Base 10-4 Joke [Every base is base 10]: The Explanation

“Every base is base 10” is probably one of the best jokes made about mathematics. The joke involves a conversation between an alien and a human astronaut about counting rocks.

Every base is base 10 joke [base 10-4], one of the best jokes in mathematics.
Every base is base 10 joke, one of the best jokes in mathematics.

The explanation of “Every base is base 10 joke”

Here’s the scene: An astronaut (a human) is talking to an alien. The alien says:

“There are 10 rocks [on the ground].”

The astronaut, seeing that there are actually 4 stones, realizes that the alien is using base 4 number system. In base 4, 10 = 1×4 + 0x1 = 4.

Details: The alien has 2 fingers on each hand, 4 fingers in total. As I will explain below, we humans use base 10 because we have a total of 10 fingers on both hands. Naturally, the alien uses base 4.

Astronaut says:

“Oh, you must be using base 4. See, I use base 10”.

At this point, the alien gets confused and says,

“No, I use base 10. What is base 4?”

Here’s why the alien is confused: In the base 10 number system, there are 10 digits in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. We don’t use a separate digit for 10; instead, 10 = 1×10 + 0x1 = 10. Similarly, 11 = 1×10 + 1×1 = 11, and so on. We express all numbers using the existing 10 digits.

If we were using a base 4, the digits would be 0, 1, 2, and 3, making a total of 4 digits. We wouldn’t have a digit for 4 because we would write 4 as 10 (1×4 + 0x1). We would write 5 as 11 (1×4 + 1×1), 6 as 12 (1×4 + 2×1), and so on.

Similarly, if we were using a base 6, the digits would be 0, 1, 2, 3, 4, and 5, making a total of 6 digits. We wouldn’t have a digit for 6 because we would write 6 as 10 (1×6 + 0x1).

This principle applies to bases greater than 10 as well. For instance, if we were using a base 16 (hexadecimal) number system (assuming we had 8 fingers on each hand), our digits would be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, and f, making a total of 16 digits. We would write 16 as 10 (1×16 + 0x1 = 16).

Therefore, in reality, every “base” is written as 10 in its own system. “Every base is base 10”.

It’s important to note that this joke makes sense in written form. When speaking, we say “ten” for the number 10, not “one-zero.” Similarly, the alien would say whatever the word for “four” is in their language, not “one-zero.”

Understanding Number Bases

A number base (or radix) is a system for representing numbers. The base determines how many digits are used and the value each digit represents.

Common number bases

  1. Base 10 (Decimal): The most widely used system, likely because humans have ten fingers. It uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each digit’s position represents a power of 10. For example, in the number 345:
    • The digit 3 is in the hundreds place (10^2), so its value is 3 × 100 = 300. The digit 4 is in the tens place (10^1), so its value is 4 × 10 = 40. The digit 5 is in the one’s place (10^0), so its value is 5 × 1 = 5.
    Therefore, 345 = 300 + 40 + 5.
  2. Base 4 (Quaternary): Uses four digits: 0, 1, 2, and 3. Each digit’s position represents a power of 4. For example, in the number 132_4 (subscript indicates base 4):
    • The digit 1 is in the sixteen’s place (4^2), so its value is 1 × 16 = 16. The digit 3 is in the four’s place (4^1), so its value is 3 × 4 = 12. The digit 2 is in the one’s place (4^0), so its value is 2 × 1 = 2.
    Therefore, 132_4 = 16 + 12 + 2 = 30 in decimal.
  3. Base 5 (Quinary): Uses five digits: 0, 1, 2, 3, and 4. Each digit’s position represents a power of 5. For example, in the number 243_5:
    • The digit 2 is in the twenty-five’s place (5^2), so its value is 2 × 25 = 50. The digit 4 is in the five’s place (5^1), so its value is 4 × 5 = 20. The digit 3 is in the one’s place (5^0), so its value is 3 × 1 = 3.
    Therefore, 243_5 = 50 + 20 + 3 = 73 in decimal.
  4. Base 16 (Hexadecimal): Uses sixteen digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A (10), B (11), C (12), D (13), E (14), and F (15). Each digit’s position represents a power of 16. For example, in the number 1A3_16:
    • The digit 1 is in the two hundred fifty-six’s place (16^2), so its value is 1 × 256 = 256. The digit A (10 in decimal) is in the sixteen’s place (16^1), so its value is 10 × 16 = 160. The digit 3 is in the one’s place (16^0), so its value is 3 × 1 = 3.
    Therefore, 1A3_16 = 256 + 160 + 3 = 419 in decimal.

Why do we use base 10

Humans naturally adopted base 10, also known as the decimal system, likely because of our ten fingers. This biological feature made it easy to count and develop a system based on ten digits.

If humans had evolved differently, with, for example, four fingers on each hand (eight in total), we might have developed a base 8 (octal) system. If we had only two fingers on each hand (four in total), a base 4 (quaternary) system might have been more natural.

The alien in the “every base is base 10” joke has two fingers on each hand (four in total). So, it uses base 4.

Other bases in use

  1. Binary (Base 2): Used in digital electronics and computer systems. It has only two digits, 0 and 1.
  2. Octal (Base 8): Sometimes used in computing, particularly in older systems.
  3. Hexadecimal (Base 16): Commonly used in computing and digital electronics because it is compact and aligns well with binary (each hex digit represents four binary digits).

Sources

M. Özgür Nevres
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