Base Converter

How general base conversion works

The standard method has two stages. First, read the input in its original base by multiplying each digit by the appropriate power of that base and summing the contributions. Second, rebuild the number in the target base by repeated division, recording remainders until the quotient reaches zero.

This is why a base converter should do more than show one final answer. The place-value table explains where the source number comes from, and the repeated-division table explains why the target digits appear in the order they do.

Worked example: converting 2F from base 16 to base 10 and base 2

To read 2F₁₆, take 2 × 16^1 + 15 × 16^0 = 32 + 15 = 47. That gives the exact decimal value. From there, converting to binary means repeatedly dividing 47 by 2, collecting remainders, and reading them in reverse order, which gives 101111₂.

That same example shows why hexadecimal is convenient: one short two-digit hex value expands into six binary digits and a two-digit decimal value, but all three representations still describe the same integer. A strong base converter makes those cross-checks visible at once.

Why base conversion is useful in programming and electronics

Binary, octal, decimal, and hexadecimal are the most common bases people switch between when reading machine instructions, memory addresses, and compact identifiers. A base converter makes that switch explicit so the same integer can be checked in whichever representation is easiest to reason about.

That is also why this page shows a place-value table and a repeated-division table. Those working steps are useful when you want to verify a conversion by hand, not just copy the answer into another tool.

Signs, prefixes, and separators

Programming-style prefixes such as 0b, 0o, and 0x are just input aids. They tell you how to interpret the following digits, but they are not extra digits themselves. This converter strips the matching prefix when it agrees with the selected source base.

Likewise, spaces and underscores are often used to group long values for readability. They do not change the arithmetic, so the calculator ignores them before validating the digits. A leading minus sign is preserved and applied after the unsigned magnitude is parsed.

Reading grouped output without changing the value

Long converted values are easier to check when they are grouped visually. Binary is commonly grouped in four-bit nibbles or eight-bit bytes, octal and decimal are often grouped in threes, and hexadecimal is often grouped in byte pairs when the value represents computer data.

Grouping is display-only. The exact output remains the continuous value shown in the copy-safe result column, while the grouped display column helps you scan the same number for byte boundaries, octal triads, decimal thousands, or likely transcription errors.

Choosing the right base conversion workflow

Use the quick examples when you want a common binary, octal, decimal, hexadecimal, or base-36 conversion without setting every field by hand. Use the source and target base controls when you need a less common radix, such as base 3, base 7, base 12, or base 32. The common-bases table remains visible either way, so a custom base conversion still gives you the familiar binary, octal, decimal, and hex cross-checks.

The swap control is useful when you are checking a reversible conversion. For example, convert decimal 255 to hexadecimal FF, swap the bases, and the input becomes FF from base 16 back to base 10. That makes the page more than a one-way number base converter: it becomes a compact worksheet for confirming that the source digits, target result, and common representations all describe the same integer.

Binary to decimal and binary fraction conversion

For binary to decimal conversion, each 1 bit activates a power of 2. Whole-number places use 2^0, 2^1, 2^2, and so on as you move left. Fractional binary places use 2^-1, 2^-2, 2^-3, and so on as you move right. That is why 1011.01₂ becomes 8 + 2 + 1 + 1/4, or exactly 45/4 = 11.25.

The binary to decimal preset keeps the old directional workflow intact while adding the broader base-converter cross-checks. It accepts grouped bits such as 0001 0101, programmer-style 0b prefixes, and binary fractions such as 1011.01. The exact-value line is especially useful for binary fraction converter searches because it shows the rational value behind the decimal string.

Binary to hex and binary to octal grouping

Binary to hex conversion is fast because one hexadecimal digit maps to exactly four binary bits. Start at the right edge, group the binary value into 4-bit nibbles, pad the leftmost group with leading zeros if needed, and map each nibble to 0 through F. For example, 11011110₂ splits into 1101 and 1110, which become D and E, so the hex result is DE.

Binary to octal uses the same idea with 3-bit triads because 8 = 2^3. The binary value 111101101₂ splits into 111, 101, and 101, which map to octal 755. This grouping context preserves the old binary to hex converter and binary to octal converter search intent without keeping separate indexable pages for each direction.

The in-page binary grouping worksheet keeps that directional detail visible on the canonical base converter. For binary-to-hex and binary-to-octal examples, it shows the original bits, the padded nibble or triad, the group value, and the output digit so left-padding can be checked without confusing it for an extra stored bit.

Decimal to binary, decimal to hex, and decimal to octal

When the source is decimal and the target is binary, hexadecimal, or octal, repeated division is the standard hand-check method. Divide the decimal value by the target base, record each remainder, continue with the quotient, and read the remainders from last to first. That is the workflow behind decimal to binary, decimal to hex, and decimal to octal conversions.

For example, decimal 156 divided repeatedly by 2 gives the binary answer 10011100. Decimal 255 divided by 16 gives FF, and decimal 493 divided by 8 gives 755. The preset chips load those common workflows directly so the page still answers long-tail queries like decimal to binary converter with steps and decimal to hexadecimal converter.

Hex to binary, hex to decimal, hex to octal, and octal back again

Hexadecimal and octal are compact ways to write binary-friendly values. Hex to binary expands each hex digit into a 4-bit nibble, while octal to binary expands each octal digit into a 3-bit group. Hex to octal and octal to hex are often easiest to reason about by expanding through binary first and then regrouping.

Hex to decimal and octal to decimal use ordinary positional notation. A hex digit such as F contributes 15 times its place value, while an octal digit such as 7 contributes seven times its power of 8. The common-bases table shows these cross-checks together, which reduces the need to run separate hex to decimal, octal to decimal, or octal to hex tools.

When the source is hexadecimal or octal and the target is binary, the worksheet separates width-preserved binary from significant binary. A value such as 0x1A is still numerically 11010₂, but the width-preserved display remains 0001 1010 so each original hex digit keeps its full nibble.

Leading zeros, signed values, and two's-complement limits

Leading zeros do not change an unsigned value, but they can matter when you are inspecting fixed-width storage. The bit string 00001111 and 1111 have the same unsigned decimal value, yet the first one visibly fits into an 8-bit byte boundary. The same caution applies to nibble and triad grouping: left-padding clarifies the group, but it is not automatically an extra stored bit.

Signed machine interpretation is a separate layer. The bit string 11111111 is 255 as an unsigned 8-bit value, but it can mean -1 under signed 8-bit two's-complement rules. This base converter reports the mathematical value of the digits you enter. It does not infer storage width, signedness, endian order, IEEE floating-point layout, packed flags, ASCII text, or programming-language parsing semantics.

Practical checks before copying a converted value

Before copying a base conversion into code, documentation, an exam answer, or a hardware note, check the selected source base, the normalized input, and the target-base result together. Many conversion mistakes come from treating a decimal-looking string as hexadecimal, or from pasting a prefixed value while the source base is still set to decimal.

For long values, the common-base outputs help catch transcription mistakes. A binary result that is one bit longer than expected, an octal value with a surprising leading digit, or a hexadecimal value that does not match a familiar byte boundary is a sign to recheck the input grouping before using the result downstream.

What this base converter does not handle

This page supports finite fractional inputs, but a fractional value can terminate in one base and repeat in another. When the target-base fractional expansion repeats, the calculator marks the repeating part in parentheses or limits very long output to a practical displayed length.

It also does not infer floating-point formats, two's-complement storage widths, packed bytes, ASCII text, endian-specific byte structure, or language-specific parsing rules. If your input belongs to a machine format, keep that semantic layer separate from the base conversion itself.