Why Two's Complement?

Question

I'm writing a tutorial to teach kids (ages 9 to 13) about programming. I started with computers themselves, they don't have that much to do with computer science, it's more about the process involved with a solution to a computational problem.

With that starting point, I'm guiding them toward an understanding that machines can help us with certain computational problems. People are great at abstract thinking and imagination, but computers are AWESOME at following a well-specified routine. They can do it again and again, at amazing speed!

Representing numbers in binary format has already been covered in my tutorial. But how do you represent negative numbers? There are so many ways to do this, in any notational system, but the system chosen for computers is for a very specific reason: to reduce the amount of machinery involved with adding signed integer values. We don't want to have to construct and build separate chips just to handle negative numbers, we want to use the same chips we have been using for natural number arithmetic!

If someone asked you on the street (as totally unrealistic as this seems) "how do computers represent negative numbers, and why do they represent them this way?"

My specific questions:

1. How do computers represent negative numbers?

2. Why do computers represent negative numbers this way?

I would guess that this many experienced developers would have to think about this a bit. Some might not even be able to come up with an answer. I'm not trying to be pompous, this is from actual experience, I've asked professional developers this question and they can't answer it. They draw a blank stare. Give them JBoss and JavaBeans and they will steamroll you with confidence. So funny! I too struggle with this question, I have to remind myself of the answers every time and I need a piece of paper or white board to work out a solution. What I'm hoping is to guide students toward a better understanding of the machine they are working with.

Answer 1

1.How do computers represent negative numbers?

Take the positive value, invert all bits and add one.

2.Why do computers represent negative numbers this way?

It makes easy to add 7 in -7 and came up with a zero. The bit operations are fast.

---

How does it make it easy?

Take the 7 and -7 example. If you represent 7 as 00000111, to find -7 invert all bits and add one:

11111000 -> 11111001

Now you can add following standard math rules:

  00000111
+ 11111001
-----------
  00000000

For the computer this operation is relatively easy, as it involves basically comparing bit by bit and carrying one.

If instead you represented -7 as 10000111, this won't make sense:

  00000111
+ 10000111
-----------
  10001110 (-14)

To add them, you will involve more complex rules like analyzing the first bit, and transforming the values.

And don't forget what @trashgod said, in 2's complement you have only one zero. To check it:

00000000
11111111 (invert all bits)
00000000 (add one)

Differently from 00000000 (0) being equal to 10000000 (-0)

Answer 2

Why do computers represent negative numbers this way?

Two big reasons I can think of:

• Simplicity of basic arithmetic operations. You don't have to worry about inspecting the sign bit to decide whether to add or subtract.
• Uniqueness of representation. It's nice to not have to worry about negative zero since there is no way to represent negative zero in two's complement.

From Wikipedia:

The two's-complement system has the advantage of not requiring that the addition and subtraction circuitry examine the signs of the operands to determine whether to add or subtract. This property makes the system both simpler to implement and capable of easily handling higher precision arithmetic. Also, zero has only a single representation, obviating the subtleties associated with negative zero, which exists in ones'-complement systems.