Number System Converter
Free online Number System Converter that runs directly in your browser.
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Numbers accompany us at every step, but the way they are written can be very diverse depending on the context. In everyday life, we use the decimal (decimal) system, based on ten digits from 0 to 9. However, in the world of computer technologies, digital electronics and programming, other systems play a key role, such as the binary system (binary - used directly by processors), hexadecimal (hexadecimal - used to record colors in CSS, MAC addresses or memory) and octal (octal). Our free online number systems converter is a universal tool that allows you to instantly convert any number between these popular systems in both directions.
Just enter the number in the appropriate field and the calculator will automatically calculate its representation in other formats, which facilitates learning computer science and the everyday work of programmers and electronics engineers.
Characteristics of the most popular positional number systems
Number systems differ from each other by their base, i.e. the number of characters (digits and letters) used to represent values. The table below presents a summary of the main recording zones:
| System name | Base (Base) | Allowed characters (system alphabet) | Main areas of application |
|---|---|---|---|
| Binary | 2 | 0, 1 | Computer logic, instructions processor machine, logical states (enabled/disabled). |
| Octal | 8 | 0, 1, 2, 3, 4, 5, 6, 7 | Permission systems in Linux systems (e.g. chmod 755), encoding older computer architectures. |
| Decimal (Decimal) | 10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | Daily math, finance, science, trade worldwide. |
| Hexadecimal (Hexadecimal) | 16 | 0-9 and the letters A, B, C, D, E, F (corresponding to values 10-15) | Saving colors in the network (e.g. #FFFFFF), RAM addressing, IPv6 addresses, data dumping (Hex Dump). |
How to convert number systems manually? Division method
Although our converter will do it in a fraction of a second, it is worth knowing the mathematical basics of converting number systems. The most popular method of converting a number from the decimal system to any other is the division method.
Example: Converting the number 13 from decimal to binary
To convert a decimal number to binary, we divide it by 2 and write down the remainder (always 0 or 1). We repeat the process for the result of integer division until we get zero:
- 13 : 2 = 6, remainder1
- 6 : 2 = 3, remainder0
- 3 : 2 = 1, remainder1
- 1 : 2 = 0, remainder1
The result is read from bottom to top:1101. Therefore, the number 13 in decimal is 1101 in binary.
Binary to hexadecimal conversion (grouping)
Conversion between binary and hexadecimal is very simple, because one hexadecimal digit corresponds to exactly four binary digits (so-called tetrad or nibble):
- Divide the binary number into groups of 4 bits from the right (e.g. 11010110 -> 1101 and 0110).
- Convert each four independently to decimal: 1101 = 13 (i.e. the letter D), 0110 = 6.
- Combine the results: 11010110 in binary is D6 in hexadecimal.
Frequently asked questions (FAQ)
Why do computers use the binary system?
Computers rely on transistors, which can be in one of two physical states: conducting current (corresponding to logic 1) or not conducting (corresponding to logic 0). The binary system is the simplest and most stable method of representing these electronic states.
What do the letters A-F stand for in hexadecimal?
In hexadecimal, the base is 16, which means we need 16 unique characters to represent the digits. Since traditional digits end in 9, subsequent values are represented by the first letters of the alphabet: A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
How to write a negative number in the binary system?
In computer science, two's complement code (U2) is most often used to write negative numbers. In this system, the oldest bit (the sign bit on the left) determines the sign of the number: 0 means a positive number and 1 means a negative number. The remaining bits determine its absolute value after appropriate transformation.
What is the octal system used for?
Nowadays, the octal system is less commonly used than the hexadecimal system, but it still plays an important role in Unix and Linux operating systems for representing file access rights (where, for example, rwxr-xr-x permissions are represented as octal 755).
Does this converter support decimal (fractional) numbers?
The presented version of the converter is intended for quick conversion of integers (both small and very large values). Converting fractional parts in non-decimal systems requires the use of special floating-point representations (e.g. the IEEE 754 standard).