Numeral Systems Converter
Convert numbers between binary, decimal, octal, hexadecimal, and custom bases from 2 to 36. This converter supports negative numbers, fractional values, uppercase or lowercase digits, instant validation, batch conversion, common base presets, conversion steps, place-value formulas, and quick outputs for the most-used numeral systems in mathematics and computer science.
Convert a Number
Result
| Output | Value | Meaning |
|---|
Batch Converter
Enter one number per line. The batch tool uses the same input and output base settings above.
Conversion Steps
What Is a Numeral Systems Converter?
A numeral systems converter changes a number from one base into another base. The most common numeral systems are binary, decimal, octal, and hexadecimal. Decimal is base 10 and uses the digits \(0\) through \(9\). Binary is base 2 and uses only \(0\) and \(1\). Octal is base 8 and uses \(0\) through \(7\). Hexadecimal is base 16 and uses \(0\) through \(9\) plus the letters \(A\) through \(F\). This converter also supports any custom base from 2 to 36.
Numeral systems matter because computers store and process data using binary, programmers often read compact binary data in hexadecimal, permissions and legacy systems often use octal, and humans usually calculate in decimal. A converter helps connect these representations so the same value can be understood in different contexts.
The central formula for a positional numeral system is:
\[ N = d_n b^n + d_{n-1} b^{n-1} + \cdots + d_1 b^1 + d_0 b^0 \]
If the number has fractional digits, the formula continues with negative powers:
\[ N = \sum_{i=-m}^{n} d_i b^i \]
Here, \(b\) is the base, \(d_i\) is the digit at position \(i\), and \(N\) is the value of the number in decimal form. This is the mathematical foundation of every base conversion.
Common Numeral Systems
| System | Base | Digits used | Common use |
|---|---|---|---|
| Binary | 2 | 0, 1 | Computer logic, bits, machine representation |
| Ternary | 3 | 0, 1, 2 | Mathematics, balanced ternary studies |
| Quaternary | 4 | 0, 1, 2, 3 | Computer science theory, compact binary grouping |
| Octal | 8 | 0–7 | Unix permissions, older computing systems |
| Decimal | 10 | 0–9 | Everyday arithmetic and measurement |
| Duodecimal | 12 | 0–9, A, B | Alternative counting systems, divisibility discussions |
| Hexadecimal | 16 | 0–9, A–F | Programming, colors, memory addresses, bytes |
| Base 36 | 36 | 0–9, A–Z | Compact IDs, encoded strings, alphanumeric systems |
How Base Conversion Works
To convert from any base to decimal, multiply each digit by its place value and add the results. For example:
\[ 1011_2 = 1\cdot2^3 + 0\cdot2^2 + 1\cdot2^1 + 1\cdot2^0 \]
\[ 1011_2 = 8 + 0 + 2 + 1 = 11_{10} \]
To convert decimal into another base, repeatedly divide the integer part by the target base and record the remainders. The remainders, read from bottom to top, form the converted number. For fractional parts, repeatedly multiply the fraction by the target base and record the integer parts that appear.
For example, convert \(11_{10}\) to binary:
11 ÷ 2 = 5 remainder 1 5 ÷ 2 = 2 remainder 1 2 ÷ 2 = 1 remainder 0 1 ÷ 2 = 0 remainder 1 Read upward: 1011
Therefore:
\[ 11_{10} = 1011_2 \]
Binary, Decimal, Octal, and Hexadecimal Chart
| Decimal | Binary | Octal | Hexadecimal |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 2 | 10 | 2 | 2 |
| 3 | 11 | 3 | 3 |
| 4 | 100 | 4 | 4 |
| 5 | 101 | 5 | 5 |
| 6 | 110 | 6 | 6 |
| 7 | 111 | 7 | 7 |
| 8 | 1000 | 10 | 8 |
| 9 | 1001 | 11 | 9 |
| 10 | 1010 | 12 | A |
| 11 | 1011 | 13 | B |
| 12 | 1100 | 14 | C |
| 13 | 1101 | 15 | D |
| 14 | 1110 | 16 | E |
| 15 | 1111 | 17 | F |
| 16 | 10000 | 20 | 10 |
Place Value Formula Explained
In decimal, the number \(472\) means:
\[ 472_{10} = 4\cdot10^2 + 7\cdot10^1 + 2\cdot10^0 \]
The same logic works in every base. For binary:
\[ 1101_2 = 1\cdot2^3 + 1\cdot2^2 + 0\cdot2^1 + 1\cdot2^0 \]
\[ 1101_2 = 8 + 4 + 0 + 1 = 13_{10} \]
For hexadecimal:
\[ 2F_{16} = 2\cdot16^1 + 15\cdot16^0 = 32 + 15 = 47_{10} \]
The letter \(F\) represents digit value \(15\), because hexadecimal needs sixteen digit symbols: \(0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F\).
