[Courses] C Programming For Absolute Beginners, Lesson 2: Fun With Printf, Scanf, Puts, and Variables

[Kevin Cole]

Oh what the heck. ;-)

Negative binary numbers:  If you only have a limited number of little
magnets (or transistors or whatever the frig they are) to represent each
bit in a number, then anything bigger just "overflows" the space
available.  Imagine you have a shelf with room for eight books, and you
keep adding a book on the right, and pushing the books to the left to make
room for the new addition. When you add book #9, the first book has been
pushed all the way to the left and falls off the shelf.

So,  in our previous 4-bit computer, what happens when we add 1 to 15
(1111)?  If we had 5 bits, we'd get 10000, a.k.a. 16. But, we don't and so
the last book fell off the shelf and we get 0000 -- and on any decent
system a warning that you've got an overflow error.

Now, for negative numbers... The sum of a positive number and its negative
cousin should equal 0.  So, for SIGNED arithmetic, we've just established
that 0001 + 1111 = 0000, given the constraint of only four bits to work
with.  Therefore, by perverse logic, 1111 must be negative 1. ;-)

The way to convert a positive binary integer to its negative, is to use the
"two's complement".  In this case, a complement isn't about telling the
number how sexy or smart it is. ;-)  It's more akin to the idea that
certain wines complement certain foods.  In this case it means the opposite
state of a bit:  The one's complement of 1 is 0 and vice versa.  So, the
one's complement of 1001 is 0110, the one's complement of 1111 is 0000,
etc.  Initially, you might be tempted to think we could use the one's
complement of a number to represent its negative value.  But there are a
few edge cases where it gets illogical.

To get the two's complement, add 1 to the one's complement:  Two's
complement of 1001 (decimal value 9) is 0110+1 = 0111 (decimal -9 in 4-bit
signed arithmetic).  If you add the two binary values together:

111
1001
0111
----
0000

1+1 = 2, which in binary is 10. Carry the 1. 1+0+1 = 10. Carry the 1. 1+0+1
= 10. Carry the 1. 1+1+ 0 = 10. Carry the 1.  Overflow. But in this case,
since we know we're dealing with signed arithmetic, the overflow doesn't
count, and the last digit drops off, giving us 0000.

Voila!

(We now return you to your regularly scheduled broadcast.)