How does one compress a key?
Use 0.6+ of the client, or software built / updated since then. More specifically:
http://bitcoin.stackexchange.com/questions/3059/what-is-a-compressed-bitcoin-keyRules that anyone might actually care about in simple terms:
Compressed public key (cpubkey):
03
9D85F2F2C4C9295ECBADB09AA118FF787B7FBDFBE528776D57C61C2E3A945F40 (Bold is the x-coordinate.)
Uncompressed public key (pubkey, I prefer upubkey for clarity):
04
9D85F2F2C4C9295ECBADB09AA118FF787B7FBDFBE528776D57C61C2E3A945F4010F812A01E8CDC71FB399EE7713EBDE44C55914172520EAD3135F84C3DCB7C1D
See the commonality?
Now, naturally, these both hash to different values. Specifically, under Bitcoin's current rules:
hash(cpubkey): E65C95EEC06C903B4CA2CA745A8245B232DE3BB7
hash(upubkey): 751B61B1646FB2AC1BB8CA34623AA29F1A821D18
Which then of course are different in the Bitcoin Base58-encoded format, also:
Compressed: 1N13L61jaUb481CWTgXCuJixJ7Sf1EXpGY
Uncompressed: 1BgCrxDUz44bU1YsBxLSkPcsep9bih1SmZ
Thus, without breaking RIPEMD-160, there's no way to guess about the pre-hashed string, and hash algorithms don't break into 1-to-1 matrices when they are considered "broken" anyway. In fact, if you notice the length difference, we're going to run into hash collisions long, long, long before address collisions. At which point we'd switch to a longer hash, and the client would switch to a new one, and it's all good.
So, despite them both corresponding to the same private key, you can't see the balance of both by checking one. You'd need to check both of them. Your private key contains the information to unlock either one. Consider it -- since it is exactly the same type of thing -- like how there is a negative and a positive root of any real number. As in, sqrt(4) can be -2, or 2. "4" is your private key, and you need to know which public key (2 or -2) your funds are in. Or both. The difference in Bitcoin is, much bigger numbers, and the curve isn't a mirror at the origin, so one "root" can be a longer number than the other. For compressed keys, we use the small one. Also the mathematical operation is much more complex than just roots. And this was an analogy, and therefore a simplification.
If you are really curious, there is always this (which includes a graph of an elliptic curve):
http://blog.cloudflare.com/a-relatively-easy-to-understand-primer-on-elliptic-curve-cryptographyAaanyway, that's why compressed keys save blockspace, and more importantly, why you shouldn't reuse addresses, because you're less protected -- once you've spent from an address, you no longer have the additional protection of RIPEMD-160 guarding your funds.