The number is the root of your wallet.  From that number, you can calculate a sequence of private keys.

Also, from the root, you can generate a public root and from the public root a matching sequence of public keys.

This means that if someone has your public root, they can generate your public key sequence.  However, they can't generate your private keys.

The public monitoring software computes the first say 1000 public keys and watches the network for those.  This is safe since it is impossible to generate matching private keys.

To spend a coin, you need the private key.  This requires the private root key.  This should be stored offline.

Once  you have that root (written on paper), all public and private keys can be generated.  So if you lose your wallet, all the keys can be regenerated from the paper backup (which never changes).