Beginner's Guide to Coding and Data

Level: Beginner

Hexadecimal explained | Higher than base-10 positional numeral systems

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Unraveling the Mystery of Bases and Digits

We're back exploring another interesting mathematical concept: positional numeral systems. Today, we'll dive into bases, digits, and what happens when we wander beyond the familiar territory of base 10.

To get started, let's break down the basic steps of understanding this concept:

1. Understand what a digit is
2. Explore different bases
3. Learn about carrying over in different bases
4. Experiment with base 16 (hexadecimal)
5. Examine how base relationships work

What is a Digit?

As we embark on this journey, it's crucial to understand what a digit is. A digit is a single symbol that represents a number. It serves as a building block in positional numeral systems. In layman's terms, digits are the symbols we use to write down numbers.

Digits and Bases

A base in positional numeral systems tells us how many distinct digits are available. For example, base `2` has `2` digits (`0` and `1`), base `3` has `3` digits, and so on. The key thing to remember is that the number of digits matches the base.

• Base 2 (Binary): 0, 1
• Base 3 (Ternary): 0, 1, 2
• Base 4 (Quaternary): 0, 1, 2, 3
• Base 5 (Quinary): 0, 1, 2, 3, 4
• Base 6 (Senary): 0, 1, 2, 3, 4, 5
• Base 7 (Septenary): 0, 1, 2, 3, 4, 5, 6
• Base 8 (Octal): 0, 1, 2, 3, 4, 5, 6, 7
• Base 9 (Nonary): 0, 1, 2, 3, 4, 5, 6, 7, 8
• Base 10 (Decimal): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

The real fun starts when we venture beyond base `10`. What happens when our base gets higher than `10`? Let's explore this by examining base `16`, also known as hexadecimal.

In base `16`, counting starts to look a bit different. After the digits 0-9, we still need additional symbols to represent numbers before carrying over. The symbols chosen are often letters. For example, the number `10` in base `16` is represented by the symbol A.

```let hexDigits = ["0", "1", "2", ..., "9", "A", "B", ..., "F"];
```

By the time we get to `15`, we represent it as F. Once we get to `16`, we carry over and start again, now with 1 in the `16s` place.

To summarize, we've seen that we can continue this pattern all the way up to base `36`, using the letters of the alphabet after running out of numerical digits.

Beyond Base `36`

We could technically keep going beyond base `36`. To do this, we would just need to invent new symbols. So, if we're feeling adventurous, the sky's the limit!

We chose to work with hexadecimal because it's prevalent in computing, mostly because of its neat relationship with base `2`.

In closing, it's vital to remember that bases using unfamiliar symbols help us understand the difference between numbers (the abstract concept) and numerals (the symbols used to represent numbers).

We're now ready to delve deeper into patterns involving bases and digits, which we'll do in the next lesson. Stay tuned!

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