Senary facts for kids

The senary numeral system, also called base-6, is a way of writing numbers using only six digits. While the standard decimal system (base-10) uses ten digits (0 through 9), the senary system uses the digits 0, 1, 2, 3, 4, and 5.

In this system, the number six is written as "10". This is similar to how the number ten is written as "10" in the decimal system. Because 6 is a number that can be divided evenly by both 2 and 3, doing math in base-6 can sometimes be easier than in base-10.

Understanding Base-6

Counting and Place Value

In the decimal system (base-10), the value of a digit depends on its place. We have the "ones" place, the "tens" place, the "hundreds" place, and so on. Each place is ten times bigger than the one to its right.

In the senary system (base-6), the places are based on the number six.

• The first place is the ones place ($6^0$).
• The second place is the sixes place ($6^1$).
• The third place is the thirty-sixes place ($6^2$).

Here is how you count from zero to twelve in senary compared to decimal:

Mathematical Patterns

Multiplication Table

Because 6 is a small number, the multiplication table for base-6 is very short and easy to learn.

Easy Division Rules

In our standard decimal system, it is easy to tell if a number can be divided by 2 or 5 just by looking at the last digit. In base-6, we have similar helpful rules:

• Divisible by 2: If the last digit is 0, 2, or 4.
• Divisible by 3: If the last digit is 0 or 3.
• Divisible by 6: If the last digit is 0.

This makes checking for factors very quick. Because 6 is made by multiplying 2 and 3 (the first two prime numbers), many fractions are also simpler to write.

Fractions and Decimals

In base-10, some fractions like 1/3 are messy. If you write 1/3 as a decimal, it repeats forever: 0.3333...

In base-6, fractions using 2 and 3 are very clean:

• 1/2 in senary is 0.3 (because 3 is half of 6).
• 1/3 in senary is 0.2 (because 2 is a third of 6).
• 1/6 in senary is 0.1.

However, fractions with 5 become repeating numbers in base-6, just like 1/3 does in base-10. For example, 1/5 in senary is written as 0.1111...

Counting with Fingers

Three fingers extended.

Four fingers extended.

Usually, people count to 10 on their fingers. But using a senary method, you can count much higher—up to 35 (which is 55 in base-6)!

Here is how it works:

• Use one hand to count the ones (0 to 5). A fist is 0, one finger is 1, and all five fingers is 5.
• Use the other hand to count the sixes. Every time you reach 6 on the first hand, you raise one finger on the second hand and reset the first hand to 0.

For example, if you have 3 fingers up on your "sixes" hand and 4 fingers up on your "ones" hand, the number is: (3 × 6) + 4 = 22.

In senary, you would write this number as 34. This system is very useful for counting large numbers without needing a pencil and paper. In some sports like NCAA basketball, players wear uniform numbers that can be shown easily with this system so referees can signal them using two hands.

Languages and Cultures

While most of the world uses base-10, a few cultures have developed base-6 systems independently.

• The Ndom language in Indonesia uses base-6. They have basic words for 6, 12 (6×2), and 36 (6×6).
• The Yam languages in Papua New Guinea also use base-6. Some of these languages have special words for very large powers of six, used for counting yams during harvest rituals.

Codes and Puzzles

Sometimes, computer programmers use a system related to senary called Base-36. This system uses numbers 0-9 and letters A-Z to represent values. Since 36 is the square of 6 ($6 \times 6$), it is mathematically related to the senary system. This helps compress large numbers into short codes, which is useful for things like website addresses or passwords.

See also

• Finger counting
• Decimal (Base-10)
• Binary number (Base-2)