Suppose I have a 5-bit floating point system with a 3 bit exponent with radix $\beta = 2$. What is the range and $\epsilon_{machine}$ of this system?
I know that I can write numbers as:
$$sign \times mantissa \times radix^{exponent} $$
So the largest mantissa is $01111 = 15$ and largest exponent is $011 = 3$. Therefore the largest positive number I can represent with this system is $15 \times 2^3 = 120$, is this correct? And smallest negative number is just $-120$?
For smallest positive value, my mantissa should be $1$. My smallest negative exponent is $100_2 = -2^4 = -16$. Hence the smallest possible value is $1 \times 2^{-16} = 1.52587891 \times 10^{-5}$
If radix was 3, all my calculations would be on base 3, right?
And how can I calculate machine epsilon?