[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.