You know that a tiny shift in place value can turn a £9.90 bargain into a £99.00 expense. This happens because our number system is a Base-10 system. Every place value column represents a different power of 10. Each column is ten times larger than the column to its immediate right and one-tenth of the value of the column to its immediate left.
Columns to the right of the decimal point represent negative powers of 10, extending into extremely small values:
A placeholder zero holds a place value column when no value exists for that power of ten. This ensures all other digits remain in their correct columns to represent the true magnitude of the number.
When you multiply or divide by powers of 10, the underlying mechanism is a place value shift. The decimal point remains fixed; it is the digits that move across the place value columns.
For example, when is divided by (), the digits shift 3 places to the right. This moves the from the tens column to the hundredths column, resulting in .
When multiplying decimals, the most reliable method is integer scaling. You convert the decimals to integers by multiplying by powers of 10, perform the calculation, and then scale back by dividing by the total power of 10 used.
Worked Example: Calculate
Step 1: Scale to integers. Multiply by to get . Multiply by to get .
Step 2: Arithmetic step.
Step 3: Scale back. The total scaling used was (). Divide the integer result by the total scaling factor: .
(Note: You can verify this using the decimal count method. The product must have the same number of decimal places as the original numbers combined: 2 DP + 1 DP = 3 DP, which matches ).
To divide by a decimal, scale the calculation up until the divisor (the number you are dividing by) is an integer. Writing the division as an equivalent fraction is the easiest way to do this. Unlike multiplication, you do not need to scale the final answer back because the ratio between the numbers remains exactly the same.
Worked Example: Calculate
Students often drop placeholder zeros when rounding large numbers. Rounding 2,530,457 to 3 significant figures gives 2,530,000, not 253 — dropping the zeros completely changes the place value of the digits.
When asked to order small decimals, the most successful strategy in Edexcel exams is to write all numbers to the same number of decimal places (by filling gaps with placeholder zeros) before comparing them.
In division calculations, only the divisor (the number you are dividing by) needs to be scaled to an integer. The numerator (the number being divided) can remain a decimal.
If a calculation involves money, always format your final answer to exactly two decimal places (e.g., £546.00), even if the last digits are zeros.
When converting to standard form, do not assume the power of 10 equals the number of zeros. The exponent tells you how many place value columns the digits shifted (e.g., 6.14 × 10⁴ = 61,400, which has 2 zeros but the digits shifted 4 places to the left).
Base-10 system
A number system where each place value column represents a power of 10, increasing by a factor of 10 to the left and decreasing by a factor of 10 to the right.
Power of 10
The result of multiplying 10 by itself a certain number of times, indicated by an exponent (e.g., 10³).
Millionths
One of a million equal parts of a whole, represented by the 6th decimal place to the right of the decimal point (0.000001).
Placeholder zero
A digit '0' used to maintain the correct place value of other digits where a specific power of ten has no value.
Leading zeros
Zeros appearing to the left of the first non-zero digit in a decimal, serving as placeholders to define magnitude but not counting as significant figures.
Precision
The exactness of a number, often indicated in decimals by trailing zeros (e.g., 0.50 is correct to 2 decimal places, while 0.5 is correct to 1).
Standard form
A way of writing numbers to replace long strings of placeholder zeros, expressed as a number between 1 and 10 multiplied by a power of 10 (A × 10ⁿ).
Significant figures
The digits in a number that carry meaning contributing to its precision, starting from the first non-zero digit. Leading zeros are never significant.
Place value shift
The movement of digits left or right across place value columns when multiplying or dividing by powers of 10, while the decimal point remains fixed.
Integer scaling
A reliable method for decimal calculations where decimals are temporarily converted to integers by multiplying by powers of 10, performing the calculation, and then adjusting the final answer by scaling back.
Put your knowledge into practice — try past paper questions for Mathematics
Base-10 system
A number system where each place value column represents a power of 10, increasing by a factor of 10 to the left and decreasing by a factor of 10 to the right.
Power of 10
The result of multiplying 10 by itself a certain number of times, indicated by an exponent (e.g., 10³).
Millionths
One of a million equal parts of a whole, represented by the 6th decimal place to the right of the decimal point (0.000001).
Placeholder zero
A digit '0' used to maintain the correct place value of other digits where a specific power of ten has no value.
Leading zeros
Zeros appearing to the left of the first non-zero digit in a decimal, serving as placeholders to define magnitude but not counting as significant figures.
Precision
The exactness of a number, often indicated in decimals by trailing zeros (e.g., 0.50 is correct to 2 decimal places, while 0.5 is correct to 1).
Standard form
A way of writing numbers to replace long strings of placeholder zeros, expressed as a number between 1 and 10 multiplied by a power of 10 (A × 10ⁿ).
Significant figures
The digits in a number that carry meaning contributing to its precision, starting from the first non-zero digit. Leading zeros are never significant.
Place value shift
The movement of digits left or right across place value columns when multiplying or dividing by powers of 10, while the decimal point remains fixed.
Integer scaling
A reliable method for decimal calculations where decimals are temporarily converted to integers by multiplying by powers of 10, performing the calculation, and then adjusting the final answer by scaling back.