KS3 · Computer Science

Representing numbers in binary

In binary, 11111111 isn't eleven million-something — it's 255. By the end of this lesson you'll know why, and you'll write any number from 0 to 255 in eight bits.

Computer Science · Data

Flip the bits. Watch the number.

8-bit binary 00010011 = 19 in denary

Decimal19

tap a bit to flip it

Each box is one bit, and the number above it is that bit's place value. Tap a bit to switch it between 0 and 1. Try this: switch every bit to 0. Then switch every bit to 1. Then flip only the left-hand bit, and after that only the right-hand bit — how far does the number jump each time?

Watch out: The place values start at 1 on the RIGHT and double as you move left. The left-hand bit of a byte is the big one: it's worth 128.

Same idea, one rule changed

Decimal (denary)vsBinary

Start with the top row — it's the one that matters most.

Focus

Each place is worth…

Decimal (denary)

10 times the place to its right

Binary

2 times the place to its right

The insight

This is the whole difference. Decimal multiplies by 10 as you move one place left; binary multiplies by 2.

Digits it uses

Decimal (denary)

Ten digits: 0 to 9

Binary

Two digits: 0 and 1

Place values, from the right

Decimal (denary)

1, 10, 100, 1000 … (ones, tens, hundreds, thousands)

Binary

1, 2, 4, 8, 16, 32, 64, 128

Its name

Decimal (denary)

Base 10

Binary

Base 2

Your turn to choose

Turn 75 into binary — you pick the missing steps

Convert the decimal number 75 into an 8-bit binary number.

Write the place values: 128, 64, 32, 16, 8, 4, 2, 1. Start with the biggest, 128, and ask: does it fit into 75?
missing step

Which line is step 2?

128 fits into 75, so write 1. 128 − 75 = 53 left.128 doesn't fit into 75, so write 0. Still 75 left.128 doesn't fit, so skip it and start at 64.

What do you think?

00000101 or 101 — which is bigger?

Amir writes a binary number as 00000101. Beth writes a binary number as 101.

Which is closest to what you think right now?

Amir's number is bigger, because it has more digits.They're the same number: one hundred and one.Beth's isn't a proper binary number, because a binary number has to have 8 bits.They're the same number, 5. The 0s on the left don't add anything.

How sure are you?

I'm sureI think soI'm guessing

Amir's number is bigger, because it has more digits.

This is a common misconception

More digits doesn't mean a bigger number here. Amir's extra digits are bits in the 128, 64, 32, 16 and 8 places — and every one of them is 0, so each adds nothing. Both numbers are 4 + 1 = 5.

They're the same number: one hundred and one.

This is a common misconception

You're right that they're the same — but not about the value. 101 in binary isn't read like a decimal number. Its places are worth 4, 2 and 1, not 100, 10 and 1. So 101 is 4 + 1 = 5, not one hundred and one.

Beth's isn't a proper binary number, because a binary number has to have 8 bits.

This is a common misconception

A byte has 8 bits, but a binary number can have any number of bits. 101 is a perfectly good 3-bit binary number. Writing it with 8 bits just adds 0s on the left, which doesn't change its value.

They're the same number, 5. The 0s on the left don't add anything.

This matches the evidence

Yes. The value comes only from the place values of the 1 bits: 4 + 1 = 5. Adding 0s on the left changes how many bits you've written, not the number.

Read every binary number by its place values, never as a decimal number. Zeros on the left — leading zeros — don't change the value.

Commit to a number

How many numbers fit in one byte?

One byte is 8 bits. How many DIFFERENT whole numbers can 8 bits represent? Slide to your answer before you look.

Your estimate

256 values

One byte is 8 bits. How many DIFFERENT whole numbers can 8 bits represent? Slide to your answer before you look. Estimate between 0 values and 512 values.

0 values512 values

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Binary works just like the numbers you already use, with one rule changed: every place is worth double the one to its right, not ten times.

What you need to know

Binary is base 2: it uses only the digits 0 and 1, and each binary digit is called a bit. A group of eight bits is called a byte.
Binary place values start at 1 on the right and double each time: 1, 2, 4, 8, 16, 32, 64, 128. A binary number's value is the place values of its 1 bits added together.
To convert decimal to binary, work from the biggest place value down: write 1 and take it away if it fits, otherwise write 0. An 8-bit number can represent every whole number from 0 to 255.

The big picture

Binary is base 2: it uses only 0 and 1 (bits), and each place value doubles as you move left — 1, 2, 4, 8, 16, 32, 64, 128 for an 8-bit byte. To read binary, add the place values of the 1 bits. To write a decimal number in binary, work from the biggest place value down: write 1 and take it away if it fits, or 0 if it doesn't. Eight bits cover every whole number from 0 to 255, and each extra bit doubles how many values are possible.

Key points

1Decimal (denary) is base 10: it uses the ten digits 0 to 9, and each place is worth 10 times the place to its right.

2Binary is base 2: it uses only 0 and 1, and each place is worth 2 times the place to its right.

3The place values of an 8-bit number, from the right, are 1, 2, 4, 8, 16, 32, 64 and 128. Eight bits make a byte.

4Binary to decimal: add together the place values of every bit that is 1. A bit that is 0 adds nothing.

5Decimal to binary: from the biggest place value down, write 1 and subtract it if it fits into what's left; if it doesn't, write 0. Read the bits from left to right.

68 bits represent 0 (all 0s) up to 255 (all 1s). Each extra bit doubles how many values can be represented, and zeros added on the left don't change a binary number's value.

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Worked example

Problem

A game stores each player's score as an 8-bit binary number. Sara scores 200. What binary number does the game store?

⚠ Watch out

Writing the place values the wrong way round, with 1 on the left and 128 on the right. Place values start at 1 on the RIGHT and double as you move left, so the left-hand bit of a byte is worth 128.

🧠

Memory hook

Start at 1 on the right and keep doubling: 1, 2, 4, 8, 16, 32, 64, 128. Add up only the places holding a 1. Switch all eight on and you get the biggest a byte can hold: 255.

✓

Check yourself

Without looking back: what is 00010110 in decimal, and what is 37 in 8-bit binary? (Answers: 16 + 4 + 2 = 22; 32 + 4 + 1 gives 00100101.)

Flashcards

(12)

What is a bit?

A single binary digit: a 0 or a 1.

What is a byte?

A group of eight bits.

Why is binary called base 2?

It uses two digits (0 and 1), and each place is worth 2 times the place to its right. Decimal is base 10: ten digits, and each place is 10 times the one to its right.

The place values of an 8-bit binary number, from right to left?

1, 2, 4, 8, 16, 32, 64, 128

How do you turn a binary number into decimal?

Write the place value above each bit, then add up the place values of the bits that are 1. Bits that are 0 add nothing.

Decimal to binary: where do you start?

At the biggest place value (128 for 8 bits), then work down to 1.

Decimal to binary: what do you do at each place value?

If it fits into what's left, write 1 and subtract it. If it doesn't fit, write 0. Stop when nothing is left and fill any remaining places with 0.

Smallest and largest number an 8-bit binary number can represent?

0 (every bit 0) and 255 (every bit 1).

How many different values can 8 bits represent?

256 — every whole number from 0 to 255.

What happens to the number of possible values when you add one more bit?

It doubles.

Do zeros added on the left change a binary number?

No. 00000101 and 101 are both 5 — leading zeros add nothing.

What is the binary number 10 in decimal?

2 — one 2 and no 1s. It is not ten.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Flip the bits. Watch the number.

Turn 75 into binary — you pick the missing steps

00000101 or 101 — which is bigger?

How many numbers fit in one byte?

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