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Binary

⏱ 6 min read

In one sentence: Binary is a way of writing numbers with only two digits, 0 and 1, and almost all modern computers use it.

Why should I care?

You have probably seen pictures of screens full of 0s and 1s. They come from a real idea: binary. Almost all modern computers and computer-based devices use binary.

Analogy

Think of four lamps in a row, each with its own light switch. Each lamp is either off or on. Nothing in between.

Give each lamp a value. From left to right, the lamps are worth 8, 4, 2 and 1. Each lamp is worth twice the lamp on its right.

To show a number, turn on the lamps that add up to it. To show 5, turn on the 4 lamp and the 1 lamp (4 + 1 = 5). Write "1" for each lamp that is on and "0" for each lamp that is off, from left to right: off, on, off, on becomes 0101. That is 5 in binary. A 0 at the front changes nothing, like in decimal, so 5 is usually written 101.

Where the analogy breaks down: you can see lamps. Inside a computer, the 0s and 1s are not things you can see. They are physical states, such as electrical signals. Also, a row of lamps does not "mean" a number by itself. It means 5 only because we agreed on the values 8, 4, 2 and 1. In a computer, too, the same pattern of 0s and 1s can mean a number or something else, such as a letter. You will see how in Text as Numbers: ASCII to Unicode.

How it actually works

Binary is base 2

In Counting Systems: Base 10 and Other Bases, you saw that our normal numbers are base 10 (decimal). Each position is worth 10 times the position on its right: ones, tens, hundreds, and so on.

Binary, or base 2, follows the same rules, with two changes:

  1. It uses only two digits: 0 and 1.
  2. Each position is worth 2 times the position on its right: 1, 2, 4, 8, 16, 32, 64, 128, and so on.

Each binary digit is called a bit (short for "binary digit"). You will learn more about it in Bit.

Reading a binary number

Take the binary number 1101. Write the worth of each position above the digits, starting with 1 on the right:

worth:         8   4   2   1
binary digit:  1   1   0   1

Diagram: the worths 8, 4, 2, 1 above the digits of 1101.

Now add the worths that have a 1 under them: 8 + 4 + 1 = 13. So 1101 in binary is 13 in decimal.

1101 in binary is read digit by digit: "one, one, zero, one", not "one thousand one hundred and one".

Writing a number in binary

To turn 13 into binary:

  1. List the worths, starting at 1 and doubling: 1, 2, 4, 8, 16. Stop before the first one that is bigger than your number. 16 is bigger than 13, so you use 8, 4, 2, 1.
  2. Start with the biggest. Is 8 bigger than 13? No. Write 1. 13 − 8 = 5 is left.
  3. Is 4 bigger than 5? No. Write 1. 5 − 4 = 1 is left.
  4. Is 2 bigger than 1? Yes. Write 0.
  5. Is 1 bigger than 1? No. Write 1. Nothing is left.

Read the digits in order: 1101.

Counting in binary

Counting uses the carrying you know from decimal. In decimal, a position goes back to 0 after 9, and you carry 1 to the next position (9 becomes 10). In binary, this happens after 1. So 1 becomes 10.

Decimal Binary
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000

Why computers use binary

A computer can store a 0 or a 1 in many physical ways. Mechanical machines can use wheel or lever positions. Electronic chips use electric charge or electrical signals, and hard drives use tiny spots of magnetism. Anything with two separate states, like a switch that is off or on, can store one binary digit.

The reason usually given for binary is that it is straightforward to build with electronic parts. How circuits work with 0s and 1s comes later, in Logic Gates: AND, OR, NOT.

How it came to be

Ideas close to binary are old. The Indian scholar Pingala (around the 2nd century BC) used a binary system to describe patterns of short and long syllables in poetry.

In Europe, Thomas Harriot studied binary, but he did not publish his results. They were found later among his papers. Decades later, Gottfried Leibniz worked on binary. He is usually given the credit for inventing the modern binary system.

Leibniz described binary counting in a handwritten paper dated 15 March 1679, called De progressione dyadica. He never published it. In 1703, he sent a paper on binary to the Academy of Sciences in Paris. Its title begins Explication de l'arithmétique binaire, "Explanation of binary arithmetic", and it says that the system uses only the characters 0 and 1.

In 1937, Claude Shannon, a student at MIT (the Massachusetts Institute of Technology), wrote a long research paper for his degree, called a master's thesis. It showed how switches connected together could do binary arithmetic.

🧪 Try it in your browser

You met the browser console in Counting Systems: Base 10 and Other Bases.

Safety rule: only type instructions in the console that you understand or that come from a source you trust. Never paste anything a stranger sends you. The instructions on this page are safe: they only do arithmetic.

  1. In Chrome, Edge or Firefox, open any ordinary web page, for example https://example.com.

  2. Open the console:

    • Windows: press Ctrl + Shift + J (in Firefox: Ctrl + Shift + K)
    • Mac: press Cmd + Option + J (in Firefox: Cmd + Option + K)

  3. You may see some messages already. Click the empty line at the bottom.

  4. Type this line yourself (some browsers block pasting in the console), then press Enter:

    (13).toString(2)

    If the browser adds a bracket or a quote mark by itself while you type, make sure the line still looks exactly like the one above.

  5. You should see 1101, maybe inside quote marks, like '1101' or "1101". The quote marks mean the answer is text. This is 13 in binary.

  6. Now go the other way. Type this line and press Enter:

    parseInt("1101", 2)

  7. You should see 13, without quote marks: this answer is a number. The 2 in both lines means "base 2".

  8. Try your own numbers: change 13 in step 4 to 255. You get 11111111: eight 1s.

  9. When you are done, close the tab.

On a phone: write the worths 8, 4, 2, 1 on paper and convert a few numbers by hand.

Common misunderstandings

  • "Binary is a special computer language." No. Binary is a way of writing numbers. The number 13 is the same amount whether you write it 13 or 1101.
  • "10 always means ten." Only in decimal. In binary, 10 means two.
  • "Inside the computer there are tiny 0s and 1s." No. The 0s and 1s are how we write down two physical states, such as two kinds of electrical signal.

Recap

  • Binary (base 2) writes numbers with only two digits, 0 and 1.
  • Each position is worth twice the position on its right: 1, 2, 4, 8, 16 and so on.
  • Almost all modern computers use binary. The reason usually given: it is straightforward to build with electronic parts.

Sources

Written with AI · Reviewed by AI · Not yet checked by a human

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