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Counting Systems: Base 10 and Other Bases

⏱ 7 min read

In one sentence: A counting system writes any number with a small set of digits, and the position of each digit tells you how much it is worth; we normally count in base 10, but other bases work the same way.

Why should I care?

Every number you have ever written uses a counting system. When you write 365, the 3 does not mean three. It means three hundred. Its position tells you that.

You already use a second counting system every day: the clock. There are 60 seconds in a minute and 60 minutes in an hour. That habit comes from a system that counted in sixties, thousands of years ago.

Computers use a third system, with only two digits: Binary. It works the same way, with fewer digits.

Analogy

Imagine you are packing beans. Each bag holds exactly ten beans, and each box holds ten bags. At the end, you have 3 boxes, 6 bags and 5 loose beans. You can write that as 365: the digits count boxes, bags and loose beans.

With five beans per bag and five bags per box, you would be counting in base 5.

Where the analogy breaks down: in real life, you could leave 14 loose beans on the table. In a written number, each position holds one digit only, from 0 to 9. When it reaches ten, you must pack them into the next position. Also, you may run out of container sizes, but a number can always add a new position on the left.

How it actually works

A digit is a single symbol for a number, like 7. The base of a counting system is how many different digits it uses, including zero.

Our normal system is base 10, also called decimal. It has ten digits: 0 1 2 3 4 5 6 7 8 9.

The value of a digit depends on its position. This idea is called place value. Each position is worth the base times the worth of the position on its right:

Position hundreds tens ones
Worth 10 × 10 = 100 10 1
Digit in 365 3 6 5
Value 300 60 5

So 365 means 3 × 100 + 6 × 10 + 5 × 1.

Counting up. You count 7, 8, 9. Then there is no digit left for ten. So you write 0 in the ones position and add 1 to the tens position: 10. This is called carrying: a full position goes back to 0, and you carry 1 to the next position. The same thing happens at 99, which becomes 100.

Zero matters. Zero marks an empty position. Without it, you could not tell 2106 from 216.

Other bases follow the same rules. Only the number of digits changes. Here is base 5. It has five digits, 0 to 4. Its positions are worth 1, then 5, then 5 × 5 = 25, and so on. Count up and carry as soon as a position reaches five:

Amount (in base 10) 4 5 6 9 10 11 24 25
Written in base 5 4 10 11 14 20 21 44 100

At 44, both positions are full. Adding one more carries twice, and you get 100: one twenty-five. So the same digits can mean different amounts, and you always need to know the base.

Base 2 has only two digits, 0 and 1. It is used in almost all computers and electronic devices. Binary explains it.

A base bigger than 10 needs more than ten digits. Base 16 uses the letters a to f as the extra digits: a means ten, b means eleven, and so on up to f, which means fifteen.

Base 60 is still with us. Each degree of an angle is divided into 60 smaller parts (arcminutes), and each of those into 60 again (arcseconds). Time works the same way: 60 seconds in a minute, 60 minutes in an hour.

How it came to be

Base 60 most likely started with the Sumerians, an ancient people, and the Babylonians inherited it. The Babylonians used place value for over 1,000 years without a sign for zero. Readers had to work out empty positions from the context. Only around 400 BC (about 2,400 years ago) did they start to put two small marks in empty positions.

Other people chose other bases. The MacTutor history of mathematics links base 10 to counting on fingers, and base 20 to counting on fingers and toes. By the year 665, the Maya of Central America used a place-value system in base 20, with a symbol for zero.

The base-10 system we use today was first developed in India. Some evidence suggests it existed as early as the first century AD (about 2,000 years ago). By around 650 AD, zero was also used as a number of its own in Indian mathematics, not only as a mark for an empty position. The system then passed to the Arabs, and later to Europe.

In 1202, the Italian mathematician Fibonacci published a book called Liber abaci. It helped bring this place-value decimal system into Europe. Today it is called the Hindu–Arabic numeral system, and it is the most common numeral system (way of writing numbers) used today.

🧪 Try it in your browser

Your browser has a hidden tool for programmers called the console. You type a short instruction, and it shows the answer. Nothing on your computer changes.

  1. Open any website in Chrome, Edge or Firefox, for example https://example.com.

  2. Open the console:

    • Chrome or Edge: press Ctrl + Shift + J (on a Mac: Cmd + Option + J).
    • Firefox: press Ctrl + Shift + K (on a Mac: Cmd + Option + K).

    A panel opens. Ignore any messages already in it. Click at the bottom of the panel, where you can type.

  3. Type this yourself, exactly as shown, including the brackets and the dot, and press Enter: (25).toString(5)

    This asks the browser to write 25 as text, in base 5. You will see 100, maybe in quote marks (they show that the answer is text). It matches the base 5 table.

  4. Now type (25).toString(2) and press Enter. You will see 11001: twenty-five written in base 2.

  5. Go the other way. Type parseInt("100", 5) and press Enter. This instruction reads 100 as a base 5 number and shows its value in base 10. You will see 25.

  6. Try your own numbers: change 25 and the base in steps 3 and 4. The base must be between 2 and 36.

  7. When you are done, close the tab.

On a phone, try it on paper instead: write 0 to 25 in base 5, and check that you reach 100.

Common misunderstandings

  • "A number and its digits are the same thing." No. A number is an amount; digits are symbols that write it. Twenty-five is 25 in base 10, 100 in base 5 and 11001 in base 2.
  • "10 always means ten." No. In any base, 10 means "one group the size of the base, and zero ones". In base 2, 10 means two. In base 5, it means five.
  • "Base 10 is the only real way to count." No. It is one choice. Your clock uses 60, and computers use base 2.

Recap

  • A digit's position tells you how much it is worth. Each position is worth the base times the one on its right.
  • Base 10 uses ten digits (0 to 9); base 5 uses five; base 2 uses two. When a position is full, you carry 1 to the next position.
  • Base 60 came from the Sumerians and Babylonians and still shapes our clocks; our base-10 system came from India.

Sources

Written with AI · Reviewed by AI · Not yet checked by a human

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