[randyrand]

It's not exactly scientific notation because a leading bit is assumed for normalized numbers.

The window and offset explanation naturally accounts for the leading bit, whereas the scientific notation explanation needs further explanation to explain how the leading bit works.

In short, 10^2 * .001 is not allowed in floating point. You can't have a mantissa that starts with 0. The leading bit means all mantissas must be greater than 1.

Without this understanding you won't have an intuitive understanding of the range and precision of floating point, which is why I think the window+offset explanation is much more natural.

[romwell]

I think this is worth saying about the leading bit (why 0.001 is not a valid mantissa):

In binary, we always know the first non-zero digit of any number - so there's no need to write it down. We know it's 1 because, well, binary.

So we don't waste space, and only write down all the other digits, and then use the exponent to put the 'point' into the right place.

We save a bit of space doing this.

[randyrand]

Thats not the right way of thinking about it. Sure, the first non-zero digit of any binary number is 1, but who is to say that the number has a bit that isn't 0? Couldn't the number be zero?

The mantissa can only be a value from [1,2).

i.e. in scientific notation: exponent * mantissa, the mantissa cannot be less than 1, or >= 2 in floating point.

So given that the mantissa can go from 000000 to 111111, what should the values represnet? Obviously it should represent the values from [1,2). Calling it a leading bit is more confusing than it needs to be. Its better to just call it an assumed minimum value.

[BeetleB]

> Thats not the right way of thinking about it. Sure, the first non-zero digit of any binary number is 1, but who is to say that the number has a bit that isn't 0? Couldn't the number be zero?

When I was taught scientific notation in middle school, high school, and at college, it was always explicitly stated that:

1. You can have only one digit before the decimal point.

2. That digit cannot be 0 (unless your number is 0, of course). So 0.3 * 10^5 is not scientific notation.

This is no different. The "twist" is that there is only one possible non-zero bit, whereas in decimal it could be [1-9].

I think this explanation is neat. However, the "usual" formulation is just scientific notation, with the optimization that one bit is redundant.

Personally, I prefer the alternative notation: M * 2^(exponent-precision+1), with M being a p-bit integer. It's easier to work with when you know that M is always an integer, and you don't need to deal with fractions in base 2. In fact, FP made a lot more sense when I took this formulation in decimal, and worked with that.