Simple math: how to extract the fractional portion from a decimal

Question

Mathematically how do I get the cents from a dollar value (ex: $21.99$)?

As a programmer, I would simply convert to a string and grab everything after the decimal... but I would think this would be doable with pure math, or maybe I have too much faith in the black magic

EDITED

I wasn't clear.
Is it possible with pure math (no computer API.. just old school paper and pencil) to find the fractional part of any given decimal with any given precision?
Is there a formula?

Given xx.yyyy is there a formula that would return yy?

Answer 1

This can't be done in general with only (finite) arithmetic, but what you're looking for is the fractional part function.

Now, if you're just interested in dollar values, and you avoid fractions of a cent, then this is less complicated, but still not reachable through basic arithmetic only. Note that $\$21.99$ is simply an alternative notation for the dollar amount $$21+\frac{99}{100}.$$ Multiplication by $100$ yields the integer $2199,$ whose remainder when divided by $100$ is $99$, which upon multiplication by $.01$ yields the fractional part.

In general, given a dollar amount $d$ with no fractions of a cent, $$.01*\text{mod}(100*d,100)$$ will be your fractional part, where $\text{mod}(m,n)$ represents the remainder when a positive integer $m$ is divided by a positive integer $n$.

Answer 2

Look at the number Mod 1. When you look at a number Mod n, you just divide by n, and look at the remainder. So here you divide by 1, and the remainder is just the fractional part.

Answer 3

The best way to do this depends on the language. For something C- or Java-based, I'd do:

 decPart = x - (int)x;

Basically, this finds the difference between the integer part and the original number.

Answer 4

You just subtract the integer part, getting $0.99$. If you intend to ask how to do it with some specified set of functions, you posting doesn't make that clear. For example, one could multiply it by $100$ (so that $21.83$ becomes $2183$), then find the remainder on division by $100$ (so that $2183$ becomes $83$), then divide by $100$ (so that $83$ becomes $0.83$). Or, if what you have is the floor function, you could find $x-\lfloor x\rfloor$, for example: $$ 21.83-\lfloor21.83\rfloor = 0.83. $$