Parsing floats with Single.Parse()

Question

Here comes a silly question. I'm playing with the parse function of System.Single and it behaves unexpected which might be because I don't really understand floating-point numbers. The MSDN page of System.Single.MaxValue states that the max value is 3.402823e38, in standard form that is

340282300000000000000000000000000000000

If I use this string as an argument for the Parse() method, it will succeed without error, if I change any of the zeros to an arbitrary digit it will still succeed without error (although it seems to ignore them looking at the result). In my understanding, that exceeds the limit, so What am I missing?

Answer 1

It may be easier to think about this by looking at some lower numbers. All (positive) integers up to 16777216 can be exactly represented in a float. After that point, only every other integer can be represented (up to the next time we hit a limit, at which point it's only every 4th integer that can be represented).

So what has to happen then is the 16777218 has to stand for 16777218∓1, 16777220 has to stand for 16777220∓1, etc. As you move up into even larger numbers, the range of integers that each value has to "represent" grows wider and wider - until the point where 340282300000000000000000000000000000000 represents all numbers in the range 340282300000000000000000000000000000000∓100000000000000000000000000000000, approximately (I've not actually worked out what the right ∓ value is here, but hopefully you get the point)

---

Number         Significand             Exponent

16777215 =   1 11111111111111111111111     2^0      = 111111111111111111111111
16777216 =   1 00000000000000000000000     2^1      = 1000000000000000000000000
16777218 =   1 00000000000000000000001     2^1      = 1000000000000000000000010
             ^
             |
             Implicit leading bit

Answer 2

That's actually not true - change the first 0 to 9 and you will see an exception. Actually change it to anything 6 and up and it blows up.

Any other number is just rounded down as float is not an 100% accurate representation of a decimal with 38+1 positions that's fine.

Answer 3

A floating point number is not like a decimal. It comprises a mantissa that carries the significant digits and an exponent that effectively says how far left or right of the decimal point to place the mantissa. A System.Single can only handle seven significant digits in the mantissa. If you replace any of your trailing zeroes with an arbitrary digit it is being lost when your decimal is converted into the mantissa and exponent form.

Answer 4

Good question. That is happening because the fact you can save a number with that range doesn't mean this type'll have enough precision to hold it. You can only store ~6-7 leading digits for floats and add an exponent to describe decimal point position.

0.012345 and 1234500 hold the same amount of informations - same mantissa, different exponents. The MSDN states only that value AFTRER EXPONENTIATION cannot be bigger, than MaxValue.