The subject in plain language
What is long subtraction?
Long subtraction is the standard written method for finding the difference between two numbers one place-value column at a time. Matching places are aligned, and the subtraction proceeds from right to left.
When the upper digit in a column is too small, regrouping exchanges one unit from the place to the left for ten units in the current place. This lets the method handle large numbers, decimals, and differences that are difficult to calculate mentally.
Long subtraction is useful for finding how much remains, measuring a decrease, or comparing two quantities. It is built on base-ten positional notation and can be checked using its inverse operation: addition.
Meaning before method
What long subtraction is doing
Subtraction finds the signed difference between a starting value—the minuend—and the amount removed—the subtrahend. Long subtraction performs that same operation one base-ten place at a time.
- m
- minuend, or starting value
- s
- subtrahend, or value subtracted
- d
- signed difference
Base-ten visual
Regrouping changes the form, not the value
If a column does not contain enough units to subtract, exchange one unit from the next place to the left for ten units in the current place.
This explains why borrowing across zeros can pass through several columns: each zero first receives a group of ten, then passes one group onward.
Worked example
Subtract 178.45 from 503.20
- Hundredths: regroup so 10 − 5 = 5.
- Tenths: after the previous regrouping, regroup again so 11 − 4 = 7.
- Ones: regroup across the zero tens column; 12 − 8 = 4.
- Tens: the regrouped column gives 9 − 7 = 2.
- Hundreds: 4 − 1 = 3.
Independent check
Add the difference back
A correct result satisfies difference + subtrahend = minuend. Here, 324.75 + 178.45 = 503.20. This inverse-operation check is stronger than repeating the same subtraction.
Boundaries
Common mistakes and calculator limits
Misaligned decimals
Align decimal points so every digit keeps its place value.
Borrowing without reducing
When ten units are moved right, the source column must lose one unit.
Negative answers
If the subtrahend is larger, the result is negative; the magnitude method does not erase the sign.
Input scope
This written layout accepts non-negative decimal inputs; each may contain up to 200 digits and 100 decimal places.
Continue with the Long Addition Calculator to study the inverse written operation.