How this instrument works
Binary is a positional counting system built on base 2 rather than base 10 — every computer's underlying arithmetic runs on it, since a digital circuit's most natural state is simply on or off, matching binary's own two digits, 0 and 1. Just like decimal place values (ones, tens, hundreds), binary place values double each step leftward: ones, twos, fours, eights, and so on.
Each bit is found the same way decimal digits are found — repeatedly dividing by the base and tracking the remainder — just done here with 2 instead of 10. Bit 3 (the eights place) comes from dividing by 8 and checking whether that result is itself even or odd; bit 2 (the fours place) does the same after dividing by 4; and so on down to bit 0, the ones place, found directly from the number's own remainder after dividing by 2.
This site's companion pages extend binary further: Binary Addition and Binary Subtraction demonstrate the carrying and borrowing that binary arithmetic needs, while Binary Fraction converts the part of a binary number that sits AFTER the point instead of before it.
- Enter a whole number from 0 to 15 into the Decimal number field.
- Read Binary bit 3 down through Binary bit 0 — the four binary digits, from the eights place down to the ones place.
- Multiply each bit by its place value (8, 4, 2, 1) and add them together to confirm you land back on the original number.
- Try 0 and 15 to see the two extremes of what four bits can represent.
Worked example — converting 13
13 splits into four binary places: bit 3 (the eights place) is 1, bit 2 (fours) is 1, bit 1 (twos) is 0, and bit 0 (ones) is 1 — read together, 1101. Checked in reverse: 1×8 + 1×4 + 0×2 + 1×1 = 13, confirming the conversion.
The two extremes: 0 converts to 0000, every bit off, and 15 (the largest value four bits can hold) converts to 1111, every bit on, since 8+4+2+1=15 exactly.
Questions
Why does binary only use two digits?
Because it's built on base 2 rather than base 10, and a digital circuit's most natural state is simply on or off — matching binary's own two digits, 0 and 1, far more directly than it could match the ten digits decimal needs.
How is a binary place value different from a decimal one?
Decimal places grow by a factor of ten moving leftward (ones, tens, hundreds); binary places instead grow by a factor of two (ones, twos, fours, eights), the same positional idea just built on a smaller, doubling base.
What's the largest number four bits can represent?
15 — every one of the four bits switched on (8+4+2+1), the binary equivalent of 9 being the largest single decimal digit before a second column becomes necessary.
How is this different from the Binary Fraction page on this site?
This page converts a whole number into binary; that page instead converts the FRACTIONAL part of a binary number (the digits after the point) back into an ordinary decimal value.
Which bit is the most significant?
Bit 3, the eights place — it contributes the largest share of the total value, while bit 0, the ones place, contributes the smallest, exactly mirroring how the leftmost decimal digit always carries the most weight.