Counting in Base 2 (Binary) · Hook

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Ever wondered how your phone stores numbers using only 0s and 1s?

Inside Chidi's phone, every photo, song and data bundle balance is stored as switches that are either OFF (0) or ON (1). Counting with just these two digits is called binary, or base 2.

Revision notes

📒 Counting in Base 2 (Binary): lesson notes

Count using only 0 and 1 — the number system every phone and computer uses — and change binary numbers to base ten.

Inside Chidi's phone, every photo, song and data bundle balance is stored as switches that are either OFF (0) or ON (1). Counting with just these two digits is called binary, or base 2.

Only two digits: 0 and 1

In base ten we use ten digits, 0 to 9. When a place reaches ten, we write 0 and carry 1: 9 + 1 = 10.

In base 2 we only have the digits 0 and 1. So as soon as a place reaches two, we write 0 and carry 1.

Counting in binary goes: 1, 10, 11, 100, 101, 110, 111, 1000, … We write 1012 to show a number is in base 2.

🔑 Key idea: In binary, 1 + 1 = 102: write 0 and carry 1.

Binary place values double

In base ten each place is 10 times the one on its right. In base 2 each place is 2 times the one on its right.

So from the right, the place values are 1, 2, 4, 8, 16, 32, … (these are 20, 21, 22, 23, …).

🔑 Key idea: Binary place values from the right: 1, 2, 4, 8, 16, 32, …

Changing binary to base ten

Multiply each digit by its place value and add. 11012 = 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 8 + 4 + 0 + 1 = 13.

To go the other way, keep dividing by 2 and read the remainders from the bottom up: 13 ÷ 2 = 6 r 1, 6 ÷ 2 = 3 r 0, 3 ÷ 2 = 1 r 1, 1 ÷ 2 = 0 r 1, so 13 = 11012.

🔑 Key idea: Binary → base ten: add the place values that have a 1.

Worked example

Convert 1110112 to base ten.

  1. Step 1: Write the place values

    Digit 1 → place value 32

    Digit 1 → place value 16

    Digit 1 → place value 8

    Digit 0 → place value 4

    Digit 1 → place value 2

    Digit 1 → place value 1

    Why? Each place is worth double the place on its right.

  2. Step 2: Multiply and add

    1110112 = 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20

    = 32 + 16 + 8 + 0 + 2 + 1

    = 59

Answer: 59

Summary

  • Binary (base 2) uses only the digits 0 and 1.
  • When a place reaches 2, write 0 and carry 1: 1 + 1 = 102.
  • Binary place values from the right are 1, 2, 4, 8, 16, 32, …
  • To change binary to base ten, add the place values that have a 1.
  • To change base ten to binary, divide by 2 again and again and read the remainders from the bottom up.

Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.