Operations and calculations
4. Multiplication
4.2. Long multiplication
Long multiplication is a written method for multiplying larger numbers, which involves simplifying the operation by breaking it into a series of smaller operations. The results of these are then added together to find the total.
Let's look at an example.
Long multiplication | Multiply Highland, 2:46
When we want to multiply two large numbers, we can use a method called long multiplication.
Let's look at 836 x 14 as an example.
To do this we first need to write the numbers out with one number on top of the other. The smaller number usually goes on the bottom and the digits need to be lined up in columns for units, tens, hundreds and thousands.
We start by multiplying 836 by the tens digit of 14 which is 1. Since it's in the tens place, we actually multiply by 10. 836 x 10 = 8,360 so we write 8,360 down as our first partial product. Next, on a new row, we will multiply 836 by the units digit of 14 which is 4.
We start by multiplying the units digit, which is 6, by 4. The result is 24. We write the 4 in the units place of the second row and carry over the two to the next column.
Next we multiply the 3 in the tens column by 4 to make 12, and add the two we carried over to make 14. We write the 4 in the tens place and carry over the 1 to the next column.
Then we multiply the 8 of the hundreds column by 4. The result is 32, plus the one we carried over, making it 33. So we write 33 in the hundreds and thousands places of this row. Now we have our second partial product 3,344.
The final step is to add the two partial products together. In the units column we have 0 + 4 which is 4. In the tens column we have 6 + 4 which is 10. We write 0 and carry over 1.
In the hundreds column we have 3 + 3, plus 1, which is 7. In the thousands column we have 8 + 3 which is 11. We write 1 and carry over 1. And in the ten-thousands column we have 1 from the carryover so we now have our answer of 11,704.
Worked example
836 x 14
In this method, starting on the right, we multiply each digit in the top row, first by the far left digit in row below, and then by the next digit, and so on.
Each set of answers is noted in a new row, and then all the rows are added together to give the final answer.
To start the process, write the smaller number below the larger number, aligning the digits in columns, according to place value, and draw a line below.
Add 0 below the line, in the units column.
Now we're ready to multiply each digit on the top row by the digit in the tens position (1) in the row below.
Starting on the right:
6 x 1 = 6, write 6 in the column to the left of the 0.
3 x 1 = 3, write 3 in the next column left.
8 x 1 = 8, write 8 in next column left.
Starting on a new line, working from right to left, we'll now multiply each digit in the top row by the digit in the units position (4) in the row below.
6 x 4 = 24
Write 4 in the units column, and note 2 below the tens column, as a reminder to add it to the result of the next multiplication.
3 x 4 = 12
Plus the 2 carried over:
12 + 2 = 14
Write 4 in this column and note 1 below the next column on the left.
8 x 4 = 32
Plus the 1 carried over:
32 + 1 = 33
Write 3 in this column, and carry 3 (3 tens) to the column on the left. As this was the final multiplication in this row, it can be written straight into the answer line.
Draw a line below the two rows created during the process so far.
We can now start adding these two rows together.
Starting on the right,
0 + 4 = 4, write 4 below the line.
6 + 4 = 10, write 0 below the line and carry 1 (1 ten) to the column on the left, noting it below.
3 + 3 = 6, plus the 1 carried over comes to 7. Write 7 below the line.
8 + 3 = 11, write 1 below the line and 1 in the next column to the left.
836 x 14 = 11704
In this process we multiplied the top number by each of the lower number's place value components (10 and 4):
836 x 10 = 8360 and 836 x 4 = 3344
These solutions were then added together:
8360 + 3344 = 11704