What is Bitcoin?

Digital Currency; no government needs to issue it and no banks need to manage accounts and verify transaction
Instead, bitcoins exist as records on a distributed ledger called the block chain
Copies of the blockchain are shared by a volunteer network of connected computers.
"Owning" bitcoin means having the ability to transfer control of it to someone else by creating a record of the transfer in the block chain.
Access to the block chain is granted by an Elliptic Curve Digital Signature Algorithm (ECDSA) private and public key pair.

Why Do BitCoins Have Value?

Bitcoins have value because they are useful as a form of money.
Bitcoins have the characteristics of money (durability, portability, fungibility, scarcity, divisibility, and recognizability) but don't rely on physical properties (like gold and silver) or trust in central authorities
Backed by mathematics: ECDSA
But like with all currency, the value of bitcoin comes only and directly from people willing to accept them as payment.

First, we start with something easy.
As a small group of people, they want to have a public ledger that records all the transaction between them
Then at the end of the month, people can calculate who owes who how much.
This will make life easier because now you don't need to pay each other multiple times but just once per month
Since it's a public ledger, anyone can add a line to it.
This creates the first rule:"Anyone can add lines to the Ledger"
For example:"A gives $100 to B." But what if B never approved the transaction?
Therefore we need a digital signatures next to the line to confirm that B has seen and approved it.

Sign( message, secret key) = a specific digital signature for this line in the ledger
The digital signature is some 256 bit signature

Verify( message, digital signature, public key) = True/False
Everyone can verify it with the public key
But there is 2^256 different signatures for this line, so it is almost impossible for people just to guess and check it

Go back to the previous example, A gives $100 to B. Can't A just copy and paste the line again and again? Because the message and signature match.
So every line has a unique ID which will also be part of the message. Then A cannot do the Copy and Paste
Thus we can have a rule "Only signed transactions are valid"

What if A claims he pay other people tons of money and run away at the end of the month?
So we add a new rule, "No overspending"
Everyone has to turn in real cash before they can spend money from others