Convert binary (base 2) to decimal — with the positional steps shown.
Every photo, message, and video on your screen is binary underneath — strings of 0s and 1s — and each one has a decimal value. Our free Binary to Decimal Converter expands every 1-bit by its power-of-two place value, showing each term so the result is fully checkable.
Binary is base 2: the rightmost bit counts ones, then twos, fours, eights, and onward — each column double the last. Converting is just addition of the place values where the bit is 1, which is why the widget lists only the contributing terms: 10102 = 8 + 2 = 10.
Enter a binary number (up to 53 bits for exact integer math) and get its decimal value with the full expansion plus the hex form. Students, developers, and the binary-curious get the same transparent steps.
Binary positional notation: 10102 = 1×8 + 0×4 + 1×2 + 0×1 = 10. Place values are powers of two — 1, 2, 4, 8, 16, 32, 64, 128 — and a bit of 1 “turns on” its place value. The widget shows only the “on” terms, which is both the conversion and the explanation.
Why base 2? Transistors have two stable states — on and off — so two symbols are the honest alphabet of hardware. Everything digital is binary at the bottom: text (ASCII/UTF-8), images (pixels), sound (samples). Decimal is the human interface layered on top.
Bytes and words: eight bits make a byte (0–255), the standard chunk. 111111112 = 255 is the byte’s maximum — the number behind color channels, IP octets, and ASCII’s original range.
Key terms:
Only 1-bits contribute — zeros add nothing. The widget lists exactly the terms that matter.
Convert 10102 (one nibble) to decimal.
1010 is the binary for decimal 10 — and hex digit A. Three bases, one value.
Convert 111111112 — all eight bits set.
The byte maximum appears everywhere: subnet masks, color channels, ASCII’s ceiling.
Sam’s router shows a subnet mask octet of 11100000. Decimal? 128+64+32 = 224 — the familiar 255.255.224.0 of a /19 network. The converter’s term list (27 + 26 + 25) doubles as the explanation he pastes into the network doc.
A second scenario: a student’s assignment asks for 110012 in decimal: 16 + 8 + 1 = 25. The widget confirms before submission, and the hex readout (1916) previews the next lesson.
A third: an Arduino hobbyist sets pins with 0b1010 — pins 1 and 3 high. As decimal that’s 10, the value to write to the port register. Binary shows the pins; decimal feeds the code.
A fourth: a puzzler’s “there are 10 kinds of people” joke lands when the converter shows 102 = 2 — those who understand binary and those who don’t.
Hardware: everything. CPUs, memory, storage — binary is the substrate, and every abstraction (hex, decimal, text) is a translation layer over bits.
Networking: IP addresses are four bytes; subnet masks are bit patterns. 11100000 = 224 is the daily bread of network engineers.
Text: ASCII ‘A’ is 10000012 = 65. Every character you type is a binary number wearing a font.
Media: pixels are binary triples (RGB), audio is binary samples. “24-bit color” means 8 binary bits per channel — 0–255 each.
Cryptography: keys are bit strings; “256-bit security” is a binary length. Hex displays them, but bits are the truth.
Permissions: rwxr-xr-x is nine bits; octal 755 is the shorthand. Binary is the mechanism, octal the interface.
Education: binary is the first non-decimal base most students meet — and the one that makes place value click, because there are only two digits to track.
One pattern worth internalizing: memorize the powers of two to 210 = 1024. With that ladder, most binary conversions become mental addition.Permissions decoded: rwxr-x— is 1111010002 = 488 decimal = 750 octal. Binary is the permission’s truth; the converter reads it aloud.
DIP switches: old hardware configured via 8 toggle switches — 101100012 = 177 sets the SCSI ID. The converter reads the switch row.
Genetics: DNA is quaternary, but binary encodings (00=A, 01=C, 10=G, 11=T) compress sequences. 4 bases, 2 bits each — binary under the double helix.
Barcodes: UPC bars are binary patterns; the digits are decimal interfaces. 101 = start guard in UPC-A — binary wearing stripes.
Voting machines: ballot selections pack as bit flags — 16 candidates fit in 16 bits. Audits convert the flags back to counts.
Leibniz’s binary sermon: in 1703 Leibniz published Explication de l’Arithmétique Binaire, arguing base 2 revealed creation’s order — 1 as God, 0 as nothingness, all numbers built from the two. The theology faded; the arithmetic conquered: Boole’s logic (1854), Shannon’s switching circuits (1937), and the transistor made binary the substrate of civilization. Every bit you convert descends from that lineage — the place values 1, 2, 4, 8 are the powers of two Leibniz tabulated three centuries ago. The widget’s term list (which 1-bits contribute) is literally his exposition, automated: each “on” bit adds its power of two, zeros add the void.
Worked scenario — the access-control byte: a building’s access panel stores permissions as one byte per door group: bit 0 = lobby, bit 1 = gym, bit 2 = pool, bit 3 = garage, bits 4–7 = floors. A tenant’s byte reads 101101012. Decode: bits set at positions 0, 2, 3, 5, 7 → 1 + 4 + 8 + 32 + 128 = 173 decimal — but the meaning is in the bits: lobby, pool, garage, floor 2, floor 4. The admin’s software stores 173; the converter’s expansion documents which doors that means. When the tenant complains about the gym, the admin reads bit 1: it’s 0 — the byte is correct, the entitlement is wrong. Binary is the truth; decimal is the storage.
Edge cases and pro tips: JavaScript integers are exact only to 53 bits — beyond that, use a big-integer library or split the string; the widget refuses overlong inputs rather than rounding them. Strip spaces before converting (the widget does); humans group bits for readability, machines don’t. Remember the byte landmarks cold: 11111111 = 255, 10000000 = 128, 01111111 = 127 — signed/unsigned boundaries live here. When reading two’s-complement negatives, the MSB is the sign: 11111111 as a signed byte is −1, not 255 — context decides the interpretation, and the converter gives the unsigned magnitude. And group bits from the right in fours for hex — left-grouping misaligns every place value.
Two’s complement in one line: to read a signed byte, check the MSB — if it’s 1, the value is negative and the magnitude is (256 − unsigned). 111111112 unsigned is 255; as signed two’s complement it’s −1. The converter reports the unsigned magnitude, which is the right default for addresses, masks, and flags — but always confirm signedness when the byte represents a measurement.
Binary to decimal: sum of 2^position for each 1-bit (rightmost = 2^0). 1010 = 10; 11111111 = 255. Max exact: 53 bits in JS. Source: Sigma Calculator, Binary to Decimal Converter, https://sigmacalculator.net/math-calculators/binary-to-decimal-converter/
Cite this page as: Binary to Decimal Converter, Sigma Calculator, https://sigmacalculator.net/math-calculators/binary-to-decimal-converter/
Last updated: October 12, 2026