GCSE · Computer Science · AQA · Spec 8525

Decimal (base 10)

You read 3705 instantly, and you knew the 7 meant seven hundred. Nobody told you that. Its column did, and that hidden column system is what 'base 10' means.

Try it first

Which column is each digit in?

Pick a digit, then put it in the column it sits in. Read the reason that appears. Is it right or wrong?

Still to sort

Thousands (1000s) (0)

Fourth column from the right. Each one is worth 1000.

Hundreds (100s) (0)

Third column from the right. Each one is worth 100.

Where the line is: The first digit is not always the thousands. In a three-digit number like 421, the first digit is in the hundreds.

Tens (10s) (0)

Second column from the right. Each one is worth 10.

Ones (1s) (0)

The right-hand column. Each one is worth 1.

Where the line is: The last digit is always in the ones column, however long the number is.

8 of 8 still to sort.

Once you've sorted them, look back at the 5s: the same digit was worth 5 in one number and 5000 in another. In base 10, a digit's value comes from its column, not from the digit alone.

Watch out: Count columns from the right-hand end, not the left. The last digit is always in the ones column.

Predict, then check

Each column can only hold one digit, from 0 to 9. Keep that rule in your head.

What is 999 + 1, written in columns?

Spot the mistake

Rebuild the number from its columns

Write 40516 as the total of what each digit is worth.

A student's answer — which line goes wrong?

Start at the right-hand end. The 6 is in the ones column: 6 × 1 = 6.
The 1 is in the tens column: 1 × 10 = 10.
The 5 is in the hundreds column: 5 × 100 = 500.
The 0 is worth nothing, so skip it and move on to the 4.
The 4 is in the thousands column: 4 × 1000 = 4000.
Total: 4000 + 500 + 10 + 6 = 4516.

What does the word mean?

'Decimal' in Computer Science

A Computer Science book says: '3705 is a decimal number.' There is no decimal point anywhere in it.

Which is closest to what you think 'decimal' means here?

It means the number is written in base 10, using the digits 0 to 9 in columns worth 1, 10, 100 and so on.A decimal is a number with a decimal point, like 3.7, so 3705 shouldn't count.Base 10 means the digits go from 1 to 10.Base 10 just means we have ten digits, and each digit is always worth its own amount: a 5 is worth 5.

How sure are you?

I'm sureI think soI'm guessing

It means the number is written in base 10, using the digits 0 to 9 in columns worth 1, 10, 100 and so on.

This matches the evidence

Yes. In Computer Science, 'decimal' is the name of the number base, not a feature of the number. Any number written in ordinary base 10 is a decimal number, whether it is 3705 or 7. You will also see it called denary. Naming the base matters because computers also use other bases, such as binary (base 2) and hexadecimal (base 16), which use the same column idea with different column values.

A decimal is a number with a decimal point, like 3.7, so 3705 shouldn't count.

This is a common misconception

That is what the word often means in everyday maths, so it is a fair guess. In Computer Science, though, 'decimal' means base 10, the number system. 3705 has no decimal point and is still a decimal number, because it is written with base 10 digits and columns. If it helps, think of it as 'denary', the other name for base 10.

Base 10 means the digits go from 1 to 10.

This is a common misconception

Close, but count them: 1 to 10 would need a single digit for ten, and there isn't one. Base 10 has ten digits, 0 to 9. When you need ten, you use two columns: 1 in the tens and 0 in the ones, which is 10. The 0 is one of the ten digits.

Base 10 just means we have ten digits, and each digit is always worth its own amount: a 5 is worth 5.

This is a common misconception

The ten digits part is right, but the second part isn't. In base 10 a digit's value depends on its column: the 5 in 50 is worth 50, and the 5 in 5000 is worth 5000. Ten digits plus columns worth ten times each other is the whole system. Leave out the columns and you can't write any number above 9.

In Computer Science, decimal means the base 10 number system (denary): ten digits, 0 to 9, in columns where each is worth ten times the one to its right.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

In Computer Science, 'decimal' means the base 10 number system (also called denary), not 'a number with a decimal point'.
Base 10 has exactly ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. There is no single digit for ten.
Reading from the right, the columns are worth 1, 10, 100, 1000 and so on. Each column is worth ten times the one to its right (10⁰, 10¹, 10², 10³).
A digit's value is the digit × its column's value, so the 7 in 3705 is worth 700.

The big picture

Decimal is the base 10 number system, also called denary. It has ten digits, 0 to 9. Each column is worth ten times the column to its right (1, 10, 100, 1000 and so on), so a digit's value is the digit multiplied by its column's value. A 0 holds the other digits in their columns, and when a column goes past 9 it rolls over to 0 and carries 1 to the left.

Key points

1Count columns from the right-hand end. The last digit is always in the ones column.

2A 0 is worth nothing, but it still fills a column and holds the other digits in place: 3705 is not 375.

3When a column goes past 9, it rolls over to 0 and carries 1 into the next column to the left: 999 + 1 = 1000.

4Move a digit one column to the left and it is worth ten times as much.

5Adding up every digit × column value gives back the number: 3705 = 3000 + 700 + 0 + 5.

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Worked example

Problem

The number 2505 has two 5s. How many times more is the left-hand 5 worth than the right-hand 5?

⚠ Watch out

Counting columns from the left. The first digit is not always the thousands: in 421 the 4 is in the hundreds. Always start from the right-hand end, where the ones column is.

🧠

Memory hook

Ten digits fill a column, and the next count carries next door. That's why every column is worth ten of the one on its right.

✓

Check yourself

What is the 8 in 1843 worth? What is it worth in 18430? Explain the change in one sentence, using the word 'column'.

Flashcards

(7)

What does 'decimal' mean in Computer Science?

The base 10 number system, also called denary. It does not mean 'has a decimal point'.

Which digits does base 10 use?

Ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. There is no single digit for ten.

How does each column's value compare with the column to its right?

It is worth ten times as much.

Name the first four column values of a decimal number, from the right.

1, 10, 100, 1000. As powers of ten: 10⁰, 10¹, 10², 10³.

How do you work out what a digit is worth?

Multiply the digit by the value of its column. The same digit is worth more the further left it sits.

What job does a 0 do inside a number?

It is a placeholder. It is worth nothing, but it fills its column so the other digits stay in the right columns.

What happens when a column needs to go past 9?

It rolls over to 0 and carries 1 into the next column to the left.

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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Which column is each digit in?

Rebuild the number from its columns

'Decimal' in Computer Science

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