Data is fragile. A scratch on a CD, a crackle on a radio wave, or cosmic radiation hitting a memory chip corrupts bits. A '0' flips to a '1'. How do you know? How do you fix it?
[ h(x) = -\log_2(p) ]
Entropy is the average amount of information produced by a source. It is also the minimum number of bits required, on average, to encode the source without losing any information. Introduction To Coding And Information Theory Steven Roman
[ H = -\sum_{i=1}^{n} p_i \log_2(p_i) ]
When your data corrupts, you are witnessing a violation of the Hamming distance. When your compression algorithm bloats instead of shrinks, you are witnessing low entropy. Data is fragile
In Shannon’s world,
When most people hear the word "code," they think of spies, secret languages, or JavaScript. When they hear "information," they think of news or data. But in the mathematical universe, these two concepts are married in a beautiful, rigorous dance that underpins every text message, every streaming video, and every photograph from Mars. How do you know
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.