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A private key is a secret, alphanumeric password/number used to spend/send your bitcoins to another Bitcoin address. It is a 256-bit long number which is picked randomly as soon as you make a wallet.The degree of randomness and uniqueness is well defined by cryptographic functions for security purposes.

This is how the Bitcoin private key looks (it always starts with 5):
 

This is another alphanumeric address/number which is derived from private keys only by using cryptographic math functions.It is impossible to reverse engineer and reach the private key from which it was generated.This is the address used to publicly receive bitcoins.This how the Bitcoin public address looks (it always starts with 1):
 

1EHNa6Q4Jz2uvNExL497mE43ikXhwF6kZm

This address is always seen and broadcasted for receiving bitcoins. Users can make as many public addresses as they want to receive bitcoins

Private keys are used for making irreversible transactions. Yes, irreversible!

They are the key to spending and sending your bitcoins to anyone and anywhere. This irreversibility is guaranteed by mathematical signatures which are linked to each transaction whenever we use the private keys to send bitcoins.

And for each transaction, these signatures are unique, even though they are generated from the same private keys. This feature makes them impossible to copy. The user can confidently use the same private key again and again.

Moreover, the signatures are mathematically related to Bitcoin addresses. This math relation helps in confirming that the signatures are only of that particular account holder who wants to transfer bitcoins.