

For Example: (3698)10

8 signifies 8 x 10⁰= 8

9 signifies 9 x 10¹ = 90

6 signifies 6 x 10² = 600

3 signifies 3 x 10³ = 3000

Upon adding them = 3000 + 600 + 90 + 8 = 3698

BINARY NUMBER SYSTEM

A number system made up of only two digits: 0 and 1, is known as the binary number system. When the binary number system is used, every number is formed using only 0 and 1. The word binary comes from ‘Bi-’ meaning two that’s why the base of the binary number is 2. It is also known as the base-2 system.

Formation of Binary numbers

As you know the binary number system consists of two digits: 0 and 1 and its base is 2. Each digit or bit in a binary number system can be 0 or 1. A combination of binary digits may be used to represent different quantities like 1001. The positional value of each digit in a binary number is twice the place value or face value of the digit on its right side. The weight of each position is a power of 2. The place value of the digits according to position and weight is as follows:

For example: 10101 or (10101)2 is

= (1 x 2⁴) + (0x 2³) + (1x 2²) + (0x 2¹) + (1x 2⁰)

= (1 x 16) + (0 x 8) + (1 x 4) + (0 x 2) + (1 x 1)

= (21)10

OCTAL NUMBER SYSTEM

A number system made up of eight digits from 0 to 7, is known as the octal number system. When the octal number system is used, every number is formed using 0,1,2,3,4,5,6 and 7. The base of the octal number system is 8. It is also known as the base-8 system. Each positioning number represents the power of base 8.

For example: (1763)8

= (1 x 8³1) + (7 x 8²) + (6 x 8¹) + (3 x 8⁰)

= (1 x 512) + (7 x 64) + (6 x 8) + (3 x 1)

= 512 + 448 + 48 + 3

= 1011 or (1011)10

HEXADECIMAL NUMBER SYSTEM

A number system made up of sixteen symbols, 0 to 9, and A to F is known as the hexadecimal number system. When the hexadecimal number system is used, every number is formed using 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F, where A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15. The base of the hexadecimal number system is 16. It is also known as the base-16 system. Each position represents a power of base 16.

For Example: (2AF)16

= (2 x 16²) + (A x 16¹) + (F x 16⁰)

= (2 x 256) + (10 x 16) + (15 x 1)

= 512 + 160 + 15

= 687

The hexadecimal number system has made the representation of large values easy. The hexadecimal numbers are used to represent colours on a webpage, that's why programmers now prefer hexadecimal numbers. Some examples of colour code are:

CONVERSION OF DECIMAL TO BINARY

To convert a decimal number into a binary number, follow the given steps:

Step Divide the number to be converted by 2 while keeping track of the quotient and remainder.

Step Continue dividing the quotient by 2 until you get a quotient of 0.

Step Then write the remainder in the reverse order.

Example: (126)10

The binary equivalent of 126 is 1111110 or (126)10 = (1111110)2

CONVERSION OF BINARY TO DECIMAL

To convert a binary number into a decimal number, follow the given steps:

Step Multiply each digit of the binary number by 2 to the power of n, where n is the position of the digit starting from 0 on the right.

Step Add the result.

Example: Convert (101001)2 to decimal number.

= 1 x 2⁵ + 0 x 2⁴ + 1 x 2³ + 0 x 2² + 0 x 2¹ + 1 x 2⁰

Sum of the products = 32 + 0 + 8 + 0 + 0 + 1 = 41

Therefore, (101001)2 = (41)10

OPERATIONS ON BINARY NUMBERS

We can perform various operations on binary numbers. Let's discuss binary addition and binary subtraction in detail.

BINARY ADDITION

Binary addition is similar to the addition of decimal numbers. When the value of addition exceeds the value 1, say 10 or 11, then 1 is carried over to the left of the current position. The rules for adding two binary digits are given below:

For example, let us add the binary numbers (101111)2 and (10111)2.

BINARY SUBTRACTION

In binary subtraction, the binary number of lower value is subtracted from the binary number of higher value. The following table explains the subtraction of digit Y from digit X. If Y is greater than X, then 1 is borrowed from the next position. When the binary digit 0 borrows 1 from the next most significant digit, it becomes 10. The