Base-N Number Converter (Bases 2–36)

Introduction to Base-N Conversion

This base-N number converter rewrites an integer in a new radix without changing the quantity behind it. Enter a value in binary, octal, decimal, hexadecimal, or any other base from 2 through 36, and the calculator will restate that same number with the digit symbols that belong to the target base. That is useful when you are reading code, checking memory addresses, working with compact identifiers, or confirming hand conversions from class or lab notes.

The idea is easiest to see when you move between common systems, but the same positional rule works everywhere from base 2 up to base 36. The decimal integer 255 becomes 11111111 in base 2, 377 in base 8, FF in base 16, and 73 in base 36. Those forms look unrelated until you translate the place values behind them; then each one is simply a different spelling of the same integer. This page walks through that logic so the output from the calculator feels predictable rather than mysterious.

Everything happens locally in your browser. Once the page loads, you can convert as many values as you like without waiting for a server round trip. That makes the tool handy for quick experiments, classroom exercises, and offline reference when you want to compare the same integer across binary, octal, decimal, hexadecimal, and higher radices.

How to Use the Base-N Converter

Using the base-N converter is mostly a matter of matching the digits to the correct source radix. Start with the Number field and type the integer exactly as it appears in the source base. Digits 0 through 9 and letters A through Z are supported, and lowercase letters are accepted as the same values as uppercase. If the number is negative, put a minus sign in front, such as -1011 or -2A3. This calculator is for whole numbers, so skip decimal points and fractional parts.

Next, set From Base to the radix that matches the number you entered. That tells the converter how to read each place value. Then choose the target radix in To Base. When you press Convert, the result area shows the rewritten value and its decimal equivalent. Seeing the decimal line helps you verify that the two notations describe the same integer.

A fast validity check saves time. In base 2, only 0 and 1 are allowed. In base 8, 7 is the largest digit. In base 16, the letters stop at F, because A through F stand for 10 through 15. In base 36, the alphabet runs all the way to Z. If you select a source base that cannot contain one of the symbols in your input, the calculator will return an error instead of guessing.

Two easy practice inputs are 1011 from base 2 to base 10 and 2A3 from base 16 to base 2. They are small enough to check by hand, but they still show how the same integer can shrink or grow depending on the radix.

Formula for Base-N Positional Notation

Base-N conversion follows the standard positional rule: each digit is multiplied by a power of the base that matches its position. Reading from right to left, the first position has weight base⁰, the next has weight base¹, then base², and so on. That is why the same sequence of symbols can mean one quantity in one base and a different quantity in another.

Suppose a number in base b has digits d_k through d₀. Its value N can be written as the sum of each digit multiplied by the correct power of the base:

Here, each digit d_i must be an integer from 0 up to, but not including, the base b. So in base 8 the digit 8 is illegal, and in base 16 the digit G is illegal. The converter first interprets your input using this positional formula in the source base, then rewrites the same integer in the target base.

The reverse step is often described with repeated division. If N is a nonnegative integer and b is the target base, repeated division by b produces remainders that become the digits of the answer. Reading those remainders from last to first gives the final representation.

That second relation explains why conversion output is never arbitrary. Every digit in the result corresponds to a remainder and therefore to a place value. Whether you do the work by hand or let JavaScript perform it instantly, the mathematical idea is the same.

Supported Bases and Digit Symbols

This converter accepts any whole-number base from 2 through 36. The symbol set is simple: values 0 through 9 use 0 through 9, and values 10 through 35 use A through Z. Lowercase letters are accepted in the input, but the output is normalized to uppercase so the answer is consistent and easier to read in code, notes, and technical documentation.

Several bases appear repeatedly in computing. Base 2 is binary, the language of bits. Base 8 is octal, which compresses binary into groups of three bits. Base 10 is decimal, the everyday human system. Base 16 is hexadecimal, a compact representation because each hex digit corresponds to four bits. Base 36 is handy when you want a short text form that uses both digits and letters.

How the Browser Converts Between Radices

Internally, the calculation follows two stages. First, the input string is parsed in the source base to recover the integer value it represents. Second, that same integer is formatted again in the target base. On this page, browser-side JavaScript handles both steps immediately after form submission, which means the converter remains responsive and private for typical use.

The most important assumption is that the input is an integer and every character belongs to the chosen source base. A leading minus sign is allowed for negative values. Once the integer value has been identified, the target representation is just another way to write it using a different set of place-value weights. That is why 11 in base 10 becomes 1011 in base 2, while 1011 in base 2 becomes 11 in base 10. The quantity stays fixed; only the notation changes.

Example: Converting Hex 2A3 to Binary

A worked example makes the idea concrete. Suppose you want to convert the hexadecimal number 2A3 to decimal and then to binary. In hexadecimal, the symbol A stands for 10. So the number is read as one digit times 16 squared, plus one digit times 16 to the first power, plus one digit times 16 to the zero power.

Expanding the value gives 2 × 16² + 10 × 16¹ + 3 × 16⁰. That becomes 512 + 160 + 3, which equals 675. Therefore 2A3 in base 16 is exactly the same quantity as 675 in base 10.

To continue into binary, divide 675 by 2 repeatedly and record the remainders. Reading the remainders from last to first yields 1010100011. So the full chain is 2A3₁₆ = 675₁₀ = 1010100011₂. If you enter 2A3 with From Base 16 and To Base 2 in the form below, the calculator should return that same binary value.

A smaller example is 1011 in base 2. Using positional weights, it equals 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰, which simplifies to 8 + 0 + 2 + 1 = 11. That is why the calculator returns 11 in decimal when you convert binary 1011. The tool is not inventing a new number; it is simply expressing the same integer in a different system.

Comparison Table for 255 Across Common Bases

Looking at the same value in several bases is a good way to build intuition. The table below shows how the decimal integer 255 appears in several common systems. Notice how smaller bases tend to need more symbols, while larger bases can express the same value more compactly.

That pattern explains why programmers often prefer hexadecimal when inspecting bytes, colors, or memory values. Hex is much shorter than binary, but it still maps cleanly to the same underlying bits.

Limitations of Base-N Conversion

This calculator is intentionally focused. It is excellent for quick whole-number base changes, but it is not a full symbolic math system. The input must represent an integer, the bases must stay between 2 and 36, and every digit must be valid for the selected source base. Those constraints keep the interface straightforward and keep the results easy to interpret.

• Integers only: fractional parts such as 3.14 or 101.101 are not supported.
• Base range 2–36: unary, base 1, and bases above 36 are outside the supported symbol set.
• Valid characters required: only digits 0–9 and letters A–Z are accepted.
• Digit limits depend on the base: for example, 8 is not allowed in base 8 and G is not allowed in base 16.
• Negative numbers are allowed: use a leading minus sign, such as -FF or -1011.
• Very large values can be difficult: the page uses standard browser number handling, which is fine for ordinary study and programming examples but may lose exactness for extremely large integers.
• No arithmetic is performed: the tool converts notation only; it does not add, subtract, multiply, or divide separate inputs.

One practical note is worth stressing. If you are working with enormous values, such as very long identifiers or cryptographic-scale integers, you may want an arbitrary-precision or BigInt-based tool instead. For classroom problems, debugging, quick sanity checks, and ordinary technical work, though, this calculator is fast, clear, and dependable.

Once those assumptions are clear, the workflow is simple: enter the digits, choose the source base, choose the target base, and read the same integer in a new notation.