Binary Numbers

A World With Two Symbols

Deep inside every computer there are billions of microscopic switches. Each switch is either on or off — nothing in between. Two states is all the hardware can reliably offer, so computing needed a number system with just two digits.

That system is binary (base 2). Its digits are 0 (off) and 1 (on), and one binary digit is called a bit.

Place Value — Doubling Instead of Tens

You already know decimal place value: each column is worth 10 times the one to its right (…1000, 100, 10, 1).

Binary works identically, except each column is worth 2 times the one to its right:

   place value:    8     4     2     1
   binary:         1     1     0     1
                   ↓     ↓     ↓     ↓
                   8  +  4  +  0  +  1   =  13

So binary 1101 is decimal 13. That is the entire trick — same place-value idea you learned in primary school, different multiplier.

Binary to Decimal

Write the place values above the bits, then add up the columns with a 1:

Note the last row: 1111 = 15 is the biggest value 4 bits can hold. Counting from 0, that gives 16 different values — with n bits you get 2ⁿ values.

Decimal to Binary

To convert decimal to binary, take the biggest place value that fits, subtract, and repeat:

Convert 6:

1. Does 8 fit in 6? No → write 0
2. Does 4 fit in 6? Yes → write 1, and 6 − 4 = 2 left
3. Does 2 fit in 2? Yes → write 1, and 2 − 2 = 0 left
4. Does 1 fit in 0? No → write 0

Result: 0110, or just 110. Check it: 4 + 2 = 6 ✓

Bits, Bytes, and Big Files

• 1 bit — a single 0 or 1
• 1 byte — 8 bits, enough for one character of text
• 1 kilobyte (KB) — about a thousand bytes
• 1 megabyte (MB) — about a million bytes (a minute of music)
• 1 gigabyte (GB) — about a billion bytes (hundreds of photos)

Those storage numbers on phones suddenly mean something: a 64 GB phone holds roughly half a trillion bits — half a trillion tiny switches set on or off.

Everything Is Numbers

Here is the deep idea, and it connects straight back to abstraction: computers store only binary numbers, yet show us text, photos, and songs. How?

• Text — every character gets an agreed number code (in one common code, A is 65, B is 66…), and the numbers are stored in binary
• Images — the picture is a grid of pixels; each pixel's colour is a set of numbers
• Sound — the sound wave is measured thousands of times per second; each measurement is a number

Layer by layer, numbers become letters, pixels, and music. When you input anything into a computer — a keypress, a photo, your voice — the very first step is always the same: turn it into binary.

Worked Example — Converting a Bigger Number

The subtract-the-biggest-place-value method scales easily to numbers larger than the 4-bit examples above. Convert decimal 83 to binary using place values 128, 64, 32, 16, 8, 4, 2, 1:

Reading the bits column top to bottom: 01010011. Check it by converting back: 64 + 16 + 2 + 1 = 83 ✓. Notice this used exactly 8 bits — one byte — which is no coincidence: a single byte holds values from 0 to 255, comfortably covering 83, and this is exactly the size used to store one ASCII character or one colour channel value in an image.

Binary Addition — Just Like Decimal, With Carrying

Binary numbers can be added using the same column-by-column method as decimal addition, just with only two digits instead of ten. The rules: 0+0=0, 0+1=1, 1+1=10 (write 0, carry 1), and 1+1+1 (with an incoming carry)=11 (write 1, carry 1).

    1 1    ← carries
    0 1 1 0   (6)
  + 0 1 0 1   (5)
  ─────────
    1 0 1 1   (11)

Working right to left: 0+1=1 (no carry). 1+0=1 (no carry). 1+1=10, write 0 carry 1. 0+0+1(carry)=1. Result: 1011, which is 8+2+1=11 in decimal — and 6+5 is indeed 11. This is exactly the mechanism a logic-gate circuit called an adder performs electronically, at billions of additions per second, using the same carrying rule you just did by hand.

Key Words

• Binary — the base-2 number system using only 0 and 1
• Bit — one binary digit
• Byte — a group of 8 bits
• Place value — the worth of each column; in binary it doubles: 1, 2, 4, 8, …
• Pixel — one dot of an image, whose colour is stored as numbers