Why Base 2 with binary numbers?

Question

History: I read from one of Knuth's algorithm book that first computers used the base of 10. Then, it switched to two's complement here.

Question: Why does the base could not be -2 in at least a monoid?

Examples:

(-2)^1 = -2 
(-2)^3 = -8

Answer 1

1's compliment does have 0 and -0 - is that what you're after?

CDC used to produce 1's compliment machines which made negation very easy as you suggest. As I understand it, it also allowed them to produce hardware for subtraction that didn't infringe on IBM's patent on the 2's compliment binary subtractor.

Answer 2

At its heart, digital logic is base two. A digital signal is either on or off. Supporting other bases (as in BCD) means wasted representation space, more engineering, more complex specification, etc.

Editted to add: In addition to the trivial representation of a single binary digit in digital logic, addition is easily realized in hardware, start half adder which is easily realized in Boolean logic (i.e. with transistors):

   (No carry) (with carry)
  |  0   1      0   1
--+--------------------
0 | 00  01     01  10
1 | 01  10     10  11

(the returned digit is (A xor B) xor C, and the carry is ((A and B) or (C and (A or B))) ) which are then chained together to generate a full register adder.

Which brings us to twos complement: negation is easy, and the addition of mixed positive and negative number follows naturally with no additional hardware. So subtraction comes almost for free.

Few other representations will allow arithmetic to be implemented so cheaply, and I know of none that are easier.

Answer 3

Optimization in storage and optimization in processing time are often at cross purposes with each other; all other things being equal, simplicity usually trumps complexity.

Anyone can propose any storage mechanism for information they wish, but unless there are processors or algorithms that support it, it won't get used.

Answer 4

In the end the decision was made because of voltage variance.

With base 2 it is on or off, no in between.

However with base 10 how do you know what each number is?

is .1 volts 1? What about .11? Voltage can vary and is not precise. Which is why an analog signal is not as good as a digital. This is if you pay more for a HDMI cable than $6 it is a waste, it is digital it gets there or not. Audio it does matter because the signal can change.

Answer 5

The problem is that with a negabinary (base -2) system, it's more difficult to understand, and the number of possible positive and negative values are different. To see this latter point, consider a simple 3 bit case.

Here the first (rightmost) bit represents the decimal 1; the middle bit represents the decimal -2; and the third (leftmost) bit represents the decimal 4

So

000 -> 0

001 -> 1

010 -> -2

011 -> -1

100 -> 4

101 -> 5

110 -> 2

111 -> 3

Thus the range of expressable values is -2 to 5, i.e. non-symmetric.

Answer 6

Please, see an example of the complexity that dmckee pointed out without examples. So you can see an example, the numbers 0-9:

0 = 0
1 = 1
2 = 110
3 = 111
4 = 100
5 = 101
6 = 11010
7 = 11011
8 = 11000
9 = 11001

Answer 7

Also, the use of the binary system has a mathematical background. Consider the Information Theory by Claude Shannon . My english skills don't qualify to explain this topic, so better follow the link to wikipedia and enjoy the maths behind all this stuff.

Answer 8

There are two reasons to choose base 2 over base -2:

First, in a lot of applications you don't need to represent negative numbers. By isolating their representation to a single bit you can either expand the range of representable numbers, or reduce the storage space required when negative numbers aren't needed. In base -2 you need to include the negative values even if you clip the range.

Second, 2s complement hardware is simple to implement. Not only is simple to implement, it is super simple to implement 2s complement hardware that supports both signed and unsigned arithmetic, since they are the same thing. In other words, the binary representation of uint4(8) and sint4(-15) are the same, and the binary representation of uint(7) and sint4(7) are the same, which means you can do the addition without knowing whether or not it is signed, the values all work out either way. That means the HW can totally avoid knowing anything about signs and let it be dealt with as a language convention.