$\color{#784}\text{If you go through life}$ $\color{#784}\text{learning and being curious, }$ $\color{#784}\text{you might be changed }$ $\color{#784}\text{by what you learn.}$ $\color{#784}\text{ And what you think you know}$ $\color{#784}\text{ might come apart.}$

• ⚜️ $\color{#784}\text{Sartre}$

Mathematics isn’t just about a fixed set of symbols—it’s about the diverse ways we represent and communicate ideas. Just as our logical language builds precise theories, our choice of number base is a tool to systematize counting.

🔢 Number Bases: How We Count and Name Numbers

🌍 The world of counting can be wild, especially when you realize that numbers can be represented in all sorts of ways. The most efficient way to systematize counting is by using something called a base.

🧮 A base is the number that marks the point where we start counting over again, but with an extra digit. The most familiar one? Our good ol' base ten! But it's not the only one!

❗️ Don't write 'base 10'! Every base is base "10"!

🚀 What is a Base?

♻️ A number base simply determines how many unique digits we use before "rolling over" and starting again, but adding an extra digit to the left. In base ten (the one we use daily), we have ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. When we hit ten, we start back at 0, but add a 1 to the left: 10.
This method of “rolling over” is fundamental to all number bases.

[!NOTE] When discussing different bases, remember that every base uses its own complete “alphabet” of digits.

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👣 Counting in a Different Base: Base 2

🖥️ Now, imagine you only had two digits: $0$ and $1$. What do you do when you run out of digits? You roll over, of course! This is the idea behind base 2, also known as binary, the language of computers.

🔳 In binary, every time you reach $2$, you start back at $0$ and add a $1$ to the left. So, $2$ in binary is written as $10$, because you've "rolled over" after reaching $1$.

• ▪️ 1 in binary = 1
• ▪️▫️ 2 in binary = 10 (one 2, no 1s)
• ▪️▪️ 3 in binary = 11 (one 2, one 1)
• ▪️▫️▫️ 4 in binary = 100 (one 4, no 2s, no 1s)
• ▪️▫️▪️ 5 in binary = 101 (one 4, no 2s, one 1)

🔲 Notice how it’s all about powers of 2.

Place values:

Just as decimal uses powers of 10 (1, 10, 100, …), binary uses powers of 2 (1, 2, 4, 8, …).

🔌 Binary is the foundation of digital systems because computers think in on/off switches, much like how binary numbers toggle between 1 and 0.

This mirrors our earlier discussions on the precision of standard notation and logical structure! The binary system is a perfect example of how a simple set of rules can underlie complex digital operations.

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💡 Why Use Different Bases?

🟥 Number bases aren't just a math curiosity. They have practical uses in different contexts. Here are some common ones:

• 👾 Base 2 (Binary):

  • The language of computers—everything in a digital device is based on binary, from data to processing.

• 🖐🏼 🤚🏼 Base 10 (Decimal):

  • The system we use every day in the real world for counting and calculations.

• 🟧 Base 16 (Hexadecimal):

  • Often used in programming and computer science because it's more compact than binary (16 digits: 0-9, A-F).
  • Colors are coded as RGB #🟥🟥🟩🟩🟦🟦 hexadecimals.

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🌍 Bases in Other Cultures:

🃏 Number bases also reflect the diversity of human thought:

• 🏺 Base 20:

  • This system groups counts by twenties.
  • Used by the Mayans.
  • Also used by the Gauls, is still used by the French in some contexts.

• Base 60**:

  • ⏳ The Babylonians used base 60, which is why we have 60 seconds in a minute and 60 minutes in an hour.

These systems highlight how different cultures have approached the challenge of counting and record-keeping using the same underlying mathematical ideas.

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🎮 Takeaway:

🐑 Number bases are simply different ways to represent quantities. Different “dialects” in the language of mathematics. Just as our logical expressions and standard notation allow for precise communication, different bases let us represent numbers in ways that best suit the task at hand.

🌌 Whether counting sheep, measuring time, or processing data in a computer, each base has its own beauty and purpose in the grand tapestry of mathematical thought.

Remember, just as we’ve built our mathematical theories on a foundation of precise language and logical rigor, our counting systems, no matter how they differ, are unified by the same core idea: a systematic way to represent quantities, making the universe’s complexity a solvable puzzle.