"Information Theory Club?"
Yuki stared at the classroom door. A notice they'd found by chance had caught their interest.
"Why don't you check it out?" Riku called from behind.
"But what is information theory?"
"I don't really know either. But it sounds interesting."
They opened the door.
Inside the club room was one senior student. Aoi. Neatly organized notebooks lined the desk.
"New members?" Aoi turned around.
"Just visiting," Yuki answered.
"Welcome. Are you interested in information theory?"
"Honestly, I don't really understand it."
Aoi smiled. "That's fine. Everyone starts that way."
Riku asked, "What does information theory study?"
"What information is. How to measure it. How to transmit it."
"That's abstract," Yuki said.
"Let's start with a concrete example." Aoi headed to the whiteboard.
"What will tomorrow's weather be?"
"Sunny or rainy," Riku answered.
"Which is more likely?"
"Sunny, maybe. About 70 percent."
"Then, would you be surprised if it rained tomorrow?"
"I'd be surprised," Yuki nodded.
"That surprise is information," Aoi explained.
"Surprise?"
"Yes. When something unexpected happens, we gain information. If it's as expected, there's little information."
Riku thought. "Then what about the sun rising in the east?"
"Zero information. Because it's certain."
"Interesting perspective," Yuki was impressed.
Aoi wrote a formula. "In information theory, we express this mathematically. I(x) = -log₂(p)"
"p is probability."
"Right. The lower the probability, the greater the information content."
Yuki copied it into their notebook. "So surprise and information are related."
"Exactly. And the average information content of all possible events is entropy."
"Entropy?" Riku asked.
"A measure of uncertainty. H(X) = -Σ p(x) log₂(p)"
Aoi gave an example. "Flip a coin. Heads or tails, 50 percent each. Entropy is 1 bit."
"1 bit?"
"A unit of information. The information needed to decide between two choices is 1 bit."
Yuki asked, "What if heads is 90 percent and tails is 10 percent?"
"Entropy is about 0.47 bits. Because it's biased, uncertainty decreases."
"It's easier to predict."
"Exactly. The higher the entropy, the harder it is to predict."
Riku's eyes lit up. "So my actions have high entropy?"
"Probably," Aoi laughed. "Because you're unpredictable."
"Is that a compliment?"
"It's a neutral fact."
Yuki thought deeply. "Information theory is a tool for measuring the world."
"Good understanding. Information theory quantifies uncertainty."
"What's good about being able to quantify it?"
"You can compare. You can optimize. And understanding deepens."
Riku raised his hand. "What do we learn next?"
"Communication. How to transmit information. Methods to deliver it accurately even with noise."
"Noise?"
"Random interference. Factors that distort messages."
Yuki laughed. "Like Riku."
"What's that supposed to mean!"
Aoi looked at the two. "Human communication can also be analyzed with information theory."
"For example?"
"Redundancy in conversation. To prevent misunderstanding, we repeat the same thing in different ways. That's the role of error-correcting codes."
Riku was amazed. "Daily life is full of information theory."
"Yes. We just don't notice it, but information theory is everywhere."
Yuki decided. "I'll join."
"Me too," Riku continued.
Aoi nodded happily. "Welcome. Let's explore the world of information together."
A bird sang outside the window. New information flowed into their daily lives. That was their encounter with information theory.