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Multiplication of positive or negative whole numbers or decimal numbers as the multiplicand and multiplier to calculate the product using long multiplication. The solution shows the work for the Standard Algorithm.
Parts of Long Multiplication
2
5
6
Multiplicand
×
3
2
Multiplier
+
5
1
2
Partial Product
+
7
6
8
Partial Product
=
8
1
9
2
Product
How To Do Long Multiplication
Long multiplication means you're doing multiplication by hand. The traditional method, or Standard Algorithm, involves multiplying numbers and lining up results according to place value. These are the steps to do long multiplication by hand:
- Arrange the numbers one on top of the other and line up the place values in columns. The number with the most digits is usually placed on top as the multiplicand.
- Starting with the ones digit of the bottom number, the multiplier, multiply it by the last digit in the top number
- Write the answer below the equals line
- If that answer is greater than nine, write the ones place as the answer and carry the tens digit
- Proceed right to left. Multiply the ones digit of the bottom number to the next digit to the left in the top number. If you carried a digit, add it to the result and write the answer below the equals line. If you need to carry again, do so.
- When you've multiplied the ones digit by every digit in the top number, move to the tens digit in the bottom number.
- Multiply as above, but this time write your answers in a new row, shifted one digit place to the left.
- When you finish multiplying, draw another answer line below your last row of answer numbers.
- Use long addition to add your number columns from right to left, carrying as you normally do for long addition.
Long Multiplication with Decimals
Long multiplication with decimals using the standard algorithm has a few simple additional rules to follow.
- Count the total number of decimal places contained in both the multiplicand and the multiplier.
- Ignore the decimals and right align the numbers one on top of the other as if they were integers
- Multiply the numbers using long multiplication.
- Insert a decimal point in the product so it has the same number of decimal places equal to the total from step 1.
Example Long Multiplication with Decimals
Multiply 45.2 by 0.21
There's 3 total decimal places in both numbers.
Ignore the decimal places and complete the multiplication as if operating on two integers.
4
5
2
×
2
1
+
4
5
2
+
9
4
=
9
4
9
2
Rewrite the product with 3 total decimal places.
Answer = 9.492
Therefore: 45.2 × 0.21 = 9.492
Long Multiplication with Negative Numbers
When performing long multiplication you can ignore the signs until you have completed the standard algorithm for multiplication. Once you complete the multiplication follow these two rules:
- If one number is positive and one number is negative make the product negative.
- If both numbers are negative or both numbers are positive make the product positive.
Long Multiplication Example: Multiply 234 by 56
Long Multiplication Steps:
Stack the numbers with the larger number on top. Align the numbers by place value columns.
Multiply the ones digit in the bottom number by each digit in the top number
6 × 4 = 24
Put the 4 in Ones place
Carry the 2 to Tens place
carry
2
2
3
4
×
5
6
4
6 × 3 = 18
Add the 2 that you carried = 20
Put the 0 in the Tens place
Carry the 2 to the Hundreds place
carry
2
2
2
3
4
×
5
6
4
6 × 2 = 12
Add the 2 that you carried = 14
This is the last number to multiply so write the whole number answer. No need to carry the 1.
carry
2
2
2
3
4
×
5
6
1
4
4
Move one place to the left. Multiply the tens digit in the bottom number by each digit in the top number.
5 × 4 = 20
Add a row to your multiplication answer
When you write your answer, shift one column to the left
Put the 0 in Ones place
Carry the 2 to Tens place
carry
2
2
3
4
×
5
6
1
4
4
5 × 3 = 15
Add the 2 that you carried = 17
Put the 7 in the Tens place
Carry the 1 to the Hundreds place
carry
1
2
2
3
4
×
5
6
1
4
4
7
5 × 2 = 10
Add the 1 that you carried = 11
This is the last number to multiply so write the whole number answer. No need to carry the 1.
carry
1
2
2
3
4
×
5
6
1
4
4
1
1
7
Add the numbers in the columns using long addition
4 + 0 = 4
0 + 0 = 0
4 + 7 = 11
write the 1 and carry 1
1 + 1 + 1 = 3
Once you add the columns you can see the long multiplication result: 234 × 56 = 13104.
2
3
4
×
5
6
carry
1
1
4
4
+
1
1
7
1
3
1
4
Related Calculators
If you need help with long addition see our Long Addition Calculator to add numbers by long addition and see the work.
For long division see the Long Division Calculator to divide numbers by using long division with remainders. This calculator also shows the work.
If you need to multiply fractions visit our Fractions Calculator. You can do fraction multiplication, addition, subtraction and division here.
References
Math is Fun shows examples of Long Multiplication in an animated video.
Long multiplication is an algorithm and you can find examples of multiplication algorithms at Wikipedia.
Goodman, Len. "Long Multiplication." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. Long Multiplication
