A private key is nothing more than a very large number picked at random that should be kept secret by the owner. The public key is then derived from the private key and can be shared with everybody. This public / private key pair can then be used to encrypt and unencrypt data or verify the authenticity of the signer of a message.
Encryption and Digital Signatures.
Well bitcoin address is kind of just a hash of the public key and you use signatures for sending transactions 
another possible use case for public key cryptography is to encrypt data
If you want to encrypt the data you use the public key which can be decrypted by using the private key.
You can also sign a message using your private key that someone can then verify using your public key
Great and very clear answer 
Not true we derive the public key from the private key using elliptic curve cryptography you do hash the public key to get a bitcoin address
Digital signatures are not really encrypted. Its like a seal to prove you signed the message that everyone can read and prove you were the one writing it using your public key
Not technically, the public key does exist in a sense (if you go philosophical all keys already exist, but not all have been calculated yet), but you must still derive it using the elliptic curve
Both keys are used in “Public key cryptography”. The private key is generated by a computer program first and then the public key is derived from it. You can not figure out what the private key is from a public key. It’s a one-way function just like a hash function. The public key is shared with anyone wanting to encrypt the message and only a private key holder can decrypt it. It’s paramount to keep the private key safe and out of reach of anyone because if it’s lost or stolen, the message can’t be decrypted or can be compromised.
Encrypted messaging and digital signatures in cryptocurrencies.
1.) In order to receive a message, one can share the public key with anybody, the private key should always be “private” and can be used to encrypt, decrypt, verify and sign tx or encrypted messages.
2.) Encryption and digital signature.
Describe the concept of public and private key with your own words.
Private Keys are a series of random numbers that are generated by a computer. They are secret and known only to the holder of the keys. These Private Keys are used to generate another specific series of numbers that is linked only to that specific Private Key. This new number is known as the Public key. It is safe to share a Public Key with the world because there is no way of discovering the Private Key using its corresponding Public Key. Public keys are used to encrypt messages which can then later be decrypted using the Private Key.
What 2 use-cases can public key cryptography be used for?
Public Key Cryptography can be used in Encryption to encrypt messages making them private. It can also be used for Digital Signatures to verify the identity of an individual or to ensure the integrity of a message.
Homework on Public and Private Keys - Questions
Describe the concept of public and private key with your own words…
The concept of Public and Private key is a widely used scientific method of securely sending a secretive message across an open or unsecured network such that any third party will not be able to decrypt the message unless the intended recipient.
Therefore, this concept use case of Public and private key is used on the Bitcoin network to make sure that transaction are secured and passed across to the right receipient.
The Private key is a computer randomly generated number which is kept secret and known only to the owner while a mathematically formula is used to derive a Public Key from the Private Key which can then be shared with others so as to use to complete the communication.And it is only a one way process such that you can never derive a Private Key from the Public one so as to ensure fool proof security but you can only generate a Public Key from a Private key.
What 2 use-cases can public key cryptography be used for?
a.)To encrypt message that needs to be sent over a network
b.)To create a digital signature to identify the source of the message.
This went kind of over my head 
What is an elliptic curve??
So ones you have your private key, you can derive a public key calculating through an elliptic curve??
Private key is my password to my email address which would be the public key. You use the private key to sign a transaction to show proof that you own the public key
Digital signature of proof of ownership and email services
Public key is can be shared to decrypt data. Private key is only for the owner. Anyone with access to private key can access so it needs to be kept secure and secret.
encrypting messages and digital signatures
The private key is randomly generated by the computer and is your fingerprint/id when making transactions on blockchain. From the private key a public key is generated by one-way cryptography function. From the public key the adress is generated.
Encryption and digital signature
If there are P=private key, R=public key, S=my signature and A=my address, there are functions from P to R and R to S and R to A. If f( R )=A, I must be able to say g( f( P ) )=A??
Private keys are randomly generated and public keys are created from it.
Public key cryptography can be used for digital signatures and cryptocurrency.
If two persons, Alice and Bob wants to communicate with end-to-end encryption, each of them generates a key-pair consisting of a private and public key; Alice pair (Ka, Pa) and Bob’s pair (Kb, Pb). K=the private key and P=the public key. The private keys are random numbers (sometimes between 1…2^256). Each public key is generated from the respective private key.
When Bob creates an encrypted message that only Alice can decrypt, he uses Alice’s public key together with his private key to encrypt the message. Alice then uses her private key to decrypt the message.
There is public key encryption as described above and public key digital signatures. A digital signature is used to prove that whoever signed a message has the private key corresponding to the public key. There is a function; SIGN, that takes as arguments the private key K, the public key P and a message MESS so that SIGN(K, P, MESS) = SIGNATURE. Then the signer (Bob) sends (MESS, P, SIGNATURE) to Alice. Then there is a function called VERIFY than takes these three as arguments;
VERIFY(MESS, P, SIGN) = TRUE OR FALSE. If the function outputs TRUE, then the signature must have been made using the message, Bobs public key and Bobs private key.
This is not definitive proof that Bob did the signing (he could have given his private key to someone else), but it’s something 