Q:

Exponential running time for factors

I'm trying to read Sipser's book chapter 10 on page 399
and there's a paragraph that says
One way to determine whether a number is prime is to try all possible 
integers less than that number and see whether any are divisors, also 
called factors. That algorithm has exponential time complexity because 
the magnitude of a number is exponential in its length

What does "the magnitude of a number is exponential in its length" exactly mean? I don't understand the difference between magnitude and length of a number

A:

Express a number $k$ in binary form.  The number of bits required to express $k$ in this way is $\lceil \log_{2}(k) \rceil$. 
In the opposite direction, using $n$ bits, you can express numbers from $0$ to $2^{n}-1$.  Thus the maximum magnitude of an $n$ bit number is $O(2^{n})$.  

