Transcript bin_dec_hexadecimal

The decimal system, also called the base 10 number system is based on ten
numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
All the numbers are formed by combination of these 10 numbers -> to form a 2digit number, the second digit is multiplied by 10.
15 is equal to (1 X 10) + 5.
To form a 3-digit number, the third digit is multiplied by 100. This concept
continues with the next digit multiplied by 1000, the next by 10,000, and so on. A
number like 8679 would be calculated as
(8 X 1000) + (6 X 100) + (7 X 10) + (9 X 1).
Or (8 103 )  (6 102 )  (7 101 )  (9 100 )
More generally, decimal or base 10 notation is based on the fact that any positive
integer can be written uniquely as sum of products of the form
d  (10n )
Where each n is a nonnegative integer (0 and above) and each d is one of the 10
digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
This concept is easy to grasp, because we use the decimal system in every day
applications.
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But the computers use base 2 or binary notation. The signals used in modern
electronics are always in one of only two states->0,1.
Just like in decimal any any positive integer can be written uniquely as sum of
products of the form
d  (10n )
where 10 is the base
Similarly in binary or base 2 notation, any positive integer can be written uniquely
as sum of products of the form
d  (2 n )
where 2 is the base
Example
27  16  8  2  1
 (1 2 4 )  (1 23 )  (0  23 )  (1 21 )  (1 20 ) 
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Converting binary to decimal
110111=
5
4
3
2
1
2 2 2 2 22
32 16 8
1
1
0
4
1
2
1
0
1
mutiply
1
1
2
4
0
16
32
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Add these to
get the equivalent
base 10 number
Fall 2002
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Converting decimal to binary
To convert a base 10 number into base 2, repeatedly divide the number by 2 till
the quotient is 0 and keep collecting the remainders at each step
Divisor
Quotient
Remainder
2
35
1
2
17
1
2
8
0
2
4
0
2
2
0
2
1
1
Collect these starting
from the bottom
So the binary equivalent
of 35 is
100011
0
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Divisor
Quotient
Remainder
2
0
1
2
1
0
2
2
0
2
4
0
2
8
1
2
17
1
Or reverse the position of quotient
And then collect the remainders
from top to bottom to get the
binary equivalent of 35 which is
100011
35
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Binary addition
Binary addition is similar to decimal addition in the sense that you start adding
the addends proceeding from right to left. But the numbers that you’re allowed to
use are only 0 and 1.
Rules for binary addition
0+0 = 0
0+1 = 1
1+0 = 1
1+1 = 0, carry over 1
1+1+1 = 1, carry over 1
Add 10101 + 11100
If the carry goes on beyond the maximum number of bits, then an
overflow has occurred.
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Negative number representation in binary notation
Earlier it was decided to make the leftmost bit the sign bit ->1 denotes a negative
number and 0 in that bit position indicates a positive number.
So 6 = 0110 (in binary) and
-6 = 1110
But this led to complications like if we add 6 and –6, we should get 0, but what do
we get? Try it out
To avoid this and many other complications like non unique representation of 0,
adding a negative and a positive number etc, an easier way called the two’s
complement was adopted in which leading 0’s means a positive number and
leading 1’s means a negative number .
So in two’s complement land, for a fixed 4-bit space,
we read for negative integers
0101 as
(1 2 2 )  (0  21 )  (1 20 )  5
1101 as (-1 23 )  (1 2 2 )  (0  21 )  (1 20 )  -3
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To find the 8-bit two’s complement of a positive integer a that is at most 255
•Write the 8-bit binary representation for a.
•Switch all the 1’s to 0’s and 0’s to 1’s
•Add 1 in binary notation.
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Two’s complement is extra handy when representing negative integers and
performing addition and subtraction on signed integers.
The 8-bit binary representation for a
The 8 - bit binary representa tion of a if a  0

The two’ s complement representa tion of a if a  0
This knowledge comes handy when we’ve to do subtraction or addition of two
different signed numbers
•Convert both integers to their 8 bit representations.
•Add using binary addition
•Truncate any leading 1’s that occur in the rightmost (2^8) position.
•Convert the sum back to decimal form interpreting leading o’s as +ve and
leading 1’s as -ve
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Hexadecimal notation
The cool thing about Hexadecimal (base 16) is that we save space writing long
binary numbers and also it is very easy to convert from hexadecimal to binary, since
almost all computer data are multiples of 4.
Similarly in hexadecimal or base 16 notation, any positive integer can be written
uniquely as sum of products of the form
d  (16n )
To convert from hexadecimal to binary and vice versa just remember table 1.5.3 on
page 71.
Use that table to convert
•E0D into binary
•011011111000101 into hexadecimal
E0D
1110 0000 1101
0011 0111 1100 0101
3
7
C
5
Juxtapose the results
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