How Binary Numbers Work, Simply

On this page

Binary numbers work on one rule: every digit is a power of two, doubling from right to left — 1, 2, 4, 8, 16 and so on — and counting rolls over at 2 instead of 10. To read a binary number, add the place values wherever you see a 1: 00010101 is 16 + 4 + 1 = 21. That single rule covers reading, writing and even adding binary numbers; the rest of this article is practice with it.

The One Rule: Every Digit Is a Power of Two

Decimal gives each digit ten possible values (0–9) and powers of ten as place values. Binary gives each digit two possible values (0 and 1) and powers of two. So the rightmost bit is worth 1, the next 2, then 4, 8, 16, 32, 64, 128 — double every step to the left. A bit set to 1 means “add this position’s value”; a 0 adds nothing. There is nothing else to learn.

Counting in Binary: Rollover at 2

Counting works exactly like decimal, just with a smaller alphabet. After 9 decimal rolls over to 10; in binary the rollover comes after 1: 0, 1, then 10 (which is 2), 11 (3), 100 (4). Every time a position would reach 2, it resets to 0 and carries 1 to the left — that is why 1 + 1 = 10 in binary. It looks wrong only until you remember the place is worth two.

Binary vs Decimal: 0 to 16 Side by Side

The rollover rhythm is easiest to see in a table. Watch how each power of two — 2, 4, 8, 16 — adds a digit:

Reading Bigger Numbers: Bytes and Beyond

Eight bits make a byte, holding 0 to 255 — 11111111 is 255, the largest byte. Bigger numbers just extend the same row of powers: 256, 512, 1024. And 1024 has a famously clean pattern — it is exactly 2¹⁰, so in binary it is a 1 followed by ten zeros: 10000000000. Powers of two always look like that, which is why computer memory sizes (512, 1024, 2048) are never round decimal numbers.

Quick Binary Arithmetic

Addition uses the same carrying you learned in school, with the rollover at 2. Try 5 + 3: 0101 + 0011. Rightmost column: 1 + 1 = 0, carry 1. Next: 0 + 1 + 1 = 0, carry 1. Next: 1 + 0 + 1 = 0, carry 1. Leftmost: 0 + 0 + 1 = 1. The result is 1000 — which is 8, and 5 + 3 really is 8. Every other operation builds on this one.

Why Computers Count in Binary

Computer hardware is built from switches that are either on or off, and two states map perfectly onto 1 and 0. Two-state circuits are cheap, fast and almost impossible to confuse electrically — so every count inside a computer happens in base 2. The broader story of why binary won is in what is binary code; this article is about the numbers themselves.

Common Mistakes

• Mistake: reading the place values from the left. 1101 read left-to-right as “1, 2, 4, 8” gives the wrong sum. Correction: the smallest value (1) is always the rightmost bit — count from the right and double going left.
• Mistake: treating a number string as text. 10101 is 21, not a letter. Correction: text needs complete 8-bit bytes; anything shorter is a number. The full distinction is in binary numbers vs binary text.
• Mistake: thinking leading zeros change the value. Correction: for numbers they do not — 0101 and 101 are both 5. Leading zeros only matter when the same bits are meant as text, where bytes must be exactly 8 bits.

One honest limit: this article covers whole numbers only. Fractions (0.5 is 0.1 in binary) and negative numbers (usually stored with two’s complement) follow extra conventions that deserve their own guide — the place-value rule here will not decode them on its own.

Practice

1. Read 1101. The 1s sit at 8, 4 and 1 → 8 + 4 + 1 = 13.

2. Write 42 in binary. 32 fits, leaving 10; 8 fits, leaving 2; 2 fits. Positions 32, 8 and 2 are 1: 101010.

3. Challenge — add 0110 + 0110. That is 6 + 6. Column by column: 0+0=0; 1+1=0 carry 1; 1+1+1=1 carry 1; 0+0+1=1 — the result is 1100 = 12. Verify any of these with the place-value table above or the translator.

Frequently Asked Questions