Beginner's Guide to Coding and Data

Understanding Numeral Systems in Different Bases

Have you ever wondered how numbers are expressed in different bases? We're here to explore the fascinating world of numeral systems and how they work across various bases, from base 2 to base 36.

To begin our journey, let's address a simple question: What numeral is used to express the number 17 in base 16?

• 17
• 1
• 11
• 2

We'll answer this question and unlock the secrets of numeral systems. Soon, you'll be able to confidently solve questions like the one above for any base. This knowledge stems from recognizing patterns inherent in numeral systems.

These patterns apply universally, from binary (base 2) to hexadecimal (base 16) to duotrigesimal (base 32) and beyond.

Pattern 1: The Universal Zero

The first pattern is all about the number 0. Interestingly, the representation of 0 remains the same across every base. It's a universal constant in numeral systems.

Pattern 2: Unity in Ones

Similar to the first pattern, the number 1 also holds a special place. Just like 0, the numeral for 1 remains consistent across positional numeral system bases.

Pattern 3: The Base Identity

Now, let's address the question we encountered earlier. For any base, the numeral representing the base itself is 10. This pattern emerges from the carryover process we discussed in previous lessons. So, whether you're in base 2 or base 36, the answer to the numeral for the base in that base is 10.

Take a moment to look at the numeral for the number 2 in base 2, and observe how the numeral 10 shifts down the list as we increment the number.

The key to answering questions like the one at the beginning of this lesson is to watch the number 10 and see who's trailing behind. If you can spot it, let us know in the comments.

Pattern 4: Less is More

When the base is higher than the number itself, only one digit is required to represent the number. It's a concise way to encode values in bases where the base exceeds the number.

Pattern 5: Magnitude and Digits

The fifth pattern relates to the magnitude of the number and the number of digits required to represent it in each base. Let's take a massive number like 1 quadrillion. As we move through different bases, we'll notice that the number of digits required to encode 1 quadrillion decreases. This phenomenon is due to the constraints of representing numbers with a finite number of digits in computer systems. Higher bases allow us to represent more significant values with a single digit.

• In base 2 (binary): 111011100110101100101000000000
• In base 8 (octal): 73425420000
• In base 10 (decimal): 1,000,000,000,000 (One Quadrillion)
• In base 16 (hexadecimal): 3B2E19C200
• In base 32: 1CKGDK0
• In base 64: 3iJpLK0

Feel free to explore this notebook on your own and see if you can discover any other intriguing patterns in numeral systems. If you do, share your findings in the comments section.