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Computer Number System
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The Number System
A number system is a way to represent numbers.
Need for different Number Systems
Intro
We are already familiar with the decimal number system.It has 10 unique digits but the computer understands only 0s and 1s (binary data).
Data is stored in computers in long strings of 0s and 1s, which becomes confusing to read and interpret.
To make it easier for the programmers, data is represented using other number systems.
Multiple Systems
So What Are The Other Number Systems?
There are multiple number systems with different base.But the 4 common number systems are:
- Decimal Number System
- Binary Number System
- Octal Number System
- Hexadecimal System
What is a Base?
Base is the number of unique digits in a particular number system.
Decimal Number System
This is the most commonly used number system. It consists of digits from 0-9 ,giving it a base of 10.
Each position in a number system is represented by a power of 10.
An expanded decimal number would look as follows:
Decimal
Binary
Binary Number System
It is a number system using which the computer saves its data. It consists of 2 digits 0 & 1. Hence the base becomes 2.
eg: 110,1011,111011
An expanded binary number would look as follows:
MSB and LSB
The digits 0 and 1 are known as binary digits or bits.
In binary numbers, the bit farthest to the left is called MSB (Most Significant Bit) as it has the highest place value and the bit farthest to the right is called LSB (Least Significant Bit) as it has the lowest place value.
Octal
Octal Number System
This number system uses 8 digits(0-7) to represent data.Its base is 8.
eg: 327,443,110
It was devised for compact representation of binary numbers.
Hexadecimal
Hexadecimal Number System
- This number system uses 16 digits(0-9 and the letters A-F) to represent data.Its base is 16.
eg: 176,DA1,110,AC
- Some areas where hexadecimal number system is used are:
- To represent the address of a memory location.
- To define color codes used on web pages.
- To display system error messages.
- An expanded Hexadecimal number would appear as follows:
Conversions
Conversions
- from decimal to the other number systems
To Convert from Decimal to Binary
To Convert From Decimal To Binary
This conversion uses repeated division method. Steps to be followed:
- The decimal number is repeatedly divided by 2 and the remainder is recorded. This process is continued till we reach a zero quotient.
- The result of the conversion is obtained by writing the remainders from the bottom to the top.
- The last number (at the bottom) is the most significant bit (MSB).
To Convert From Decimal To Octal
To Convert From Decimal To Octal
This conversion also uses repeated division method.
- In this method, the decimal number is repeatedly divided by 8 and the remainder is recorded.
- This process is continued till we reach a zero quotient. The result of the conversion is obtained by writing the remainders from the bottom to the top.
To Covert from Decimal to Haxadecimal
To covert from Decimal To Hexadecimal
This conversion also uses repeated division method.
- In this method, the decimal number is repeatedly divided by 8 and the remainder is recorded.
- This process is continued till we reach a zero quotient.
- The result of the conversion is obtained by writing the remainders from the bottom to the top.
Conversions
Conversions
continued...
- from Binary to Decimal
- from Binary to Octal
- from Octal to Binary
To Convert from Binary to Decimal
To Convert From Binary to Decimal
- Find the positional values of all the digits of the given number.
- Add all the positional values.
To Convert From Binary To Decimal
For this example lets convert the binary number (10011011)2 to decimal number.
Multiply the individual numbers in the binary number from right to left with 2 and increasing the power from right to left starting from the power 0. After multiplying all add the products and you will obtain the converted decimal number.
Binary equivalents for 0-7
To Convert From Binary To Octal
To Convert From Binary To Octal
To Convert From Octal to Binary
To Convert From Octal to Binary
Computer Arithmetic
- In a computer, CPU converts the data into binary code and then performs arithmetic operations on it.
- The four types of arithmetic operations using the binary digits are:
- Binary Addition
- Binary Subtraction
- Binary Multiplication
- Binary Division
Binary Addition
Following rules are followed by a computer for binary addition
Binary Addition
Binary Subtraction
Following rules are followed by a computer for binary subtraction
Binary Subtraction
Binary Multiplication
Following rules are followed by a computer for binary multiplication