The numbers in a floating-point system are defined by a base B, a mantissa length t, and an exponent range [L, U]. A nonzero floating-point number x has the form
x = +/-(o.b1b2.....bt)B^e ---1
then how to prove
x = +/-( b1 + b2/(B) + b3/(B^2).......+bt/(B^(t-l)))B^e ---2
I'm actually confused about the relationship between these two equations. I tried to understand it for several days but I couldn't. Can you explain this? Is there any fault with these equations?
If you assume $B=10$ and think about the place values of a decimal it will probably become clear. The significance of each digit is divided by the base because it is one position lower. It looks like you are one power of $B$ off because $b_1$ is behind the decimal point in the first.