What Is The Gap Between 2 And the first IEEE number larger then 2

Question

What is the gao between 2 and the first IEEE single format number larger than 2?
What is the gap between 1024 and the first IEEE single format number larger than 1024
What is the differece between calculating the gap of int between double.

I dont know how to answer on these questions, I read IEEE Documentation.
I need some hints.
Thanks.

Answer 1

Positive single precision floating point numbers (except for small subnormal numbers that we can disregard for this question) are stored in the format $$2^{\textrm{e}}\cdot (1 + \textrm{n}\, 2^{-23})$$ where $\textrm{e}$ is an integer and $\textrm{n} \in \{0, 1, \dotsc, 2^{23}-1\}$. The number $2$ is represented as $$2 = 2^1 \cdot (1 + 0)$$ and the next number is therefore $2^1 \cdot (1 + 2^{-23})$. The gap is $2^{-22}$. The gap for $1024$ can be computed in the same way (note that $1024 = 2^{10}$).