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Year 8 · Spring 1 · Representations: from clay to silicon

Binary to denary and back

TheoryPresent ▶

What this lesson covers

Topic 3 Binary, denary and hex

KS3 Computing · They're three ways of writing the same numbers. Each one uses place value; only the number of digits changes.

Computer SystemsCambridge Lower Secondary Computing 0860

You will learn to

  • Convert an 8-bit binary number to denary by adding the columns that hold a 1.
  • Convert a denary number from 0 to 255 to binary by taking away the biggest column that fits.
  • Check each answer by converting it back the other way.

Success criteria

  • I can convert binary to denary, like 01101011 = 107.
  • I can convert denary to binary, like 156 = 10011100.
  • I can explain each step using the words 'fits' and 'left over'.

Key words

Say each word, then read what it means. Every one is in the glossary too.

most significant bit
Most significant bit: the leftmost bit, worth the most.
least significant bit
Least significant bit: the rightmost bit, worth 1.
Denary
Base 10. Our everyday numbers.
Binary
Base 2. Digits 0 and 1. Columns 128 64 32 16 8 4 2 1.
Hexadecimal
Base 16: digits 0–9 then A–F. One digit = 4 bits.
Overflow
A result too big for the bits available.

The explanation

Read it here, or open the study page below to go one step at a time with things to try.

Binary place values

1286432168421
★Remember

Start at 1 on the right. Double each time you move left. 1, 2, 4, 8, 16, 32, 64, 128. Learn this row by heart.

The leftmost bit is the most significant bit. The rightmost is the least significant bit.

Binary → denary

  1. Write the place values above the bits.
  2. Find every column with a 1.
  3. Add those column values together.
e.g.Example

01101011 → the 1s are under 64, 32, 8, 2, 1 → 64 + 32 + 8 + 2 + 1 = 107.

★Remember

A 1 means 'count this column'. A 0 means 'skip it'.

Denary → binary

Go from the biggest column to the smallest. Ask: does this column fit into what is left?

  1. 156: does 128 fit? Yes → 1. 156 − 128 = 28 left.
  2. 64 into 28? No → 0.
  3. 32 into 28? No → 0.
  4. 16 into 28? Yes → 1. 12 left.
  5. 8 into 12? Yes → 1. 4 left.
  6. 4 into 4? Yes → 1. 0 left.
  7. 2 into 0? No → 0.
  8. 1 into 0? No → 0.
e.g.Example

Answer: 10011100. Check: 128 + 16 + 8 + 4 = 156.

Study it

Open the study page and go one step at a time.

Topic 3 · Year 8

Binary, denary and hex

Computers store numbers in binary (base 2). We use denary (base 10). Programmers use hex (base 16) as a short way to write binary. All three use place value.

The steps for this lesson

Pages to explore

Years 8–11

Bit Maths Lab

All the binary maths, one small step at a time and colour-coded. Tap anything; nothing can break.

Check yourself

One question at a time. The reason shows after each answer.

Saved on this device.