General programming: Decimal numbers, floats

Question

I'm probably completely wrong, and I don't really know anything about it, but I have a question about decimal number data types in programming languages. I understand that floats aren't completely precise, because they're stored in binary with a power or something, but I always wondered why decimal number data types don't just store a number as if there was no decimal, so do calculations as if there wasn't a decimal, and then add it in after. Like in this situation:

2.159 * 3.507 --> 2159 * 3507 = 7571613
  ^^^     ^^^
  123     456

6 decimals in total... 7571613 -> 7.571613
                        ^^^^^^
                        654321

so 2.159 * 3.507 = 7.571613

Why can't it just work like that?

Answer 1

That's exactly what they do. A floating-point number is stored in exponent form. Let's assume that we're working on a decimal-based computer so I don't have to change all these numbers to binary.

You're multiplying 2.159 * 3.507, but in actuality 2.159 is stored as 2159 * 10^-3 and 3.507 is stored as 3507 * 10^-3. Since we're working on a decimal-based system, the 10 is assumed, so we only really have to store -3 without the 10, like this: 2159,-3 or 3507,-3. The -3 is the location of the "floating point": as the point moves left the floating point decreases (.3507 is stored as 3507,-4) and as the point moves right the floating point increases (35.07 is stored as 3507,-2).

When you multiply the two together, the decimal number (or the binary number on a binary computer) is the only thing that gets multiplied. The floating point gets added! So behind the scenes what happens is:

2.159 * 3.507
2159,-3 * 3507,-3
2159 * 3507,-3 + -3
7571613,-6

7571613,-6 is just 7571613 * 10^-6 (remember we can assume the 10 because we're working on a decimal computer) which is the same as 7.571613.

Of course, the floating point doesn't have to be -3, it could be anything that fits into the storage:

21590 * .3507
2159,1 * 3507,-4
2159 * 3507,1 + -4
7571613,-3
7571.613

And of course, most computers don't store things in decimal, so the actual numbers would be all in binary, and the floating point would be something like 2^-9 -> -9 rather than 10^-3 -> -3. But you get the idea.

Answer 2

This is called "fixed-point arithmetic" People do it all the time.

See http://gameprogrammer.com/4-fixed.html

Answer 3

There are quite a few implementations of fixed-point arithmetic. However, we often run out of decimal places very, very quickly with fixed-point storage. It's ideal for monetary transactions, where we know that we aren't going to store/care about any irrational numbers.

Additionally, for a lot of other things, fixed-point arithmetic just isn't worth the overhead. Floating point is just a lot faster.

Things to read:

• Floats
• Fixed-point arithmetic
• Decimal vs Double