Touchpad Plus Ver. 3.1 Class 7: Linux & LibreOffice

1 NUMBER SYSTEM

Although the words data and information are often used interchangeably, but there is an important distinction between the two words. In the strict sense, data consists of the raw numbers that computers organise to produce information.

From early age, we are introduced to the concept of numbers and counting. Toddlers learn at an early age that they can carry two cookies, one in each hand. Kindergarteners start counting by twos and fives. Invariably, we use the decimal number system. Our number system is based on 10, most likely because we have 10 fingers. Let us learn more about the number system.

Number System

A number system is simply a method of counting. Many different number systems exist. Consider a clock. Clocks have 24 hours, each hour composed of 60 minutes. Each minute is in turn composed of 60 seconds. When you learnt to count, you used the numbers, like 1, 2, 3, etc. Similarly, computers also have their own number system, known as the binary number system.

The digital computer represents all kinds of data and information like audio, graphics, video, text and numbers in binary form. The total number of digits used in a number system is called its base or radix. Therefore, when someone says that they are working with number system of radix 2, it means base 2, that is, binary number system. The base is written after the number as subscript such as (512)10. In this example, the number 10 is written as the subscript to express the number in decimal number system.

There are four types of number systems which are as follows:

Decimal number system

Binary number system

Octal number system

Hexadecimal number system

The decimal number system is used in general. However, the computers use binary number system. The octal and hexadecimal number systems are also used in the computer.

Decimal Number System

The decimal number system is a standard number system for denoting numbers. It consists of ten digits from 0 to 9. Only these digits can be used to represent any numeric value in decimal number system. Hence, the base of decimal number system is 10. The decimal number system is the most widely used number system. The value represented by individual digit depends on the weight and position of the digit.

The position of first digit towards left side of the decimal point is 0. The position of second digit towards left side of the decimal point is 1. Similarly, the position of first digit towards right side of decimal point is –1. The position of second digit towards right side of decimal point is –2, 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 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 Most Significant Digit (MSD). The digit 7 in the number 724 is the most significant digit and 4 is the least significant digit.

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

Binary Number System

The word binary comes from 'Bi-' meaning two. We see 'bi-' in words such as ‘bicycle‘ (two wheels) or ‘binocular’ (two eyes). The binary numbers have the base of 2.

A computer is a machine made up of transistors, switches and other components. All these electronic components are in two mutually exclusive states, either ON or OFF. The two binary digits represent these two states. Every instruction given to a computer is converted into 0’s and 1’s so that it can be understood and implemented by the computer. Binary language is therefore known as the machine language.

A binary number is made up of only 0s and 1s.

Example of Binary Number: 110100

There is no 2, 3, 4, 5, 6, 7, 8 or 9 in Binary!

For Example: 1101 represents an odd number (13); 10010 represents an even number (18)

Let us first learn how to form binary numbers.

As the binary number system consists of two digits 0s and 1s hence, its base is 2. Each digit or bit in 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 binary number is twice the place value or face value of the digit of 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:

Octal Number System

The octal number system consists of eight digits from 0 to 7. Hence, the base of octal number system is 8. In this system, the position of each digit represents a power of 8. Any digit in this system is always less than 8. Octal number system is used as a shorthand representation of long binary numbers. The number (841)8 is not valid in this number system as 8 is not a valid digit.

Hexadecimal Number System

The hexadecimal number system consists of 16 digits from 0 to 9 and A to F. The letters A to F represent decimal numbers from 10 to 15. The base of this number system is 16. Each digit position in hexadecimal number system represents a power of 16. The number (764)16 is valid hexadecimal number. It is different from (764)10 which is seven hundred and sixty four. This number system provides shortcut method to represent long binary numbers.

Number System Conversion

Value of one number system can be converted into another. For example, decimal numbers can be converted into binary numbers and vice-versa.

Decimal to Binary Conversion

To convert a decimal number into a binary number, follow these steps:

Step 1 Divide the decimal number by 2 (the base of the binary number system).

Step 2 Note down the quotient and the remainder.

Step 3 Divide the quotient obtained again by 2 and note down the resulting quotient and remainder.

Step 4 Repeat the procedure till you reach a quotient less than 2.

Step 5 List the last quotient and all the remainders (moving from bottom to top). You will get your binary number.

Look at the given examples to better understand the conversion process.

Example 1: Convert the decimal number 26, i.e., (26)10 to binary.

Example 2: Convert the decimal number (64)10 to binary.

Binary to Decimal Conversion

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

Step 1 Start from the right-most digit known as the LSD before the fractional point, and move leftwards.

Step 2 While doing so, multiply each digit by 2 raised to a particular power. The powers of 2 start from 0 and increase to 1, 2, and so on as you move leftwards.

Step 3 Add up all the resulting products. You will get your decimal number.

The following examples will help you to understand the conversion.

Example 1: Convert (1111)2 to decimal.

= 1 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰

= 8 + 4 + 2 + 1

= 15

Example 2: Convert (10111)2 to decimal.

= 1 × 2⁴ + 0 × 2³ + 1 × 2² + 1 × 2¹ + 1 + 2⁰

= 16 + 0 + 4 + 2 + 1

= 23

(1111)2 = (15)10 (10111)2 = (23)10

Octal to Decimal Conversion

Similar to binary number, you can convert an octal number into decimal number by multiplying each digit of the number with the increasing powers of 8 starting from right-hand side. For example,

Example 1: Convert (37)8 into decimal.

(37)8 = 3 × 8¹ + 7 × 8⁰

= 24 + 7

(37)8 = (31)10

Example 2: Convert (467)8 into decimal.

(467)8 = 4 × 8² + 6 × 8¹ + 7 × 8⁰

= 4 × 64 + 6 × 8 + 7 × 1

(467)8 =(311)10

Hexadecimal to Decimal Conversion

You can convert a hexadecimal number into decimal number by multiplying each digit of the number with the increasing powers of 16 starting from right-hand side. For example,

Example 1: Convert (3B)16 into decimal.

3B16 = 3 × 16¹ + 11 × 16⁰

= 48 + 11

(3B)16 = (59)10

Example 2: Convert (3A4)16 into decimal.

(3A4)₁₆ = 3 × 16² +10 × 16¹ + 4 × 16⁰

= 768 + 160 + 4

(3A4)₁₆ = (932)₁₀

Operations on Binary Numbers

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, 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.

For example, let us subtract the binary number (111)2 from (1000)2.

Binary Multiplication

Binary numbers are multiplied in the same manner as decimal numbers. When two binary numbers are multiplied, for each digit of the second number, the product of that digit of the first binary number is calculated. Then it is shifted leftwards so that its rightmost digit lines up with the digit in second number that