Finding the Smallest Integer Not in Floating Point Definition

[Codezion]

This should be an easy one, but my PC is bugging me! Based on the floating point definition:

F = $$\pm$$( $$\stackrel{m}{\overline{B^{t}}}$$)$$B^{e}$$

Where B is the base (usually 2), m is the mantissa and varies from 1 $$\leq$$ $$B^{t}$$ - 1. e is the exponent (1024 for double and 128 for single precision machines).

Quetion: what is the smallest intenger that DOES NOT belong to this floating point definition.

Solution: Since we want to minimize the denominator term we will let t=1, This would again let m =1. Hence, we can conclude that this said integer is 2$$^{1023}$$. However, I cannot verify this on the computer. In matlab, if I have this number and I add 1 to it, it gives me back the same number instead of Inf as I expect it. Can someone tell me how to verify this on a computer, or if my analysis is wrong?

Thanks!

 

[lippyka]

The smallest integer that does not belong to this floating point definition is 2^1024. This is because the mantissa (m) can range from 1 to 2^(t)-1, and since t=1 in this case, the maximum value of m is 1. 2^1024 is larger than any value of m in this floating point definition, so it cannot be represented. You can verify this on a computer by using a programming language like Python or Java to represent the floating point number.