Welcome to Chapter 4! If you recognize the title of this post, you might be a fan of old-school video games. But today, "All your base" refers to something fundamental in mathematics: Number Systems.

Why Base 10?

Have you ever stopped to ask why we use the digits $0$ through $9$? It is not a universal law of the universe; it is biology. We have ten fingers, so humans naturally developed a Base 10 (Decimal) system. But what if we were cartoons with only four fingers on each hand? We would likely count in Base 8 (Octal).

In this lesson, we are exploring how changing the "base" changes the value of a number. This concept is crucial for understanding everything from ancient history to modern computing.

The Power of Position

Our number system is a positional system. The value of a digit depends on where it sits. In Base 10, positions correspond to powers of 10:

  • $10^0 = 1$ (The Ones place)
  • $10^1 = 10$ (The Tens place)
  • $10^2 = 100$ (The Hundreds place)

However, when we switch to a different base, such as Base 5, the "containers" or place values change size. Instead of groups of 10, we group by powers of 5:

  • $5^0 = 1$ (The Ones place)
  • $5^1 = 5$ (The Fives place)
  • $5^2 = 25$ (The Twenty-fives place)

Example: Reading a Base 5 Number

Let's look at the number $123_5$ (read as "one-two-three base five"). To convert this to our familiar Base 10, we expand it using the powers of 5:

$$123_5 = (1 \times 5^2) + (2 \times 5^1) + (3 \times 5^0)$$$$123_5 = (1 \times 25) + (2 \times 5) + (3 \times 1)$$$$123_5 = 25 + 10 + 3 = 38_{10}$$

So, $123$ in Base 5 is actually equal to the number $38$ in our standard counting system!

Common Bases We Study

In the attached notes and video lecture, we cover several specific systems:

  1. Binary (Base 2): The language of computers, using only $0$ and $1$.
  2. Mayan (Base 20): A distinct historical system that counts vertically rather than horizontally.
  3. Babylonian (Base 60): The reason we have 60 seconds in a minute and 360 degrees in a circle.

Action Items

Please review the Chapter 4 Class Notes linked below to see detailed examples of converting from Base 10 to other bases (using division and remainders). Also, don't forget to check the Project Link for your assignment details regarding number systems.

Math is a language—learning new bases is just like learning a new dialect!