Babylonian Numbers Converter

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD and Steven Wooding
Last updated: Mar 31, 2022

Discover the math of one of the first civilizations with our ancient Babylonian numbers converter!

Take a clay tablet and stylus in hand, and dive in with us on a journey in time to the four thousand years old civilization where it (almost) all began. Here you will learn about:

• Mesopotamian math;
• Babylonian numbers; and
• How to convert from decimal numbers to Babylonian, and vice-versa.

And much more!

If Babylonians aren't enough, you can try our other ancient math tools: visit our Mayan numerals converter for another numerical system with another base; our Mayan calendar converter to learn the Mayans' style of counting days; or our Egyptian fractions calculator to learn how to write decimal numbers like an Egyptian!

Babylonian math: calculating with stylus

Mesopotamia, the fertile crescent, was the perfect ground for the development of math. With the creation of a writing system, and the first material used to support such writing, it was finally possible to compute in an abstract fashion.

The first traces of Babylonian math dates back to 3000 BC. Sumerians mastered pretty complex metrology while Babylonians grew a particular interest in numbers and arithmetic. Their clay tablets are preserved enough to give modern historians a wide and varied repertoire of mathematical notions.

Babylonians used their mathematical clay tables mainly for two reasons:

• Help with calculation thanks to computed values; and
• Statements and solutions to problems.

They complemented this knowledge with some basic geometry, which included one of the first attempts to compute the value of $\pi$.

There are two fundamental differences between modern and Babylonian math:

• Babylonians wrote in clay tablets 😁
• Their numeration system was sexagesimal: the base was $60$.

The base of Babylonian numbers

Babylonians used a sexagesimal positional numerical system. This means that they counted in base $60$, using $59$ different symbols ($60$ if we count the zero) and that the position of a digit in a number is nothing but the multiplier of the relative power of $60$.

🙋 Babylonians had a complicated relationship with zero: they didn't know it "existed" for a long time. In later years, though, zeros started appearing, but only in the middle of a number. Babylonians didn't write zeros at the leftmost end: for them, $1$ and $100$ were the same!

How to convert decimal numbers to Babylonian numbers

To convert a decimal number to a Babylonian number, we must change its base from $10$ to $60$. How to do that? Follow these steps!

1. Take the number in decimal base and apply an integer division by $60$.
2. Save the remainder of this operation: it is the first digit of the base 60 number.
3. Repeat the two steps before on the quotient of the integer division.
4. Stop when the quotient is smaller than $60$.

Let's see this with an example: we will convert the number $19281295$ into Babylonian!

• $19281295/60 = 321354$ with remainder $55$;
• $321354/60=5355$ with remainder $54$;
• $5355/60 = 89$ with remainder $15$;
• $89/60=1$ with remainder $29$; and
• $1/60=0$ with remainder $1$.

The Babylonian number is then $1.29.15.54.55$. As you can see, we need to use a period to separate the digits: Arabian numerals are not the best choice to represent numbers in base $60$.

How to write numbers in Babylonian

Babylonians didn't use Arabic numerals (the digits so familiar to us): their math used the Babylonian cuneiform numbers. Here you will learn how to write numbers as a Babylonian!

Babylonians used a stylus and clay tablets instead of pen and paper. This system reduced the possible set of characters available to a scribe while at the same time allowing for a quick correction of mistakes and solid and durable support. The wedged stylus's jagged end could leave some distinct triangular markings; hence the name cuneiform writing system.

Babylonian cuneiform numbers used a combination of two symbols plus the symbol for zero. Each digit uses two of those symbols to represent each number from $1$ to $59$

The first nine digits correspond to an increasing number of vertical wedges:

The other symbol, an open triangle with a "heavier" tip, represented the tens: $10$, $20$, $30$, $40$, and $50$.

As you can easily see, combining two of these 14 symbols allow you to create every number from $1$ to $59$. Zero had its specific placeholder:

Take the number we saw in the previous example: $1.29.15.54.55$. To write it using Babylonian cuneiform numbers, we have to:

• Write the single digits using the Babylonian symbols; then
• Join them in the correct order, ensuring that enough separation allows for identifying the various positions in the number.

You can see the result below. Note how numbers like $29$ use a combination of the symbols seen in the previous section!

How to convert from ancient Babylonian numbers to decimal numbers

What about converting Babylonian numbers to decimal numbers?

First thing, in case you have a Babylonian cuneiform number in front of you, we need to translate it into Arabic numerals.

To do so, count the horizontal and vertical wedges: each of them equals, respectively, $10$ and $1$.

Once your number is written more familiarly, the real power of a positional system comes into play. Each "digit" gets multiplied by the power of $60$ corresponding to its position. The rightmost position equals $0$, and the value increases going left. We then sum the results together.

Take the number $12.9.35.0.22$ in base $60$. Starting from the right, we have:

\footnotesize \begin{align*} &12.9.35.0.22_{60}=22\cdot60^0+0\cdot 60^1+\\ &+35\cdot 60^2+9\cdot 60^3+12\cdot 60^4=\\ &=22+0+126,000+1,944,000+\\ &+ 155,520,000= 157,590,022 \end{align*}

🙋 Since the root of the decimal system ($10$) is smaller than the one of the sexagesimal system ($60$), the same number occupies more positions.

How to use our Babylonian numbers converter

Whether you want to convert from or to Babylonian cuneiform numbers, our tool can help you! Here we will teach you how to use it.

First, choose if you want to convert from or to the Babylonian numbers. The variable direction serves this purpose.

• If you choose to convert from base $10$ to base $60$, insert a positive number. Do not insert symbols or negative numbers. Choose if you want to use zero in the Babylonian style or as in the modern numerical system. We will show you both the numerical conversion and the Babylonian cuneiform number.

🔎 You can choose to visualize or not the expansion: this may be helpful to understand the conversion to Babylonian numbers.

• If you chose to convert from Babylonian numbers to decimal numbers, insert a number in base $60$. Separate the digits using a period, and remember that each digit can't be bigger than $60$. We will translate the number you inserted in cuneiform and then show you the correspondent in base $10$.

FAQ

What are Babylonian numbers?

Babylonian numbers are ancient numbers that used base 60 to perform arithmetic operations. Babylonians developed this numerical system more than four thousand years ago and used them intensively. They were originally written using the Babylonian cuneiform script.

How to convert from base 10 to Babylonian numbers?

The conversion to Babylonian numbers requires converting a number from base 10 to base 60. To do so:

1. Divide the number by 60, and note the remainder.
2. Repeat the previous step on the quotient.
3. Repeat the steps before until the quotient of the division is 0.

The remainder is written from the last to the first, forming the base 60 number.

How do I write 13451 in Babylonian?

To convert the number 13451, follow these steps:

1. Divide 13451 by 60: 13451/60 = 224 with remainder 11.
2. Repeat: 224/60 = 3 with remainder 44.
3. Repeat: 3/60 = 0 with remainder 3.

The resulting base 60 number is 3 44 11.

How do you write numbers in Babylonian?

Babylonians used a combination of two symbols to represent every possible number.

• A vertical wedge to indicate the numbers from 1 to 9; and
• An open triangle indicates the tens: 10, 20, 30, 40, and 50.

With these symbols, you can write every number from 1 to 59, corresponding to a single base 60 digit.

Davide Borchia
Direction
From decimal to Babylonian
Number
₁₀
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