What does 44' mean in BIP44? After reading BIP44, I saw 


  Purpose is a constant set to 44' (or 0x8000002C) following the BIP43
  recommendation.


I believe 0x8000002C is 128 as an integer, how does this equate to 44' however?

I also believe that the ' means that it is a "hardened key", is it hardened because 128 is between 0 and 2^32 - 1 ? Which means that the only key that can derive it, is the private key?

To further elaborate my thinking, which may or may not be right:

These are the levels:


  m / purpose' / coin_type' / account' / change / address_index


purpose, coin_type and account, all have to be derived with the private key. They will always be between 0 and 2^32 -1

I have also come to believe that if we have derive a change address with the xpub, then if we derive the same path with xpriv, we can spend from that change address?

For example, if I receive some funds to the change address with path:

m / 44' / 0' / 0' / 0 / 0 

Then to spend from this specific address, I would need the private key that corresponds to it. To get it, I would use the xpriv on this path?
I believe 0x8000002C is 128 as an integer


... No, it isn't. First of all, it already is an integer, just represented in hexadecimal. In decimal it is 2147483692. How did you get 128?


  how does this equate to 44' however?


44' means that hardened keys should be used. The distinguisher for whether a key a given index is hardened is that the index is greater than 2^31, which is 2147483648. In hex, that is 0x80000000. That is what the apostrophe (') means.

The 44 comes from adding it to 2^31 to get the final hardened key index. In hex, 44 is 2C, so 0x80000000 + 0x2C = 0x8000002C.


  I also believe that the ' means that it is a "hardened key", is it hardened because 128 is between 0 and 2^32 - 1 ?


No, hardened means that the key index is between 2^31 and 2^32 - 1. Between 0 and 2^31 - 1 are unhardened keys.


  Which means that the only key that can derive it, is the private key?


Yes.