General programming: Decimal numbers, floats
decimal-point, floating-point
Solution
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.
Problem
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?