If I tell you something you already know (e.g., "The sun will rise tomorrow"), I have transmitted very little information. If I tell you something shocking (e.g., "The sun did not rise today"), I have transmitted a massive amount of information.
Think of entropy as the "randomness temperature." High entropy (like white noise or scrambled text) means high information density. Low entropy (like a repeating loop of silence or a predictable string of zeroes) means you can compress it down to almost nothing. Coding Theory: The Art of Reliable Imperfection If information theory is about efficiency , coding theory is about survival . Introduction To Coding And Information Theory Steven Roman
Why the logarithm? Because information is additive. If you flip two coins, the total surprise is the sum of the individual surprises. The logarithm turns multiplication of probabilities into addition of information. The most famous equation in information theory is Entropy ( H ): If I tell you something you already know (e
Mathematically, the information content ( h(x) ) of an event ( x ) with probability ( p ) is: Low entropy (like a repeating loop of silence