First I present the problem and then my workings and thoughts:
Given the normalized floating point number system
$(\beta, t, L, U) = (10, 7, -6, 4)$ where $\beta$ is the base and $t$ is the number of digits Calculate the smallest possible value of $y - x$ for which $\bf x < 100 < y$
My workings:
$100 = 0.1 \times 10 ^2 = 0.1000000 \times 10^2$
I can calculate that the smallest possible number greater than $100$ (i.e. $y$) is equal to: $$0.1000001 \times 10^2$$ But, I think I miscalculated the greatest possible number greater than $100$ (i.e. $x$); I calculated it to be$$ 0.9999999 \times 10^1 $$ I know the result should be $y - x = 0.11 \times 10^{-3}$, whereas I get $0.11 \times 10^{-4} $
Any hints on how to solve my mistake?
$$100= 0.1 \cdot 10^3= 0.1000000 \cdot 10^3$$
The smallest possible number greater than $100$ is equal to : $$y=0.1000001 \cdot 10^3 $$
The greatest possible number smaller than $100$:
$$0.0999999 \cdot 10^3=0.999999 \cdot 10^2$$
$$y-x=0.1000001 \cdot 10^3 -0.999999 \cdot 10^2=100.00001-99.9999=0.00011=0.11 \cdot 10^{-3}$$