Subtraction with regrouping

Type a subtraction like `352 − 187` or `4,005 − 1,237`. The answer appears with the sum drawn in columns: each digit that lends is crossed out with its new value above it, and the column that borrows shows its new number. Underneath, every column is explained in words.

It handles numbers of any size up to ten digits, decimals like `5.2 − 1.75` (the decimal points line up), and subtractions where the second number is bigger, which give a negative answer. For adding, use addition with regrouping.

Work from the right, one place at a time. When the top digit is smaller than the one below it, you can't take it away, so you regroup. Take 1 from the place to the left, which is worth 10 in this place, and add it to the top digit.

In `352 − 187`, the ones are 2 − 7. Regroup a ten: the 5 tens become 4, and the 2 ones become 12. 12 − 7 = 5. Now the tens are 4 − 8, so regroup a hundred: 3 becomes 2, and 4 tens become 14. 14 − 8 = 6. Then 2 − 1 = 1 in the hundreds, and the answer is 165.

Regrouping, borrowing and decomposition are three names for the same method.

Zeros make it harder because there's nothing next door to borrow. In `4,005 − 1,237`, the ones need to regroup but the tens and hundreds are both 0. Go left until you find a digit that isn't zero, the 4 thousands, and take 1 from it, leaving 3.

That thousand becomes 10 hundreds. One hundred goes on to the tens, leaving 9 hundreds; that hundred becomes 10 tens, and one ten goes on to the ones, leaving 9 tens. The ones become 15. It reads as 4,005 becoming 3, 9, 9, 15, and from there each column subtracts without trouble, giving 2,768. The drawing shows each zero becoming 10 and then 9.

The crossings-out have a name: crutches. Borrowing was in American textbooks before then, but the marks spread through American schools after the education researcher William A. Brownell published a study claiming they helped, and they displaced the other methods in use.

Some European schools teach a different method with no borrowing at all, often called the Austrian or equal additions method. When a top digit is too small, add 10 to it and add 1 to the bottom digit in the next place instead. Both numbers grow by the same amount, so the difference is unchanged. For 352 − 187: 12 − 7 = 5, then 15 − 9 = 6, then 3 − 2 = 1, giving 165 again.