Fractional Base Conversion
Numeral systems can also include fractions. For example:
\[ 101.101_2 = 1\cdot2^2 + 0\cdot2^1 + 1\cdot2^0 + 1\cdot2^{-1} + 0\cdot2^{-2} + 1\cdot2^{-3} \]
This becomes:
\[ 101.101_2 = 4 + 0 + 1 + 0.5 + 0 + 0.125 = 5.625_{10} \]
Fraction conversion may sometimes repeat forever. For example, \(0.1_{10}\) cannot be represented exactly with a finite binary fraction. This is why calculators often allow a selected number of fraction digits. This converter lets you choose how many fractional digits to generate.
Binary to Decimal
Binary to decimal conversion is one of the most important base conversions in computer science. Binary uses only two symbols: \(0\) and \(1\). Each position represents a power of \(2\). Starting from the right side of the integer part, the place values are \(2^0,2^1,2^2,2^3,\ldots\).
Example:
\[ 11101_2 = 1\cdot2^4+1\cdot2^3+1\cdot2^2+0\cdot2^1+1\cdot2^0 \]
\[ 11101_2 = 16+8+4+0+1=29_{10} \]
Decimal to Binary
To convert decimal to binary, divide by \(2\) repeatedly and record the remainders. Example:
29 ÷ 2 = 14 remainder 1 14 ÷ 2 = 7 remainder 0 7 ÷ 2 = 3 remainder 1 3 ÷ 2 = 1 remainder 1 1 ÷ 2 = 0 remainder 1
Read the remainders from bottom to top:
\[ 29_{10}=11101_2 \]
Decimal to Hexadecimal
Hexadecimal is base \(16\). It is compact and maps neatly to binary because one hexadecimal digit represents four binary bits. This is why hex is common in programming, memory addresses, color codes, byte dumps, Unicode values, and low-level computing.
Example:
255 ÷ 16 = 15 remainder 15 15 ÷ 16 = 0 remainder 15
The digit value \(15\) is written as \(F\), so:
\[ 255_{10}=FF_{16} \]
Hexadecimal to Binary Shortcut
Hexadecimal converts to binary very quickly because each hex digit equals exactly four binary digits:
| Hex | Binary | Hex | Binary |
|---|---|---|---|
| 0 | 0000 | 8 | 1000 |
| 1 | 0001 | 9 | 1001 |
| 2 | 0010 | A | 1010 |
| 3 | 0011 | B | 1011 |
| 4 | 0100 | C | 1100 |
| 5 | 0101 | D | 1101 |
| 6 | 0110 | E | 1110 |
| 7 | 0111 | F | 1111 |
For example:
\[ A7_{16}=1010\ 0111_2 \]
Octal and Unix Permissions
Octal is base \(8\), using digits \(0\) through \(7\). It was historically common in computing because one octal digit represents exactly three binary bits. Octal is still visible in Unix and Linux file permissions. For example, permission \(755\) is often used for executable directories or scripts:
\[ 7 = 111_2,\quad 5 = 101_2,\quad 5 = 101_2 \]
So:
\[ 755_8 = 111\ 101\ 101_2 \]
In permission language, \(7\) means read, write, execute; \(5\) means read and execute.
Custom Bases from 2 to 36
This converter supports custom bases from \(2\) to \(36\). For bases greater than \(10\), letters are used as digit symbols. Base \(16\) uses \(A\) through \(F\). Base \(36\) uses all digits \(0\) through \(9\) and all letters \(A\) through \(Z\). The digit \(Z\) represents value \(35\).
A digit is valid only if its value is less than the base. For example, \(8\) is not valid in octal because octal digits only go from \(0\) to \(7\). \(G\) is not valid in hexadecimal because hexadecimal ends at \(F\). The calculator validates input before converting.
