I think of a float as an integer with a shift. Take the mantissa, stick on the implicit leading 1, put it in the middle of an infinitely big string of zeroes wi

I think of a float as an integer with a shift. Take the mantissa, stick on the implicit leading 1, put it in the middle of an infinitely big string of zeroes with a decimal (well, binary) point at the right-hand end, then shift it by the exponent (which can be positive or negative). If you're comfortable thinking about shifting integers, then perhaps this is a simple way to think about floats. For me, it's easier than thinking about them as fractions.