Negative Numbers in Base 2, 16, etc.

Date: 01/24/2003 at 23:38:20
From: Jay (Dad) and Nathaniel Wasson
Subject: Negative numbers in base 2, 16, etc.

Are there negative numbers in alternate bases?  Our calculator says 
that 1-2 in base 16 is FFFFFFFFFF. This can't be right!

Our TI-34 calculator also says that 1-3 is FFFFFFFFFE, etc. Why does 
the calculator consistently do the "negatives" in this pattern if it's 
wrong instead of saying there is an error?  

Date: 01/25/2003 at 22:27:25
From: Doctor Peterson
Subject: Re: Negative numbers in base 2, 16, etc.

Hi, Jay and Nathaniel.

Great question! I like to see calculators used to explore numbers 
(rather than to replace learning to calculate), but sometimes they 
can raise questions like this.

In any base you can just do what we do in base ten, using a negative 
sign to indicate negative numbers. so 1 - 2 = -1 in base 16 as well 
as in base 10.

But calculators that do binary or hexadecimal arithmetic are designed 
for programmers - I am one, and I do hexadecimal arithmetic in my head 
all the time! Computers use binary because they can only store ones 
and zeroes; so some way had to be invented to store negative signs as 
ones and zeroes. There are several possibilities for doing this, such 
as using one bit (digit) as the sign, with 1 meaning negative. But the 
method that is almost universally used is called "twos-complement," 
and what you are seeing is the twos-complement representation of 
negative numbers.

The basic idea is to make use of the fact that the numbers stored in 
a computer are restricted in their range anyway. To use the smallest 
example found in real life, let's suppose that numbers are stored as 
eight bits. If we treat them as unsigned numbers, then they will vary 
from 0 (00000000) to 255 (11111111). If we are to store negative 
numbers, we'll have to use half of those bit combinations for 
negative numbers. We take those with a 0 in the first bit as positive:

    00000000 = 0
    00000001 = 1
    ...
    01111111 = 127

Now, to make the computer design as simple as possible, we'd like to 
be able to use the same addition circuits for signed numbers that we 
use for unsigned. So the number we use for "-1" ought to be the one 
that gives 0 when we add it to 1. That turns out to be 11111111:

    11111111
  + 00000001
  ----------
   100000000

That doesn't look as if it adds up to 0, until you notice that the 
answer, 256, has nine digits. If we simply drop the overflow, the 
answer is 00000000. And in fact, if we represent the number -n by 
256-n (as in this case we use 255 to represent 1), then (256-n) + n 
= 256, which is treated as zero. So the negative numbers are

    10000000 = -128
    10000001 = -127
    ...
    11111110 = -2
    11111111 = -1

If you convert these to hexadecimal (which programmers use essentially 
as a shorthand for the binary notation), it looks like this:

    80 = 10000000 = -128
    81 = 10000001 = -127
    ...
    FE = 11111110 = -2
    FF = 11111111 = -1
    00 = 00000000 = 0
    01 = 00000001 = 1
    ...
    7F = 01111111 = 127

The same idea can be extended to larger storage; the numbers you 
reported appear to be 5-byte numbers, which seems odd; but 4-byte 
(32-bit) numbers are typical, and follow the same pattern.

Here are some pages that explain this notation further:

   Negative Numbers in Binary
   http://mathforum.org/library/drmath/view/55924.html 

   Two's Complement
   http://mathforum.org/library/drmath/view/54344.html 

   Twos Complement Notation
   http://mathforum.org/library/drmath/view/54372.html 

   Binary Division and Negative Binary Numbers
   http://mathforum.org/library/drmath/view/54375.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/