Negative Numbers
This converter supports negative numbers with a leading minus sign. For example:
\[ -1010_2 = -10_{10} \]
This is standard signed notation. It is different from fixed-width two’s complement representation, where a binary pattern may represent a negative value depending on bit width. For example, in 8-bit two’s complement, \(11111111_2\) means \(-1\), but as an unsigned binary number it means \(255\). This converter uses direct signed notation unless you manually enter the interpretation you need.
Common Mistakes
| Mistake | Why it happens | Correct approach |
|---|---|---|
| Using digit 2 in binary | Binary only has 0 and 1 | Use base 2 digits only: \(0,1\) |
| Using 8 or 9 in octal | Octal ends at 7 | Use digits \(0\) through \(7\) |
| Reading remainders in the wrong order | Decimal-to-base conversion remainders must be reversed | Read remainders from bottom to top |
| Assuming every decimal fraction has a short binary form | Some fractions repeat in another base | Use enough fractional digits and understand approximation |
| Confusing hexadecimal letter values | A through F represent 10 through 15 | Remember \(A=10\), \(B=11\), \(C=12\), \(D=13\), \(E=14\), \(F=15\) |
Worked Examples
Convert \(1011_2\) to decimal
\[ 1011_2 = 1\cdot2^3+0\cdot2^2+1\cdot2^1+1\cdot2^0 \]
\[ 1011_2 = 8+0+2+1=11_{10} \]
Convert \(255_{10}\) to hexadecimal
\[ 255 \div 16 = 15 \text{ remainder } 15 \]
\[ 15 \div 16 = 0 \text{ remainder } 15 \]
Since \(15=F\), the answer is:
\[ 255_{10}=FF_{16} \]
Convert \(7A_{16}\) to decimal
\[ 7A_{16}=7\cdot16^1+10\cdot16^0 \]
\[ 7A_{16}=112+10=122_{10} \]
Convert \(13.5_8\) to decimal
\[ 13.5_8=1\cdot8^1+3\cdot8^0+5\cdot8^{-1} \]
\[ 13.5_8=8+3+\frac{5}{8}=11.625_{10} \]
How to Use This Converter
- Enter the number you want to convert.
- Choose the original base, such as binary, decimal, octal, hexadecimal, or a custom base from 2 to 36.
- Choose the target base.
- Select how many fractional digits should be generated for fractional conversions.
- Click “Convert number.”
- Review the main result, quick outputs, conversion steps, and common-base equivalents.
- Use the batch converter if you have multiple numbers to convert.
Why This Page Does Not Include Exam Score Tables
A numeral systems converter is a mathematics and computer-science tool, not an exam score calculator. Therefore, score guidelines, score tables, and next exam timetables do not apply directly to this page. The equivalent high-value material for this topic is correct base-conversion logic, formulas, examples, charts, validation rules, binary/decimal/octal/hex shortcuts, fractional conversion support, and clear explanations for students and programmers.
Numeral Systems Converter FAQs
What is a numeral system?
A numeral system is a way to represent numbers using a base and a set of digits. Decimal is base 10, binary is base 2, octal is base 8, and hexadecimal is base 16.
What is the formula for converting a base to decimal?
Use \(N=\sum d_i b^i\), where \(b\) is the base, \(d_i\) is the digit value, and \(i\) is the position power.
How do I convert decimal to binary?
Repeatedly divide the decimal integer by 2, record the remainders, and read the remainders from bottom to top.
Why does hexadecimal use letters?
Hexadecimal is base 16, so it needs sixteen digit symbols. After 0 through 9, the letters A through F represent 10 through 15.
Can this converter handle fractional numbers?
Yes. It supports fractional values such as \(101.101_2\) or \(13.5_8\). Fractional outputs may be rounded to the selected number of digits.
What bases are supported?
This converter supports bases from 2 through 36.
Why is 8 invalid in octal?
Octal is base 8, so valid digits are only 0 through 7. A digit must always be less than the base.
What is binary used for?
Binary is used in computers because digital circuits naturally represent two states, commonly interpreted as 0 and 1.
What is hexadecimal used for?
Hexadecimal is used in programming, memory addresses, color codes, byte values, debugging, and compact binary representation.
Can this converter do batch conversion?
Yes. Enter one number per line in the batch converter and it will convert each line using the selected bases.
Suggested internal links: binary to decimal converter, decimal to binary converter, hexadecimal converter, octal converter, ASCII converter, bit calculator, bytes converter, and programming tools.
