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Converting to/from hexadecimal (base 16)
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-Ian! D. Allen - idallen@idallen.ca - www.idallen.com
The following link is a very good explanation and procedure for converting
hexadecimal to decimal, including handling both unsigned numbers and
two's complement negative numbers using bit flipping and adding one:
http://www.madsci.org/posts/archives/2000-02/950277263.Cs.r.html
Converting a negative two's complement number to decimal usually means
flipping all the bits (doing a one's complement) and then adding one,
and putting a minus sign in front of the positive result.
Since it's tedious to convert hex to binary, flip the bits, add one,
then multiply it out to get decimal, here is a nice table that lets you
"flip" (do a one's complement) on any hex digit directly:
One's Complement Table for Hexadecimal
0 <--> F
1 <--> E
2 <--> D
3 <--> C
4 <--> B
5 <--> A
6 <--> 9
7 <--> 8
Find the hex digit you are complementing (bit flipping) in the table,
and replace it by the other number on the same line. 4A23 becomes B5DC.
(Note that you can get this table by simply "folding" a list of 16 hex
digits in half - write the hex digits down one side and then up the
other side. The sum of any row is "F" [15].)
Once you've flipped the bits, you can complete the conversion process
by adding one to the bit-flipped result and converting to decimal.
EXAMPLE 1: Convert two's complement 16-bit hex A2C9 to decimal.
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Step 1: Check the sign. If negative, make it positive and save the sign.
A2C9 is negative (the sign bit is on, since "A" = 1010 binary)
Use the table on A2C9 to flip the bits in each hex digit,
giving 5D36 which is the one's complement,
and then add one +1
giving 5D37 (a positive number, since "5" is 0101 binary)
Step 2: multiply out the powers of 16 for each hex digit
5 times 16**3 = 5 times 4096 = 20480
D times 16**2 = 13 times 256 = 3328
3 times 16**1 = 3 times 16 = 48
7 times 16**0 = 7 times 1 = 7
Add them all up and get 23863
Step 3: put back the sign (saved from Step 1)
Answer -23863 (negative)
EXAMPLE 2: Convert unsigned 16-bit hex A2C9 to decimal.
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Step 1: Unsigned numbers are all positive. No need to check any sign.
Step 2: multiply out the powers of 16 for each hex digit
A times 16**3 = 10 times 4096 = 40960
2 times 16**2 = 2 times 256 = 512
C times 16**1 = 12 times 16 = 192
9 times 16**0 = 9 times 1 = 9
Add them all up and get TOTAL 41673
Step 3: put back the sign (saved from Step 1)
Unsigned numbers are always positive.
Answer 41673
EXAMPLE 3: Convert two's complement 20-bit hex A2C9 to decimal.
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Step 1: Check the sign. If negative, make it positive and save the sign.
A2C9 is actually 0A2C9 in hex, since we are told it is 20-bits wide.
Because the sign bit is off (since "0" = 0000 binary), this is
a positive number. No need to flip any bits.
Step 2: multiply out the powers of 16 for each hex digit
0 times 16**4 = 0 times 65536 = 0
A times 16**3 = 10 times 4096 = 40960
2 times 16**2 = 2 times 256 = 512
C times 16**1 = 12 times 16 = 192
9 times 16**0 = 9 times 1 = 9
Add them all up and get TOTAL 41673
Step 3: put back the sign (saved from Step 1)
This is a positive number.
Answer 41673
EXAMPLE 4: Convert two's complement 16-bit hex 1A8C to decimal.
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Step 1: Number is positive, since "1" is 0001 binary.
Step 2: multiply out the powers of 16 for each hex digit
1 times 16**3 = 1 times 4096 = 4096
A times 16**2 = 10 times 256 = 2560
8 times 16**1 = 8 times 16 = 128
C times 16**0 = 13 times 1 = 13
Add them all up and get TOTAL 6796
Step 3: put back the sign (saved from Step 1)
This is a positive number.
Answer 6796
EXAMPLE 5: Convert two's complement 16-bit hex A123 to decimal.
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Step 1: Check the sign. If negative, make it positive and save the sign.
A123 is negative (the sign bit is on, since "A" = 1010 binary)
Use the table on A123 to flip the bits in each hex digit,
giving 5EDC which is the one's complement,
and then add one +1
giving 5EDD (a positive number, since "5" is 0101 binary)
Step 2: multiply out the powers of 16 for each hex digit
5 times 16**3 = 5 times 4096 = 20480
E times 16**2 = 14 times 256 = 3584
D times 16**1 = 13 times 16 = 208
D times 16**0 = 13 times 1 = 13
Add them all up and get 24285
Step 3: put back the sign (saved from Step 1)
Answer -24285 (negative)
EXAMPLE 6: Convert unsigned 16-bit hex A123 to decimal.
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Step 1: Unsigned numbers are all positive.
Step 2: multiply out the powers of 16 for each hex digit
A times 16**3 = 10 times 4096 = 40960
1 times 16**2 = 1 times 256 = 256
2 times 16**1 = 2 times 16 = 32
3 times 16**0 = 3 times 1 = 3
Add them all up and get 41251
Step 3: put back the sign (saved from Step 1)
Answer 41251
See Also
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Any Base Converter: Convert numbers in any base up to 32:
http://www.cut-the-knot.org/binary.shtml