Long Multiplication Calculator
Enter any two numbers to multiply them using the standard long multiplication algorithm. The calculator shows every partial product and the final addition step, so you can follow the working digit by digit. Decimals and negative numbers are both supported.
What is long multiplication?
Long multiplication is the standard written algorithm for multiplying any two numbers, no matter how many digits they have. Instead of trying to multiply the whole numbers at once, you break the second number (the multiplier) into its individual digits, multiply the first number (the multiplicand) by each digit separately, account for the place value of each digit by shifting the result left, and then add all the partial products together. The technique works for positive and negative integers, and for decimals once you temporarily remove the decimal points and reinsert them at the end.
How to do long multiplication step by step
Write the larger number on top (the multiplicand) and the smaller on the bottom (the multiplier). Starting with the rightmost digit of the multiplier, multiply it by each digit of the multiplicand from right to left, carrying any tens into the next column. Write the result as the first partial product. Move to the next digit of the multiplier and repeat, but indent this partial product one place to the left to account for the fact that it represents tens rather than ones. Each successive digit gets one more place of indentation. Finally, add all the partial products using long addition. For decimals: count the total decimal places in both numbers, do the multiplication ignoring the decimal points, then move the decimal point that many places from the right in the answer.
Multiplying negative numbers
Sign rules for multiplication are simple: positive times positive is positive, negative times negative is positive, and positive times negative (or negative times positive) is negative. When one or both factors are negative, strip the negative signs, complete the standard algorithm on the absolute values, and then apply the sign rule to the final product. This calculator handles sign logic automatically and shows it as the first step in the working.
Multiplying decimals using the long method
Count the total number of decimal places across both numbers. For example, 2.5 has one decimal place and 3.14 has two, giving a combined total of three. Remove both decimal points and multiply the resulting whole numbers (25 and 314) using the standard algorithm. The product of 25 and 314 is 7,850. Now move the decimal point three places from the right to get 7.850, which simplifies to 7.85. The key insight is that multiplying 2.5 by 3.14 is identical to multiplying 25 by 314 and then dividing by 1,000.
Place value reference
| Digit position | Place name | Shift (zeros appended) | Example: digit 3 at this position |
|---|---|---|---|
| 1st from right | Ones | 0 | 3 × any = 3 × any |
| 2nd from right | Tens | 1 | 3 × any × 10 |
| 3rd from right | Hundreds | 2 | 3 × any × 100 |
| 4th from right | Thousands | 3 | 3 × any × 1,000 |
| 5th from right | Ten-thousands | 4 | 3 × any × 10,000 |
Each digit position in the multiplier creates one partial product shifted by that many places.
Frequently asked questions
What is a partial product in long multiplication?
A partial product is the result of multiplying the multiplicand by a single digit of the multiplier. If your multiplier has three digits, you get three partial products. Each partial product is shifted one extra place to the left compared to the one before it, because each digit represents a higher place value. When you add all the partial products together, you get the final answer.
How do I multiply large numbers using long multiplication?
The process is the same regardless of how many digits are involved: work through each digit of the multiplier from right to left, multiply the entire multiplicand by that digit (carrying as needed), indent the row by one extra place for each successive digit, and finally add all the partial product rows. Keeping the columns neatly aligned is the main challenge with very large numbers, which is why graph paper or a column-based layout helps.
How does long multiplication handle decimal numbers?
Count the decimal places in both numbers and add them together. Remove the decimal points and multiply the resulting whole numbers normally. Move the decimal point in the product leftward by the combined count. For example, 12.5 times 4.2 has a combined two decimal places: multiply 125 by 42 to get 5,250, then shift the decimal two places left for 52.50.
What is the standard algorithm and is it the same as long multiplication?
Yes, the standard algorithm and long multiplication refer to the same procedure taught in most schools worldwide. Some teachers distinguish between the short multiplication method (when one factor is a single digit) and long multiplication (when both have multiple digits), but the underlying principle is identical: multiply digit by digit, account for place value, and sum the partial products.
Can I use long multiplication with negative numbers?
Yes. Apply the sign rule first: two negatives make a positive, one negative makes the product negative. Then ignore the signs and work through the standard algorithm on the absolute values. Attach the correct sign to the final answer. This calculator shows the sign step explicitly so you can see exactly when it is applied.