Skip to the page
Year 8 · Spring 1 · Representations: from clay to silicon

Tally marks to number bases

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

  • Compare tally marks with place value, and explain why place value needs fewer symbols.
  • Explain that each denary column is worth 10 times the one on its right.
  • Explain that each binary column is worth 2 times the one on its right.
  • Use the byte toy to switch bits on and off and read the value.

Success criteria

  • I can write the binary place values 128, 64, 32, 16, 8, 4, 2, 1 from memory.
  • I can explain why 9 is followed by 10 in denary, and 1 is followed by 10 in binary.
  • I can make a given number with the byte toy and read back its value.

Key words

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

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.

How normal numbers already work

We use ten digits, 0 to 9. When we run out, we start a new column on the left. That is why 9 is followed by 10.

ThousandsHundredsTensOnes
1000100101
3507

3507 = (3 × 1000) + (5 × 100) + (0 × 10) + (7 × 1). Binary works exactly the same way, with 2 instead of 10.

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'.

Bonus: negative numbers

Two's complement makes the leftmost bit worth −128. To write −5: write 5 (00000101), flip every bit (11111010), add 1 (11111011).

★Remember

This is A-level content. KS3 and GCSE use positive whole numbers only.

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.

Topic 2 · Year 7

Bits and bytes

A bit is one switch: 0 or 1. A byte is 8 bits. Every file is measured by counting these.

The steps for this lesson

Try it

Pages to explore

Years 8–11

Binary, Denary and Hex

Three ways of writing the same numbers. Place value column by column, with every conversion broken down.

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.