OneTaskBy Tim King · Updated
Long Multiplication Calculator
Column Multiplication With Partial Products for Decimals and Large Numbers
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How the Long Multiplication Calculator works
This long multiplication calculator multiplies two numbers the column way taught at school. It writes one row for each digit of the second number, shows how many zeros that row needs, and adds the rows up. Decimals and negative numbers work, and there is no practical size limit, so 18-digit numbers come out exact.
Long multiplication splits a hard product into easy ones. Instead of multiplying 347 by 28 in one go, you multiply 347 by 8, then by 20, and add the two results. Each digit of the second number gives one row, and each row only needs a single-digit times table plus carrying.
How the long multiplication calculator works
The formula is the distributive law: a × (b₁ + b₂ + …) = a × b₁ + a × b₂ + …, where each b is one digit of the second number at its place value. The calculator works from the ones digit leftwards. A tens digit row gets one zero on the end, a hundreds digit row gets two, and so on. Then the rows are added. For decimals it first ignores the points, multiplies the whole numbers, and puts the point back by counting the decimal places in both numbers.
First example. 347 × 28. The ones row is 347 × 8 = 2,776. The tens row is 347 × 2 = 694 with one zero added, 6,940. Adding gives 9,716.
Second example. 4.5 × 1.25. Multiply 45 by 125 to get 5,625. There are three decimal places between the two numbers, so the answer is 5.625.
How many tiles cover a floor 23 tiles wide and 47 long? 47 × 3 = 141 and 47 × 20 = 940, so the floor takes 1,081 tiles.
Long multiplication of 347 × 28, one row per digit of the second number
| Place | Digit | Multiply | Zeros added | Row |
|---|---|---|---|---|
| ones | 8 | 347 × 8 = 2,776 | none | 2,776 |
| tens | 2 | 347 × 2 = 694 | 1 zero | 6,940 |
| Total | Add every row | 9,716 |
Squares from 11 to 30 by long multiplication: the tens row plus the ones row
| Number | × tens | × ones | Product |
|---|---|---|---|
| 11 | 11 × 10 = 110 | 11 × 1 = 11 | 121 |
| 12 | 12 × 10 = 120 | 12 × 2 = 24 | 144 |
| 13 | 13 × 10 = 130 | 13 × 3 = 39 | 169 |
| 14 | 14 × 10 = 140 | 14 × 4 = 56 | 196 |
| 15 | 15 × 10 = 150 | 15 × 5 = 75 | 225 |
| 16 | 16 × 10 = 160 | 16 × 6 = 96 | 256 |
| 17 | 17 × 10 = 170 | 17 × 7 = 119 | 289 |
| 18 | 18 × 10 = 180 | 18 × 8 = 144 | 324 |
| 19 | 19 × 10 = 190 | 19 × 9 = 171 | 361 |
| 20 | 20 × 20 = 400 | 20 × 0 = 0 | 400 |
| 21 | 21 × 20 = 420 | 21 × 1 = 21 | 441 |
| 22 | 22 × 20 = 440 | 22 × 2 = 44 | 484 |
| 23 | 23 × 20 = 460 | 23 × 3 = 69 | 529 |
| 24 | 24 × 20 = 480 | 24 × 4 = 96 | 576 |
| 25 | 25 × 20 = 500 | 25 × 5 = 125 | 625 |
| 26 | 26 × 20 = 520 | 26 × 6 = 156 | 676 |
| 27 | 27 × 20 = 540 | 27 × 7 = 189 | 729 |
| 28 | 28 × 20 = 560 | 28 × 8 = 224 | 784 |
| 29 | 29 × 20 = 580 | 29 × 9 = 261 | 841 |
| 30 | 30 × 30 = 900 | 30 × 0 = 0 | 900 |
Tips
- Put the longer number on top. Fewer rows means less adding.
- Zero digits are free. A 0 in the second number gives a row of zeros you can skip.
- Estimate first. 347 × 28 is about 350 × 30 = 10,500, so 9,716 is the right size.
FAQ
What are the steps of long multiplication?
Write the larger number on top. Multiply it by the ones digit of the bottom number and write that row. Move to the tens digit, put one zero at the end of the new row, and multiply again. Keep going for each digit, adding one more zero each time, then add all the rows together.
How do you do long multiplication with decimals?
Take the decimal points out and multiply as whole numbers. Then count the digits after the point in both original numbers and put the point that many places from the right of the answer. 2.4 × 0.36 becomes 24 × 36 = 864, and three decimal places makes it 0.864.
Why do you add a zero on the second row?
The second digit of the bottom number is a tens digit, so it is really worth ten times its face value. Multiplying by 2 in the tens place means multiplying by 20, and the zero on the end is the multiplying by 10. The hundreds row gets two zeros for the same reason.
Is the grid method the same as long multiplication?
They give the same answer by the same idea, splitting numbers by place value. The grid method splits both numbers, so 347 × 28 becomes six small boxes such as 300 × 20. Long multiplication only splits the bottom number, which gives fewer, longer rows and is quicker once carrying is easy.
See also: all Arithmetic tools, Number Theory, Algebra and Geometry, Statistics, Trigonometry and Calculus.
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