Long Multiplication Calculator

Q: How do you check a multiplication answer without a calculator?

Try one of these three checks: Estimate. Round both numbers and multiply. 137 × 23 is near 140 × 20 = 2,800, so 3,151 looks right. Swap and redo. Multiply 23 × 137 instead. You should get the same product. Casting out nines. Add the digits of each number until one digit is left (137 → 11 → 2; 23 → 5). Multiply those: 2 × 5 = 10 → 1. The digit sum of 3,151 is 10 → 1. They match, so the answer is likely correct.

Introduction

This Long Multiplication Calculator multiplies two numbers and shows you every step. Type a multiplicand (the top number) and a multiplier (the bottom number), then press Calculate. You get the answer plus the full written method, just like you would do it on paper.

The calculator splits the problem into partial products. It multiplies the top number by each digit of the bottom number, shifts each answer one place to the left, then adds them all up. Carried digits sit in small boxes above the columns, and every digit sits in its own place-value column, so you can trace the ones, tens and hundreds straight down the page.

It works with whole numbers, decimals, and negative numbers. If you use decimals, the calculator counts the decimal places and puts the point back in the right spot. If one number is negative, it explains the sign rule. You also get a summary grid, a step-by-step walkthrough, a final addition grid, and a bar chart that shows how much each partial product adds to the total.

Use it to check homework, learn the long multiplication method, or practice with the Random button. The steps show for numbers up to 9 digits each. Bigger numbers still give an exact answer.

How to use our Long Multiplication Calculator

Type the two numbers you want to multiply. The calculator shows the answer, a place-value grid, and step-by-step long multiplication with each partial product.

Multiplicand (top number): Enter the first number here. You can use whole numbers, decimals, or negative numbers, up to 15 characters.

Multiplier (bottom number): Enter the second number here. Each digit of this number becomes one step in the solution below.

Calculate: Click this button to get the product and the full worked solution. You can also just press Enter.

Clear: Click this to empty both boxes and remove the results so you can start again.

Random: Click this to fill both boxes with example numbers and solve them right away. Great for practice.

See the Solution: Click this to jump straight down to the grids and the step-by-step long multiplication steps.

What Is Long Multiplication?

Long multiplication is a way to multiply big numbers by hand. You break the problem into small parts, multiply one digit at a time, and then add the parts together. It works for any two numbers, no matter how many digits they have.

How Long Multiplication Works

Write the first number (the multiplicand) on top and the second number (the multiplier) below it. Line the digits up by place value: ones over ones, tens over tens. Then follow these steps:

Multiply the top number by the ones digit of the bottom number. Carry when a product is 10 or more.
Move to the tens digit. Multiply again, but shift the answer one spot to the left (write a 0 in the ones column).
Keep going for each digit, shifting one more spot left each time.
Add all the partial products. That sum is your answer.

Example: 137 × 23

Multiply 137 by 3 to get 411. Multiply 137 by 2 to get 274, then shift it one place left to make 2,740. Add them: 411 + 2,740 = 3,151.

Key Words to Know

Multiplicand: the top number being multiplied.
Multiplier: the bottom number that tells how many times.
Partial product: the answer from one digit of the multiplier.
Carry: the extra digit you move to the next column.
Product: the final answer.

Decimals and Negative Numbers

For decimals, ignore the decimal points and multiply like whole numbers. Then count the decimal places in both numbers, add them, and move the point that many spots from the right. For example, 1.3 × 0.2 has two decimal places in all, so 13 × 2 = 26 becomes 0.26.

For signs, follow one simple rule: two negatives make a positive, and one negative makes a negative.

Tips to Avoid Mistakes

Keep your columns straight so digits stay in the right place.
Write carries above the next column and cross them out after you use them.
Don't forget the zeros when you shift each partial product left.
Check your answer by estimating: 137 × 23 is close to 140 × 20 = 2,800, so 3,151 makes sense.

