Number system converter

What is a number system

A number system is a way of writing numbers using a specific set of symbols and rules. All the numbers we commonly use are written in the decimal number system, which uses 10 digits (from 0 to 9). However, there are many other systems, each having its own base (or radix). The base of a system shows the number of distinct symbols used to represent numbers.

For example:

• In the binary system — 2 symbols: 0 and 1. Used in computing.
• In the octal system — 8 symbols: from 0 to 7.
• In the decimal system — 10 symbols: from 0 to 9. Used in everyday life and is the most common system.
• In the hexadecimal system — 16 symbols: from 0 to 9 and from A to F, where A = 10, B = 11, …, F = 15. Common in modern computers. For example, colors are often specified in hexadecimal. The color blue is #0000FF.

In more extended systems (for example, base-36) digits and Latin letters are used, where: A = 10, B = 11, …, Z = 35.

How conversion between number systems works

To convert a number from decimal to a base $b$ system:

1. Divide the source number by the base $b$.
2. Record the remainder of the division.
3. Repeat division on the integer quotient until it becomes zero.
4. Write the recorded remainders in reverse order — that is the result.

To convert a number from one base to another, it is common to first convert the number to decimal, and then to the desired base.

How to convert step by step

Step 1. Convert to the decimal system

Suppose we have the number $10110_2$.

Calculate using the formula:

Step 2. Convert from decimal to octal

Let’s convert $22_{10}$ to octal.

Result:

Main number systems

Table of symbols for bases up to 36

Example 1. Convert a decimal number to hexadecimal

Divide 120 by base 16 and write down the remainders until the quotient is zero. Write the remainders in reverse order:

Example 2. Convert 12345₁₀ to base-36

Now write the sequence of remainders in reverse order:

Example 3. Conversion between arbitrary bases

Convert $1101_2$ to hexadecimal.

1. First find the decimal value:

2. Convert 13₁₀ to hexadecimal: Remainder of dividing $13 ÷ 16 = 13 → D$

Result:

Historical fact

The first number systems appeared long before our era.
The ancient Sumerians used a sexagesimal system (base 60) — this is why there are 60 minutes in an hour and 60 seconds in a minute.
Later, the Egyptians and Romans used decimal and vigesimal (base-20) systems in their records, and the idea of positional notation was fully developed in India and transmitted to Europe by Arabic scholars.

Notes

• When entering a number, use only the symbols allowed for the chosen base.
• Letter values for digits start with A=10, B=11 up to Z=35.
• The converter automatically checks the validity of the entered data and instantly provides the result with a detailed explanation in tabular form.

Frequently Asked Questions

How to convert the number 255 from decimal to hexadecimal?

Result:

How to convert 101010₂ to decimal?

How to convert 42₁₀ to octal?

Result:

How to represent 999₁₀ in base-12?

Result:

What is the maximum base supported by this converter?

This converter supports conversions for number systems from 2 to 36.
This covers all possible combinations of digits and Latin letters (0–9, A–Z).