### Floating point precision

I've been reading up a bit on the standard representation of floats, and I kind of understood that numbers with a zero fractional part, e.g. 5.0, 9.0, can be exactly represented. Is this correct?
 `5 = 5*2^(0), 2.5 = 5*2^(-1) , 1.25 = 5*2^(-2), ...`

In general, multiples of powers of two, given that the sign, significand and exponent can each be represented within the number of bits allotted to them, can be represented exactly.
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 numbers with a zero fractional part, e.g. 5.0, 9.0, can be exactly represented. Is this correct?

5.0 and 9.0 are exactly represented, but it's not true of all integers:

`float f = 123456789; // f is 123456792 `
That's because 123456789 is too large to fit in the number of bits used as the significand for a 32 bit floating point number. It's also true of 16777217. Because the significand cannot store the number alone using an exponent of 0, you are forced to approximate it as a number with a smaller significand multiplied by some power of 2.
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So any x/1, x/2, x/4, etc. can be represented exactly as long as x fits in the bits of the mantissa?
Assuming x is an integer, and the exponent is between -126 and 127, then yes. + or - x*2^exponent is the exact representation.
Alright, thanks for the help.
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