Hexadecimal (also base 16, or hex) is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F (or alternatively a–f) to represent values ten to fifteen. Each hexadecimal digit represents four binary digits (bits), and the primary use of hexadecimal notation is a humanfriendly representation of binarycoded values in computing and digital electronics.
The Octal Number System
Data Representation
Hexadecimal
Sketches
Data Representation
The Digital number System
Data can be simply said as information stored on a computer.A computer cpu is not the same as a human brain. Hence, we must introduce data to the system in a language it understands and that is where data representation comes into scene.
4 common number systems
Binary
Octal
Decimal
Hexadecimal
Binary
Each memory location is like a switch
"
On
" is 1 and "
off
" is 0
Octal is used in computing instead of hexadecimal, perhaps most often in modern times in conjunction with file permissions under Unix systems. It has the advantage of not requiring any extra symbols as digits (the hexadecimal system is base16 and therefore needs six additional symbols beyond 0–9). It is also used for digital displays.
The number System that starts from 0 and ends at 7
Binary is used in electronics and gadgets. Since it has only 2 values that is 0 and 1. This can be manipulated to describe high and low electrical voltage to transmit data from one end to another
It is also referred to as the language of computers
I bring good luck to this presentation
: )
Binary
Binary to decimal Conversion
Decimal to Binary conversion
Binary to decimal conversion can be easily done using the exponential method
For Example:
And the same can be done using the exponential method of 2
Octal to decimal conversion
This process will be some what similar to that we see for binary conversion but the only difference will be powers of 8 instead of 2 as in Binary
Decimal to octal conversion
= 217
Hexadecimal to decimal
Converting hexadecimal values to decimal can be done using the exponents of 16
Converting Binary to hexadecimal
Binary
Decimal
Hexadecimal
Octal
Divide by 2
Multiply by 2
Divide by 16
Multiply by 16
Multiply by 8
Divide by 8
Conversions Recap
Addition and Subtraction of Number Systems
Now that we know a little about the number systems let us understand how to do simple math calculations using the various number systems
For Binary
The addition of binary numbers is simple but we must keep one thing in mind 1+1= 0 carry 1
Subtraction of Binary
Binary subtraction is performed in similar manner as that of decimal subtraction.
To subtract binary numbers we must keep 4 things in mind
Case 1: 00= 0
Case 2: 10= 1
Case 3: 1 1= 0
Case 4 101= 1
For example:
111
101
010

Addition and Subtraction using 1's
Compliment
The ones' complement of a binary number is defined as the value obtained by inverting all the bits in the binary representation of the number (swapping 0's for 1's and viceversa). The ones' complement of the number then behaves like the negative of the original number in some arithmetic operations. To within a constant (of −1), the ones' complement behaves like the negative of the original number with binary addition.
In 1's complement arithmetic, the two numbers are added including the sign bits. Let us see the various cases to get a clearer picture of binary arithmetic using one's compliment
Case 1
Positive number added to another positive number
Eg: 7+3=13
11
0,0111
0,0110
0,1101
+
Sign Bit
This type of addition is straight forward
Case 2
Addition of a positive and negative number(Subtraction)
For Eg: 73=4
1 1
0,0111
0,0110
10,1101

1
0,0100
In this example there is an overflow in the sign bit, this overflow bit is added to the least significant bit of the sum to get the correct result. This process is called end around carry
Case 3
Addition of 2 negative numbers
1
1,1000
1,1100
11,0101
1
1,0101
1's compliment representing 10
Eg: 73= 10
Adding and Subtracting Octal Numbers
In order to add two octal numbers, we proceed by adding lower order digits individually,generating carries if any and moving to higher order digits. For instance,consider the following example
2764+6435 (Both in base 8)
2764
6435
1 1 1
Carries
11421
+
Subtraction
Octal subtraction is similar to the decimal subtraction
Hexadecimal Addition
Hexadecimal calculations are similar to decimal keeping in mind the extra numbers 1015 or AF in the hexadecimal number system.
For Example:
Carry
Hexadecimal Subtraction
3 Digit Binary equivalent
4 Digit Binary equivalent
3 Digit Binary
4 Digit Binary
Thank You
Topics Covered
1. A brief Intro on major number systems other than Decimal
That includes
1. Binary
2. Octal
3. Hexadecimal
2. Conversions for all four major number systems
3. And lastly
Addition and subtraction on all major number systems
excluding the decimal number system
Number Systems
We humans have 10 fingers and that is why we mostly use a number system of base 10 which is because our Ancestors have been counting using their fingers
But computers have other number systems that are used to represent data
We can convert Decimal to octal using the division by 8 method
Decimal can be converted to hexa decimal using the division method
Binary Multiplication
Binary multiplication is a bit similar to the decimal multiplication but before we start that we must know these 4 things
0 x 0=0
0 x 1=0
1 x 0=0
1 x 1=1
For Example:
1011
101
x
0000x
1011xx
110111
101 1
+
Binary Division
Thomas Cherian
Class X1 A
Roll no: 10
Binary Division is again similar to decimal
Points to remember
0/1 = 0
1/1 =1
For Example
11001
101
101
101
101
101
0
4. One's and twos compliment
5. Data transfer and parity Checking
6. ASCII and Unicode
Addition and subtraction using 2's compliment
Now that we know 1's compliment lets see 2'd compliment
The two's complement of a number behaves like the negative of the original number
Adding 2 positive values
5+3=8
0,0101
0,0011
1 11
0,1000
Subtracting
73=4
0,0111
1, 1101
1 0,0100
Extra sign bit removed
Data transfer and Parity Checking
In communications, parity checking refers to the use of parity bits to check that data has been transmitted accurately. The parity bit is added to every data unit (typically seven or eight bits ) that are transmitted. The parity bit for each unit is set so that all bytes have either an odd number or an even number of set bits.
ASCII
The American Standard Code for Information Interchange (ASCII is a characterencoding scheme originally based on the English alphabet. It supports 256 characters (0255) included in 2 lists.
The Indian Standard Code for Information Interchange (ISCII)
is a primarily Indian based coding scheme which spans over 10 different indian scripts including Devanagari, Tamil, Assamese, Bengali etc.
Made of ASCII Values
Unicode
Unicode is the most recent and best char encoding scheme and has 110000 chars over 100 scripts. it stores chars with easy indexing of hex numbers and is the best scheme.
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