Binary Converter

How to use the Binary Converter

What this calculator does. This converter translates whole integers between ordinary decimal notation and a fixed-width binary representation. It also separates the three representations that matter when the decimal number is negative: the binary magnitude of the positive opposite, its one's complement, and the final two's-complement bit pattern used by most modern computers. The tool is an exact integer converter; it does not handle fractions, floating-point encodings, text characters, or arbitrary-precision values beyond 64 signed bits.

When to use it. It is useful when checking a programming exercise, interpreting a register or byte dump, learning signed integer storage, or verifying the bit pattern that should be written to a device. It also helps compare what the same bits mean as an unsigned number and as a signed two's-complement number.

How to calculate. The calculator opens with a complete 8-bit example for decimal – 87, so the results and the Excel workbook are immediately available. Follow these steps:

1. Choose Binary number representation. The selected width determines the signed decimal range and pads every output to exactly that many bits.
2. Replace the value in Decimal to convert base 10 to binary, or replace Binary to convert a bit pattern back to decimal. Editing either field makes it the active source and updates the other field automatically.
3. Read Two's complement binary as the stored signed pattern. Compare it with Positive magnitude, One's complement, Decimal value, and Unsigned value.
4. Use the bit-position table to audit each bit and its power-of-two contribution. Select Download Excel to export the current controls, outputs, and bit rows. Reset clears the demonstration data and may disable export until you enter a complete valid value again.

Input guide. Binary number representation is required and accepts 4, 8, 12, 16, 32, or 64 bits. For example, 8-bit signed integers range from – 128 through 127. Increasing the width preserves the numerical value while adding leading sign-extension bits. A common mistake is choosing too few bits for the desired decimal value. Decimal is a required whole base-10 integer when used as the source; signs are accepted, but grouping commas, decimals, and scientific notation are rejected. An example is – 87. A value outside the displayed range is invalid. Binary is a required string of zeroes and ones when used as the source. It may contain spaces for readability, but no prefixes such as 0b. Up to the selected width is accepted; shorter values are treated as positive and left-padded with zeroes, while a full-width leading 1 is interpreted as negative two's complement.

Output guide. Two's complement binary is the exact fixed-width signed bit pattern. Decimal value is its signed base-10 interpretation. Positive magnitude is the selected-width binary form of the absolute value. One's complement flips every magnitude bit for a negative input; for a nonnegative input it matches the ordinary binary pattern. Unsigned value interprets the stored bits without a sign, so it can be much larger than the signed value when the leading bit is 1. The summary pills show width, signed range, and interpretation mode. In the table, Bit position counts from the least-significant bit at zero, Bit is the stored digit, Unsigned weight is 2 raised to that position, and Unsigned contribution is the weight when the bit is 1 or zero otherwise.

Worked example. With 8 bits and decimal – 87, the positive magnitude is 87 = 0101 0111. Flipping every bit gives the one's complement 1010 1000. Adding one gives the two's-complement pattern 1010 1001. Interpreted unsigned, that pattern equals 169; interpreted as signed 8-bit two's complement, it equals 169 – 256 = – 87. This matches the first-open results and workbook.

Learn more. The University of California, Riverside provides a concise guide to binary, octal, hexadecimal, and decimal conversion. For the broader theory of positional notation, see the Wolfram MathWorld overview of positional number systems.