Binary and Hexadecimal Number
Systems
A number system of base r, is a system that uses distinct symbols for r digits.
Deecimal system (base 10) uses 10 digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Binary system (base 2) uses 2 digits: {0 and 1}
Hexadecimal system (base 16) uses 16 digits:
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}
The value of a number in base r is obtained by multiplying each digit by an integer power
of r and adding all the weighted digits. For representations of positive integers, the
rightmost digit has weight r0, the next digit has weight r1, the next digit has weight r2 and
so on.
Examples:
Base 10: 1234 = 1X103 + 2X102 + 3X101 + 4X100
Base 2: 10010=1X24 + 0X23 + 0X22 + 1X21 + 0X20 = 16+0+0+2+0=(18)10
01101101= 0X27+1X26+1X25+0X24+1X23+1X22+0X21+1X20=
0+64+32+0+8+4+0+1 = (109)10
Base 16: 1A6 = 1X162 + 10X161 + 6X160 = 256+160+6 = (422)10
6D = 6X161 + 13X160 = 96 +13 = (109)10
Converting from positive decimal integer to binary/hexadecimal
The conversion of a decimal integer to binary is done by successive divisions by 2 and
accumulation of the remainders. This gives the magnitude of the integer. Similarly,
conversion of a decimal integer to hexadecimal is done by successive divisions by 16 and
accumulation of the remainders.
Examples
Converting between binary and hexadecimal (positive integers)
Each hexadecimal digit can be represented by a 4-bit binary number whose value is equal
to that of the digit. To convert from hexadecimal to binary, simply replace each digit by
the corresponding 4-bit binary number. To convert from binary to hexadecimal, divide
the bits into groups of four (starting with the rightmost) and replace each 4-bit binary
number by the corresponding hexadecimal digit. Additional 0's may be added to the left
of the number.
For example
2's complement arithmetic
In order to represent both positive and negative binary numbers, we must include a sign
bit. This can be done by using the 2's complement representation as follows:
(i) For positive integers, simply put a 0 in front of the magnitude. For example, +29 = 0
11101, +87 = 0 1010111 and +142 = 0 10001110.
(ii) For negative integers, first obtain the corresponding positive integer and then negate it
by inverting and adding 1. For example, the 2's complement representations for the
decimal -29, -87 and -142 are given below.
If we negate a negative number by inverting and adding 1, we will get back the original
positive number.
Rules for addition
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An n-bit number in 2's complement representation contains a total of n bits,
including the sign bit. For example 1 0101001 is the 8-bit 2's complement
representation of -87.
When adding two n-bit numbers, the result should also be n bits. Any extra bits to
the left are discarded.
The position of the sign bit does not change.
If after adding two positive numbers, the result is negative or after adding two
negative numbers, the result is positive, then the corresponding result is invalid.
Extra 0's may be added to the left of a positive number and extra 1's may be added
to the left of a negative number without changing its value.
Subtraction is equivalent to adding the negation (2's complement) of a number.
Examples:
The divisor in a CRC calculation is called the polynomial, or poly. In the case of CRCs,
the poly is nothing more than a binary number. It gets its name because the number
represents a polynomial with binary coefficients. For example:
1*(x7)+0*(x6)+0*(x5)+1*(x4)+1*(x3)+0*(x2)+1*(x1)+1*(x0) = 10011011
Polys come in various sizes. The more popular polys use 16 or 32 bit lengths. The length
of a poly is determined by the position of the leftmost 1 bit.
There are many popular polys in use, or you can create your own. However, some polys
are better at identifying errors or differences in data than others. With this in mind, it's a
good idea to use one of the more time-tested polys than one you create.
A popular 32 bit polynomial is (0x04C11DB7), which is used in PKZip, Ethernet and
FDDI. By searching on the Internet, you can find more polys that you can use in your
CRCs.
Four versions of polynomials are widely used:
CRC-12
= X 12 + X 11 + X 3 + X 2 + X + 1
CRC-16
= X 16 + X 15 + X 2 + 1
CRC-CCITT = X 16 + X 12 + X 5 + 1
CRC-32
= X 32 + X 26 + X 23 + X 22 + X 16 + X 12 + X 11 + X 10 + X 8 + X 7 +
X 5+ X 4 + X 2 + X + 1
Error:
An error E(x) will only be undetectable if it is divisible by P(X). It can be shown that all
of the errors are not divisible by a suitably chosen P(x) and hence are detectable:
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All single-bit errors
All double-bit errors, as long as P(x) has at least three 1s.
Any odd number of errors, as long as P(x) contains a factor (X + 1)
Any burst error for which the length of the burst is less than the length of the
divisor polynomial; that is , less than or equal to the length of the FCs
Most larger burst errors
How CRCs Work
The purpose of error detection is to notify the receiver that the message received is
different than the message transmitted by the sender. A simple way to detect errors after a
file transfer is to compare a checksum calculated before the transfer with a checksum that
is generated after the transfer. The advantage of a CRC is that it is virtually impossible
for a random change in a block of data to still generate the same checksum.