MadSci Network: Computer Science

Re: How do I convert the number A432 from hexadecimal to decimal?

Date: Thu Feb 10 18:21:52 2000
Posted By: Mike Westerfield, Staff, Computer Science, Byte Works, Inc.
Area of science: Computer Science
ID: 949878043.Cs

There is a possible complication when converting hexadecimal values to 
decimal. Lets start with the simple case, when the hexadecimal value 
represents an unsigned decimal value, and deal with the possibility of a 
signed number later.

Hexadecimal numbers work the same way as the familiar decimal numbers, but 
there are 16 digits instead of 10, so each column goes up by a power of 16.

For a four-digit number in base 10 we multiply the rightmost digit by 1, 
which is 10 raised to the power 0. Working towards the left, the next digit 
is multiplied by 10, or 10 raised to the power 1. The third digit is 
multiplied by 100, or 10 squared, and the final digit by 1000, or 10 cubed.

Transferring that knowledge to base 16 numbers, the rightmost digit is still 
multiplied by 1, although technically it’s 16 raised to the power 0 this 
time. The second digit from the right is multiplied by 16, the third from 
the right by 256 (16*16), and the leftmost digit by 4096 (16*16*16).

The additional piece of knowledge you need is the values for each of the 
hexadecimal digits. Digits 0 through 9 represent the same values as their 
decimal counterparts. For the digits 10-15, we use the letters A-F, so A is 
10 and F is 15.

Incidentally, using letters is becoming a standard for bases larger than 10. 
Some languages, like Ada, allow you to specify numbers in pretty much any 
base you like. For base 20 numbers, Ada uses A-J for the values 10-19. This 
convention allows you to use any base up to base 36.

Getting back to your specific example, then, A432 is

   10 * 4096 + 4 * 256 + 3 * 16 + 2

which works out to 42034 decimal.

There is one possible gotcha. If the number you are converting is a 
hexadecimal representation of a number extracted from the bowels of a 
computer’s memory, you have to know if the number is signed or unsigned. For 
a four-digit unsigned hexadecimal value, the possible values of 0000 to FFFF 
convert to the decimal numbers 0 to 65535. That doesn’t allow for negative 
numbers, though. The shortest integer in most programming languages is still 
represented by four hexadecimal digits, but a special way of coding the 
numbers is used. It’s called two’s complement notation, and for a four-digit 
hexadecimal value you get decimal numbers from -32768 to 32767. Either way, 
there are 65536 possible values, but one way they are all positive or zero, 
and the other way half are negative.

Here’s how two’s complement notation works:

Any hexadecimal number with the most significant bit set is a negative 
number. As you probably know, each hexadecimal digit is actually four bits 
in the computer. The decimal, hexadecimal, and binary values for the 16 
hexadecimal digits are:

    0    0    0000
    1    1    0001
    2    2    0010
    3    3    0011
    4    4    0100
    5    5    0101
    6    6    0110
    7    7    0111
    8    8    1000
    9    9    1001
   10    A    1010
   11    B    1011
   12    C    1100
   13    D    1101
   14    E    1110
   15    F    1111

Any number from 0000 to 7FFF converts to decimal just like I described 
earlier. Numbers from 8000 to FFFF have the high bit set (which means the 
leftmost bit in the binary representation is one), so they are actually 
negative numbers.

There are two ways to convert negative two’s compliment hexadecimal numbers 
to decimal.

The first is the way the computer does it. To convert negative hexadecimal 
numbers to decimal, start by flipping all of the bits--each 0 becomes a 1, 
and each 1 becomes a 0. The result is the "complement" of the starting 
number. You’ll need a table to figure out what digits to substitute. You can 
work this table out for yourself from the one above, but I’ll save you the 

    0    F
    1    E
    2    D
    3    C
    4    B
    5    A
    6    9
    7    8

Find the number you are complementing in the table, and replace it by the 
other number on the same line. A432 becomes 5BCD.

The second step is to add one. 5BCD becomes 5BCE.

Convert the resulting hexadecimal number to decimal, and put a minus sign in 
front. 5BCE is 23502 decimal, so the value A432 is -23502.

The second method is to convert the original number to decimal, then 
subtract 65536, which is 2 raised to the power 16. Sixteen is also, you’ll 
notice, the number of bits in a four-digit hexadecimal number. A432 converts 
to decimal 42034, and 42034 - 65536 is, you guessed it, -23502.

Ok, so which is it? is A432 really 42034 or -23502? The actual answer 
depends on how you are using the value. The hexadecimal value can represent 
either number; you have to know some other way whether you are using 
unsigned numbers, which means the decimal value is 42034, or two’s 
compliment numbers, in which case the decimal value is -23502.

As an obscure aside, two’s compliment notation is really just a tricky way 
to handle negative numbers in a binary world that really deals only with 
positive values and zero. It’s a code, if you like. It’s far and away the 
most common code used to represent signed integers on modern digital 
computers, but it isn’t the only one. It’s been a long time since I used a 
computer that used something else, though, so unless you have some reason to 
believe your situation is really odd, it’s pretty safe to assume that any 
hexadecimal value that represents a signed decimal integer is using two’s 
compliment notation to do it.

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