How can I represent any negative value mathematically using summation?

Question

For example, I know that I can represent the value -1 in this way, where "n" is the bit size of the type, n-1 is the most significant bit, and the value -2, which gives the complement representation of 2, plus the sum of the sum described below, now, I would like to know how I can use the sum, or any other type of calculation, to represent any negative value, in the form of a mathematical demonstration.

Assuming n has 8 bits, 8 - 1, 7, that is -2^7 = -128 + 127 = -1, I'm saying the summation will go through all the bits, such as 2^0 + 2^1, ..., 2^6, ie I ignore the most significant bit.

What do you mean by "in the form of a mathematical demonstration"? This is a computer programming site, not a mathematics site. Your summation already demonstrates how to express any value in a suitable range. — Raymond Chen, Mar 25 '22 at 02:52
@chux-ReinstateMonica Assuming the type has 8 bits, I can represent 0 to -128, how can I represent any other negative value, other than -1 ? — Yuri Bittencourt, Mar 25 '22 at 02:55
Your comments are not adding clarity to the problem you're trying to solve. Twos complement with 8 bits will represent every integer in the range [-128, 127]. — paddy, Mar 25 '22 at 02:57
I know this, but I want to know how to represent mathematically for any negative value, is there a way to do this, using summation with the sum of -2 raised to the size of the type, ignoring the sign bit? @paddy — Yuri Bittencourt, Mar 25 '22 at 03:03
It is not "-2 raised to the size of the type", but -"2 raised to the (size of the type - 1)". Negation after power. — chux - Reinstate Monica, Mar 25 '22 at 03:09
Yes, that's right, but how can I represent other negative values, other than -1 ? What would I have to modify in this formula? — Yuri Bittencourt, Mar 25 '22 at 03:11
The various `b sub-i` are not all 1, some are 0. That added to the -"2 raised to the (size of the type - 1)" results in the various negative values. — chux - Reinstate Monica, Mar 25 '22 at 03:12

Gene · Accepted Answer · 2022-03-25T13:27:29.720

Iiuc, the formula you gave does exactly what you're asking to do. You have answered your own question.

Let's do examples with n=4. For a reference, we can count in the negative direction to get all possibilities:

-1 = 1111
-2 = 1110
-3 = 1101
-4 = 1100
-5 = 1011
-6 = 1010
-7 = 1001
-8 - 1000

So now let's see if the formula gives correct results. By convention, a 4-digit binary number representation is four bits numbered 3 down to 0: b_3 b_2 b_1 b_0.

Example: 1111

So we have b_0 = b_1 = b_2 = b_3 = 1.

Substituting, v = -2^3x1 + 1x1 + 2x1 + 4x1 = -8 + 7 = -1.

Example: 1010

So we have b_0 = 0, b_1 = 1, b_2 = 0, b_3 = 1.

Substituting, v = -2^3x1 + 1x0 + 2x1 + 4x0 = -8 + 2 = -6.

You can work all the other values. They'll be correct.

The formula works for positive numbers, too. In that case b_3 = 0, so the -8 term goes away.

How can I represent any negative value mathematically using summation?

1 Answers1