Numbers in Binary

Numbers in binary use only the digits 0 and 1, with each digit position worth double the one to its right. Counting works like decimal counting with faster carrying: zero is 0, one is 1, and two carries over to become 10, the same way ten does in decimal.

The table on this page lists numbers alongside their binary forms so you can see the machinery move. Before you scan it, here’s what to look for.

How binary counting works

Watch the ones column as you read down the table: it alternates 1, 0, 1, 0 the whole way, because odd and even numbers strictly take turns. The twos column flips every two rows, the fours column every four. Each column is a gear spinning half as fast as the gear to its right. That’s all counting is, in any base; decimal’s gears just have ten teeth instead of two.

The rows worth stopping at are the powers of two: 2, 4, 8, 16. Each one is written as a single 1 followed by zeros, like 10000, and each marks the moment binary numbers grow a digit. Their neighbors one step down (3, 7, 15) are solid ones across, like 1111. The pair is binary’s version of 99 rolling over to 100, and it happens so often that you develop a feel for it within minutes of scanning the table.

Every number from 0 through 20 also has its own page with a full place-value breakdown; 10 in binary and 16 in binary are good ones to start with. For numbers beyond the table, the decimal to binary converter handles anything, and the repeated-division technique in how to convert decimal to binary lets you do it yourself. Going the other direction, how to convert binary to decimal covers reading any row of this table from right to left.

One warning that saves confusion later: number encoding and text encoding are different systems. The number 7 in binary is 111, three digits. The character “7” typed in a text file is stored as a full ASCII byte, which is a different, longer pattern. This page and table are about numbers. For character codes, see the binary chart, and for the theory underneath all of it, the binary number system article.

FAQ

How do you count in binary?

Count like decimal, but carry after 1 instead of after 9. Start 0, 1, then you’re out of digits, so two becomes 10 and three is 11. Four carries again to 100. Every column flips half as often as the column to its right.

Why do binary numbers get long so fast?

Each binary digit only carries two possibilities, so you need more of them to say the same thing. Roughly ten binary digits do the work of three decimal digits. Twenty already needs five digits (10100). Computers don’t mind the length; two-state digits are what their circuits do best.

What are the place values in binary?

From the right: 1, 2, 4, 8, 16, 32, 64, 128, doubling forever. A number is the sum of the place values where its digits are 1. So 1101 means 8 + 4 + 1, which is 13. Decimal works identically with places of 1, 10, 100.

Is 10 in binary the same as ten?

No, and this catches everyone. The digits “10” read as binary mean two: a 1 in the twos place, nothing in the ones place. The number ten in binary is 1010. When the base is ambiguous, people write subscripts or say “one zero” aloud to avoid exactly this mix-up.

How high can 8 bits count?

To 255. Eight doubling place values sum to 255 when every digit is 1, giving 256 total values counting zero. That single fact explains many familiar limits: RGB color channels, IPv4 address segments, and classic game stats that cap at 255 are all one byte wide.