Duodecimal facts for kids

The duodecimal system, also known as base twelve or dozenal, is a numeral system that uses twelve as its base. Think of it like our regular decimal system, which uses base ten. In duodecimal, the number twelve is written as "10". This means 1 group of twelve and 0 single units. In our everyday decimal system, "10" means one group of ten and zero units.

To count in duodecimal, we need special symbols for the numbers ten and eleven. On this page, we will use 'A' for ten and 'B' for eleven, similar to how hexadecimal (base sixteen) works. So, counting from zero to twelve in duodecimal would look like this: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, and then 10.

Twelve is a very special number! It's the smallest number that can be divided evenly by four different numbers (2, 3, 4, and 6), not counting 1 and 12 itself. This makes it very useful for dividing things into equal parts. For example, one-half (0.6), one-third (0.4), and one-fourth (0.3) all have short, exact representations in duodecimal. This is often seen as an advantage over the decimal system, where fractions like one-third go on forever (0.333...).

Understanding the Duodecimal System

What is Base Twelve?

A number system's "base" tells you how many unique digits it uses before it rolls over to the next place value. Our common decimal system uses base ten, with digits 0 through 9. When you reach ten, you write "10". The duodecimal system uses base twelve. It needs twelve unique digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and two new symbols for ten and eleven.

In duodecimal:

• "10" means twelve (like "12" in decimal).
• "100" means twelve times twelve, which is 144 in decimal.
• "1,000" means twelve times twelve times twelve, or 1,728 in decimal.
• "0.1" means one-twelfth (about 0.08333... in decimal).

Why Twelve is a "Friendly" Number

Twelve is considered a "friendly" number for calculations because it has many factors. Factors are numbers that divide evenly into another number. The factors of 12 are 1, 2, 3, 4, 6, and 12. Compare this to 10, whose factors are only 1, 2, 5, and 10.

Having more factors means that many common fractions can be written neatly in duodecimal. For example, if you want to divide something into halves, thirds, or quarters, duodecimal handles these easily. This makes calculations simpler for many everyday tasks.

A Look at History: Where Twelve Was Used

Counting on Fingers and Ancient Times

Some historians believe the duodecimal system might have started with a unique way of finger counting. Imagine using your thumb to point to the three bones (knuckles) on each of your other four fingers. This lets you count up to 12 on one hand! This method is still used in some parts of Asia today.

Ancient civilizations also used twelve in their measurements of time. The Babylonians, for example, had twelve hours in a day. Our modern calendar still uses twelve months in a year, and there are twelve signs of the zodiac.

Twelve in Everyday Life and Measurements

Even today, we see the number twelve everywhere!

• There are 12 inches in an imperial foot.
• We often buy things by the dozen (12 items).
• A gross is 12 dozens, or 144 items.
• A great gross is 12 gross, or 1,728 items.
• Our day has 24 hours (which is 2 x 12).

The ancient Romans even used a system of fractions based on 12. Their "uncia" became our words "ounce" and "inch." Many parts of Europe historically used a money system where 12 pence made a shilling.

New Symbols for New Numbers

How We Write Ten and Eleven

Since the duodecimal system needs symbols for ten and eleven, many ideas have been suggested. As mentioned, on this page, we use 'A' for ten and 'B' for eleven. These are easy to type and understand, similar to how they are used in hexadecimal.

Other groups, like the Dozenal Society of Great Britain, use special symbols that look like turned-around digits 2 and 3 for ten and eleven. These symbols were even added to the Unicode Standard in 2015, so computers can now easily display them.

Saying Duodecimal Numbers Aloud

There isn't one official way to say duodecimal numbers. However, the Dozenal Society of America suggests pronouncing ten as "dek" and eleven as "el."

We already have words for some powers of twelve in English:

• Twelve (10 in duodecimal) is a "dozen".
• Twelve squared (100 in duodecimal, or 144 in decimal) is a "gross".
• Twelve cubed (1000 in duodecimal, or 1,728 in decimal) is a "great gross".

Why Some People Love Duodecimal

Easier Math with More Factors

Many people believe the duodecimal system is better than decimal for everyday math. Alexander Aitken, a famous mathematician, said that duodecimal tables are easier to learn than decimal ones. He also thought that calculations could be done much faster in base twelve.

The main reason for this is the number of factors. Since 12 can be divided by 2, 3, 4, and 6, it makes working with fractions like 1/2, 1/3, 1/4, and 1/6 much simpler. In the decimal system, 1/3 is a repeating decimal (0.333...), but in duodecimal, it's a neat 0.4. This means fewer messy repeating numbers in many common calculations.

A duodecimal clockface, like the logo of the Dozenal Society of America, showing musical keys.

Organizations like the Dozenal Society of America (founded in 1944) and the Dozenal Society of Great Britain (founded in 1959) actively promote using the duodecimal system. They believe it would simplify mathematics and measurements.

Comparing Twelve to Other Bases

How Fractions Behave in Different Systems

When you write a fraction in any number system, it will either end exactly (terminate) or repeat forever. A fraction terminates if the prime factors of its denominator are also prime factors of the base.

• Decimal (Base 10): The prime factors of 10 are 2 and 5. So, fractions like 1/2, 1/4, 1/5, 1/8, 1/10, 1/20, etc., terminate. But 1/3, 1/6, 1/7, 1/9, 1/11, etc., repeat.
• Duodecimal (Base 12): The prime factors of 12 are 2 and 3. This means fractions like 1/2, 1/3, 1/4, 1/6, 1/8, 1/9, 1/12, etc., terminate. Fractions with 5 or 7 in their denominator (like 1/5 or 1/7) will repeat.

Because 3 is a very common factor in real-life division (think of dividing things into three equal parts), duodecimal often avoids repeating decimals more often than the decimal system.

Here's a quick look at how some fractions appear in decimal and duodecimal:

The Duodecimal Multiplication Table

Learning a new number system means learning new multiplication facts! Here is the multiplication table for duodecimal. Remember, 'A' is ten and 'B' is eleven.

Converting Between Duodecimal and Decimal

Simple Steps for Changing Bases

Converting numbers between duodecimal and decimal might seem tricky at first, but it follows a logical pattern. Each digit in a number system has a "place value" based on the system's base.

For example, in decimal, the number 123 means (1 x 100) + (2 x 10) + (3 x 1). In duodecimal, the number 123 means (1 x 12²) + (2 x 12¹) + (3 x 12⁰). So, duodecimal 123 would be (1 x 144) + (2 x 12) + (3 x 1) = 144 + 24 + 3 = 171 in decimal.

To convert a decimal number to duodecimal, you repeatedly divide the decimal number by 12 and keep track of the remainders. The remainders, read from bottom to top, form the duodecimal number. Remember to use 'A' for a remainder of 10 and 'B' for a remainder of 11.

See also

In Spanish: Sistema duodecimal para niños

• Vigesimal (base 20)
• Sexagesimal (base 60)