AQA GCSE Mathematics (8300) — Number
Every time you check a price tag of £4.99, you are already using place value. The position of a digit determines its actual worth. Place value is the value represented by a digit based on its position within a number.
In our Base-10 system, each column to the left is 10 times larger than the column to its right, and each column to the right is (divided by 10) of the column to its left. The decimal point acts as a fixed anchor between the Ones and Tenths columns. It never moves; instead, digits undergo a digit shift across the decimal point when scaled by a power of 10.
| Column Name | Value (Digits) |
|---|
| Power of 10 |
|---|
| Millions | 1,000,000 | |
| Hundred Thousands | 100,000 | |
| Ten Thousands | 10,000 | |
| Thousands | 1,000 | |
| Hundreds | 100 | |
| Tens | 10 | |
| Ones (Units) | 1 | |
| [Decimal Point] | . | Fixed |
| Tenths | 0.1 | |
| Hundredths | 0.01 | |
| Thousandths | 0.001 | |
| Ten-thousandths | 0.0001 | |
| Hundred-thousandths | 0.00001 | |
| Millionth | 0.000001 |
Large numbers are traditionally read by grouping digits in threes using spaces (e.g., ). Digits after the decimal point are read individually (e.g., is read as "zero point five two").
Imagine trying to button a shirt but starting with the wrong hole; nothing lines up. When adding or subtracting decimals, the golden rule is precise vertical alignment. Decimal points must be perfectly lined up so that digits of the same place value occupy the exact same column.
If numbers have different lengths, use a placeholder zero to fill empty gaps so they have the same number of digits after the decimal point. You may need to carry a '10' to the next higher column during addition, or exchange (borrow) a '1' from a higher column as '10' into a lower column during subtraction.
Worked Example: Addition
Calculate
Step 1: Align decimal points and add a placeholder zero to match the number of decimal places.
Step 2: Add column by column from right to left, carrying numbers where necessary. Answer:
Worked Example: Subtraction
Calculate
Step 1: Write as to align the decimal points perfectly.
Step 2: Exchange from the units to allow subtraction in the hundredths and tenths columns (; ; ). Answer:
You might assume all decimal arithmetic works the same way, but multiplication and division have completely different rules to addition. You do not line up the decimal points for multiplication.
To find the product, ignore the decimal points and multiply as whole integers. Then, count the total number of decimal places in the original numbers and place the decimal point in your final answer so it has that same number of decimal places.
When dividing, never divide by a decimal directly. Always use the equivalent fractions method: multiply both the dividend (the inside number) and the divisor (the outside number) by the same power of 10 until the divisor is a whole integer. Then you can calculate the quotient. Decimal answers may be terminating (they end) or recurring (they repeat infinitely).
Worked Example: Multiplication
Calculate
Step 1: Ignore decimals and multiply as integers.
Step 2: Count the total decimal places in the question. (2 d.p.) + (1 d.p.) = 3 decimal places in total.
Step 3: Put 3 decimal places into the integer answer. Answer:
Worked Example: Division
Calculate
Step 1: Write as a fraction and multiply the numerator and denominator by 10 to make the divisor a whole number.
Step 2: Use the bus stop method to calculate . Answer:
Multiplying by , , or makes a number larger by shifting all digits to the left (by 1, 2, or 3 places respectively). Dividing shifts digits to the right.
Decimals can act as operators too. Scaling by decimals acts inversely to what you might intuitively expect:
Rounding is approximating a number to a specific degree of accuracy. The first of the significant figures (s.f.) is always the first non-zero digit reading from left to right.
Once you find the first significant figure, every following digit (including zeros) is significant. Look at the "decider" digit immediately to the right of your required accuracy: if it is or more, round up; if it is less than , keep the digit the same.
In AQA exams, if accuracy is not explicitly specified, always give your answer to 3 significant figures. Money must always be given to 2 decimal places, and angles to 1 decimal place.
Worked Example: Decimal Rounding
Round to 2 s.f.
Step 1: Identify the first significant figure. The first non-zero digit is . The second is .
Step 2: Look at the decider digit. The digit after the is . Since is less than , we round down (keep the the same). Answer: (Note: You must keep the trailing zero to show the 2 s.f. degree of accuracy).
Students often mistakenly try to line up decimal points for multiplication as they do for addition, which leads to completely wrong place value in the final answer.
In non-calculator papers, AQA frequently awards a Method Mark (M1) for correctly setting up column addition or subtraction with aligned decimal points, even if you make a small arithmetic error later.
Avoid 'adding zeros' when multiplying a decimal by 10 (e.g., is not ). You must apply the digit shift so it correctly becomes .
Always estimate your calculation first (e.g., ) to quickly check if your final calculated answer is a plausible magnitude.
When working with money contexts, always write answers to 2 decimal places; if your calculator display shows £1.4, you must manually write £1.40 to secure the final mark.
Do not round numbers too early in multi-step problems (keep at least 4 s.f. in your intermediate workings) to prevent final-answer truncation errors.
Place value
The value represented by a digit based on its position within a number.
Power of 10
A number formed by (e.g., ).
Digit shift
The physical movement of digits across place value columns when multiplying or dividing by a power of 10.
Placeholder zero
A zero used to maintain the position of other digits and the correct magnitude of a number.
Millionth
The sixth decimal place to the right of the decimal point, representing or .
Carry
Moving a '10' from a lower place value column to the next higher column (left) during addition.
Exchange
The process of taking '1' from a higher place value column and moving it to a lower column as '10' during subtraction.
Product
The result of a multiplication calculation.
Dividend
The number being divided (the 'inside' number in a calculation).
Divisor
The number dividing into the dividend (the 'outside' number).
Quotient
The final result of a division calculation.
Terminating
A decimal that ends (e.g., ).
Recurring
A decimal that repeats infinitely (e.g., ).
Rounding
Approximating a number to a specific degree of accuracy.
Significant figures (s.f.)
Digits that contribute to a number's precision, starting from the first non-zero digit.
Put your knowledge into practice — try past paper questions for Mathematics
Place value
The value represented by a digit based on its position within a number.
Power of 10
A number formed by (e.g., ).
Digit shift
The physical movement of digits across place value columns when multiplying or dividing by a power of 10.
Placeholder zero
A zero used to maintain the position of other digits and the correct magnitude of a number.
Millionth
The sixth decimal place to the right of the decimal point, representing or .
Carry
Moving a '10' from a lower place value column to the next higher column (left) during addition.
Exchange
The process of taking '1' from a higher place value column and moving it to a lower column as '10' during subtraction.
Product
The result of a multiplication calculation.
Dividend
The number being divided (the 'inside' number in a calculation).
Divisor
The number dividing into the dividend (the 'outside' number).
Quotient
The final result of a division calculation.
Terminating
A decimal that ends (e.g., ).
Recurring
A decimal that repeats infinitely (e.g., ).
Rounding
Approximating a number to a specific degree of accuracy.
Significant figures (s.f.)
Digits that contribute to a number's precision, starting from the first non-zero digit.