Formulas used

Partial product for each multiplier digit

p_j = A \times b_j \times 10^{j}

Product as the sum of partial products

A \times B = \sum_{j=0}^{n-1} A \times b_j \times 10^{j}

Column digit and carry in a single-digit multiplication

a_i \times d + c_i = 10\,c_{i-1} + r_i

Decimal places in the product

d_{total} = d_A + d_B

Restoring the decimal point

A \times B = \frac{(A \cdot 10^{d_A}) \times (B \cdot 10^{d_B})}{10^{\,d_A + d_B}}

Sign rule for the product

\mathrm{sign}(A \times B) = \mathrm{sign}(A) \cdot \mathrm{sign}(B)

Full signed result

A \times B = \mathrm{sign}(A)\,\mathrm{sign}(B) \cdot \frac{|A|_{int} \times |B|_{int}}{10^{\,d_A + d_B}}

Frequently asked questions

Why do you put a zero when you multiply by the tens digit?

Because that digit is not worth ones. In 137 × 23, the 2 stands for 20, so 137 × 2 = 274 is really 2,740.

Writing a 0 in the ones column pushes the partial product one place to the left, which keeps every digit in the right column. For the hundreds digit you write two zeros, for the thousands digit three zeros, and so on.

Does it matter which number goes on top in long multiplication?

The answer is the same either way. 23 × 137 equals 137 × 23.

But the work is easier if you put the number with more digits on top and the shorter number on the bottom. The bottom number decides how many steps you do, so a short multiplier means fewer partial products to write and add.

How many partial products does a long multiplication problem have?

One for each digit in the bottom number (the multiplier).

1-digit multiplier → 1 partial product
2-digit multiplier → 2 partial products
4-digit multiplier → 4 partial products

The top number's size does not change how many rows you get. It only changes how long each row is.

What do you do when there is a zero in the multiplier?

Anything times 0 is 0, so that whole partial product is a row of zeros.

You have two safe choices. You can write the zero row and add it like normal, or you can skip it and shift the next partial product two places left instead of one. For 412 × 302, the middle step is 0, so you just add 824 and 123,600 to get 124,424.

How do you check a multiplication answer without a calculator?

Try one of these three checks:

Estimate. Round both numbers and multiply. 137 × 23 is near 140 × 20 = 2,800, so 3,151 looks right.
Swap and redo. Multiply 23 × 137 instead. You should get the same product.
Casting out nines. Add the digits of each number until one digit is left (137 → 11 → 2; 23 → 5). Multiply those: 2 × 5 = 10 → 1. The digit sum of 3,151 is 10 → 1. They match, so the answer is likely correct.

What grade do kids learn long multiplication?

Most students multiply 2-digit numbers by 1-digit numbers in grade 3 or 4. They learn full long multiplication, like 2-digit by 2-digit, in grade 4, and become fluent with bigger numbers in grade 5.

Before starting, a student should know the times tables up to 10 and how to add numbers in columns with carrying.

What is the lattice method of multiplication?

The lattice method uses a grid with a diagonal line in each box. You write one digit of the top number above each column and one digit of the side number beside each row.

Multiply the two digits for each box and write the tens above the diagonal and the ones below it. Then add along the diagonals, starting at the bottom right, and carry to the next diagonal. The digits down the left side and across the bottom spell out the answer. It helps students who lose track of carries.

How do you multiply numbers that end in zeros, like 4,000 × 300?

Drop the zeros, multiply the leftover digits, then put all the zeros back.

For 4,000 × 300: multiply 4 × 3 = 12. There are three zeros in 4,000 and two zeros in 300, so add five zeros. The answer is 1,200,000.

The same shortcut works for 10, 100 and 1,000: 46 × 100 = 4,600.

How do you multiply three numbers together?

Multiply two of them first, then multiply that answer by the third number.

For 12 × 5 × 8: do 12 × 5 = 60, then 60 × 8 = 480. The order does not change the answer, so pick the pair that is easiest. Here, 5 × 8 = 40 first, then 40 × 12 = 480, which is even quicker.

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Introduction

How to use our Long Multiplication Calculator

What Is Long Multiplication?

How Long Multiplication Works

Example: 137 × 23

Key Words to Know

Decimals and Negative Numbers

Tips to Avoid Mistakes

Formulas used

Frequently asked questions

Why do you put a zero when you multiply by the tens digit?

Does it matter which number goes on top in long multiplication?

How many partial products does a long multiplication problem have?

What do you do when there is a zero in the multiplier?

How do you check a multiplication answer without a calculator?

What grade do kids learn long multiplication?

What is the lattice method of multiplication?

How do you multiply numbers that end in zeros, like 4,000 × 300?

How do you multiply three numbers together?

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