Understanding the Two's Complement System

[ehrenfest]

Homework Statement

let me check my understanding of what a twos complement

when you use the rs complement system that means you use that system to express negative numbers only; in order to express the rs complement of a number you take r^n - that number; however if the number you want to express is positive then you do not really use the twos complement system, so it is somewhat of a misnomer.

I mean that even if we are working in the two's complement system and we want to express 5 we do not take r^n - 5,we just write 5

does all that make sense

Homework Equations

The Attempt at a Solution

 

[kuba]

r's complement

r's complement is a way to express negation in a given radix r, when the number of available digits is limited to n-1.

So, by obtaining r's complement, you're really obtaining a negative number.

So, obviously, it doesn't apply to positive numbers unless you want to turn them negative :)

A complement, in given radix, always behaves as a negative number as long as the number of digits you're considering is less than n.

 

[piercas]

Yes, your understanding of the two's complement system is correct. The two's complement system is a method for representing both positive and negative numbers in binary form. In this system, the most significant bit (leftmost bit) is used as a sign bit, where 0 represents a positive number and 1 represents a negative number. This allows for a wider range of numbers to be represented using a fixed number of bits.

To find the two's complement of a number, you take the one's complement (inverting all the bits) and then add 1 to the result. This is equivalent to subtracting the number from 2^n, where n is the number of bits being used.

For example, in an 8-bit system, the two's complement of 5 would be 11111011. This is found by taking the one's complement of 5 (00000101) and adding 1, resulting in 11111010 + 1 = 11111011.

However, as you mentioned, when working with positive numbers, we do not need to use the two's complement system. We can simply write the number in binary form without any additional steps. This is because the leftmost bit is always 0 for positive numbers in the two's complement system.

Overall, your understanding of the two's complement system is correct. It is an important concept in binary arithmetic and is commonly used in computer systems for representing both positive and negative numbers.