How Long Subtraction Works
Long subtraction is the column method taught in school for subtracting one multi-digit number (the subtrahend) from another (the minuend) to get the difference: Difference = Minuend − Subtrahend. Instead of subtracting the whole numbers at once, you align them by place value and subtract one column of digits at a time, from the rightmost (ones) column to the leftmost, borrowing from the next column whenever the top digit is smaller than the bottom digit.
The borrow-and-regroup algorithm, step by step
Write the minuend above the subtrahend, lining up ones under ones, tens under tens, and so on (pad the shorter number with leading zeros so both have the same number of digits). Starting from the rightmost column, subtract the bottom digit from the top digit. If the top digit is smaller, "borrow" 1 from the digit immediately to its left — that borrowed 1 is worth 10 in the current column, so you add 10 to the top digit before subtracting, and reduce the digit you borrowed from by 1. Repeat for every column moving left. For example, subtracting 178 from 452: in the ones column 2 is smaller than 8, so borrow from the tens digit (5 becomes 4) and compute 12 − 8 = 4; in the tens column 4 is smaller than 7, so borrow from the hundreds digit (4 becomes 3) and compute 14 − 7 = 7; in the hundreds column 3 − 1 = 2. The result is 274, found using 2 borrows.
Common mistakes in long subtraction
- Forgetting to reduce the borrowed-from digit: after borrowing, the digit to the left must drop by 1 — skipping this step is the most common source of wrong answers.
- Misaligning place values: numbers with different digit counts or decimal places must be lined up by place value (ones under ones, tenths under tenths) before subtracting, not by their rightmost printed character.
- Subtracting the smaller digit from the larger one in a column regardless of position: always subtract bottom from top in each column (using a borrow if needed) — never just flip the two digits to avoid borrowing.
Checking your answer
Subtraction and addition are inverse operations, so the fastest way to verify a long-subtraction answer is to add the difference back to the subtrahend: Difference + Subtrahend should equal the Minuend. In the example above, 274 + 178 = 452, which confirms the subtraction was done correctly.
Real-world uses
Long subtraction with borrowing shows up whenever you need an exact difference between multi-digit quantities: balancing a checkbook, calculating change due, finding the gap between a budget and actual spending, working out how many days remain until a deadline, or subtracting measurements in a recipe or construction plan. It's also the foundation for subtracting decimals (align the decimal points first) and for the borrowing logic used inside long division.