# Babylonian Numbers Converter

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!

- Take the number in
**decimal base**and apply an**integer division**by $60$. - Save the
**remainder**of this operation: it is the first digit of the base 60 number. - Repeat the two steps before on the
**quotient**of the integer division. - 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 $60$.

🙋 Try our binary converter, decimal to hexadecimal converter, or binary to hexadecimal converter to learn more about conversion between numerical bases!

## 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:

🙋 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:

**Divide**the number by 60, and note the remainder.- Repeat the previous step on the quotient.
- 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:

- Divide
`13451`

by`60`

:`13451/60 = 224`

with remainder`11`

. - Repeat:
`224/60 = 3`

with remainder`44`

. - 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.