# Why does using a Real data type in a mathematical equation return an incorrect result?

I was trying to figure out the source of what appeared to be a rounding error today, and discovered that one of the columns used in a mathematical equation was using the real data type, and this is causing my equation to return an incorrect value.

Can someone explain to me why

``````SELECT 776.2384 * cast(100 as real) / 100
``````

equals `776.2385` while

``````SELECT 776.2384 * 100 / 100
``````

equals `776.2384`?

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And this one which I find lighter read and in much better formatting: http://floating-point-gui.de/ : What Every Programmer Should Know About Floating-Point Arithmetic. Plus, it has a xkcd – ypercubeᵀᴹ Sep 13 '12 at 14:20
Why are you explicitly using `REAL`? `REAL/FLOAT` should be reserved for cases where they specifically need to be used. In most business applications `DECIMAL/NUMERIC` are a better (and more predictable) choice. – Aaron Bertrand Sep 13 '12 at 14:22
@AaronBertrand I honestly have no idea. The database is part of some 3rd party software, and I don't like making changes to it unless I absolutely have to – Rachel Sep 13 '12 at 14:24
@ypercube Thanks for the links :) – Rachel Sep 13 '12 at 14:47

In your second query, `776.2384` is a decimal number (and so are the two other values). So the calculation is done in that type, and the result is mathematically exact.
In the first, you introduce a `real`. The other literals are promoted to that type, and the calculation is done according to IEEE 754 rules, in round up mode. IEEE 754 floating points are not exact, so you will generally not obtain the "mathematically exact" result you're expecting.
So.... its actually storing the number as something like `776.238400000000001`, and rounding up to `776.2385`? Strangely enough, doing the division before the multiplication actually returns the correct result. I also get the correct result when only doing the multiplication, or only doing the division. Thank you though, I think I have a better understanding of what `Real` is and why its not accurate (although I had to Google `IEEE 754 rules`) :) – Rachel Sep 13 '12 at 14:41