Regrouping refers to the process of re-arranging numbers to form groups of 10 when adding or subtracting two digit (or larger) numbers. It is also referred to as carrying (in addition) and borrowing (in subtraction).
Regrouping in addition
Performing the standard addition algorithm involves positioning the numbers being added vertically, aligned based on place value, then adding the digits one column (place value) at a time starting from the right. In cases where the sum of a column is less than 10, the sum is simply written below the numbers being added, separated by a line.
Regrouping becomes necessary when the sum of a column exceeds 10. This has to do with how the decimal numeral system works, and is discussed below. To perform the addition algorithm however, simply "carry" a 1 over to the next column, then add it along with the next column, as in the example below.
Because 7 + 5 = 12, which is a 2 digit number, we cannot just write 12 below the line, but instead need to carry the 1 over to the next column. The 2 in 12 is written below, and the 1 is carried to the top of the next column, then added to 4 and 2 to get 7, for a final answer of 72.
Regrouping in subtraction
Regrouping in subtraction is similar to regrouping in addition, except that instead of "carrying" a 1 to the next column, in cases where the number on top is smaller than the one on the bottom, a 1 is "borrowed" from the next column, as in the example below.
3 is smaller than 6, so although it is possible to subtract 6 from 3 to get a negative number, since the 3 and 6 are part of larger numbers, this is not necessary, since we can borrow a 1 from the following column. Subtracting 1 from the 4 in the next column is equivalent to having 10 more in the first column, so the 3 + 10 becomes 13, which we can subtract 6 from to get 7. In the next column, we are left with 3 - 2, which equals 1, leaving us with a final answer that 43 - 26 = 17.
How regrouping works
To understand how regrouping works, it is necessary to understand the decimal numeral system. Regrouping has to do with place value and the way the decimal numeral system works. In the decimal numeral system, each "place" or position represents a power of 10. The value of the digit in each position is determined by multiplying the digit by the place value denoted by that position.
The figure below is known as a place value chart, and can be used to determine the value of each digit in a number. Note that there are many more place values on both sides than what is shown, these are just a few.
The value of each digit is the digit multiplied by the power of 10 represented by the place. The 1 in the ones place is therefore 1 × 100, or 1. The 8 in the tens place is 8 × 101, where the power increases by 1 as we move left each place value. As we move right, the power decreases by 1, so the power of the tenths place is 10-1. The number shown in the place value chart, fully expanded, si as follows:
8(10000) + 7(1000) + 4(100) + 8(10) + 1(1) + 2(0.1) + 3(0.01) = 87481.23
Moving back to regrouping, when we carry a 1 to the next column, we are essentially adding 1 to the next place value. So in the addition example above, 7 + 5 = 12, we are writing the 2 below, meaning that we are left with 10 in the ones place, which is the same as having 1 in the 10s place. For larger numbers, the concept is still the same. 10 in the tens column is 1 in the hundreds column. 10 in the hundreds column is 1 in the thousands column, and so on.
Similarly, when borrowing a 1 from the next column in subtraction, we are adding 10 to the preceding column. In the subtraction example above, we subtracted 1 from the 4 to leave 30 in the tens column, and the 1 that is borrowed is 10 in the ones column, which we added to 3 to make 13 so that we could subtract 6.