Multiplication Calculator
This multiplication calculator multiplies any two numbers and lays the long multiplication out one partial product at a time, naming the place value behind every digit. It also shows the sum, difference and quotient of the same pair, so you can check a whole set of answers at once.
How to use the multiplication calculator
- Type the first number into the top box. It can be whole, decimal or negative.
- Type the number you are multiplying by into the second box.
- Read the product at the top of the results. Everything updates as you type, so there is nothing to press.
- Scroll the working underneath to see each partial product, the column it belongs to, and the addition that finishes the sum.
The panel beside the product also gives you the sum, the difference and the quotient of the same two numbers. If you only wanted the quotient, the division calculator walks that one out digit by digit instead.
How long multiplication works
Long multiplication is one hard problem broken into several easy ones. You never multiply by 28. You multiply by 8, then by 2, and the 2 is worth twenty because it sits in the tens column, so its answer gets shifted one place to the left.
That shift is the whole trick, and it is where most mistakes happen. Written out as a column sum you show it with a zero in the units place. Written out in a line, as the working here does, you show it as an explicit × 10.
Example: 347 × 28
The second number has two digits, so there are two partial products. The 8 is in the units column and the 2 is in the tens column.
So 347 × 28 = 9,716. As a rough check, 347 is close to 350 and 28 is close to 30, and 350 × 30 = 10,500 — the same size of answer.
More worked examples
Example: 426 × 35
Two digits again, so two partial products.
Example: 214 × 306, with a zero in the middle
A zero digit still gets a line. Skipping it is the fastest way to shift every later partial product into the wrong column.
Example: 3.6 × 1.25
Ignore both decimal points, multiply the whole numbers, then put one point back.
Multiplying decimals: count the decimal places
Decimals do not need a new method. Strip the points out, multiply the digits as whole numbers, then count how many digits sat after the point in the two numbers you started with. The answer gets that many decimal places.
This is why multiplying two numbers smaller than 1 gives something smaller than both of them. Each decimal place you add pushes the answer one column further to the right.
| Problem | Whole-number multiply | Decimal places | Answer |
|---|---|---|---|
| 0.4 × 0.25 | 4 × 25 = 100 | 1 + 2 = 3 | 0.100 = 0.1 |
| 1.2 × 0.5 | 12 × 5 = 60 | 1 + 1 = 2 | 0.60 = 0.6 |
| 3.6 × 1.25 | 36 × 125 = 4,500 | 1 + 2 = 3 | 4.500 = 4.5 |
| 2.5 × 40 | 25 × 40 = 1,000 | 1 + 0 = 1 | 100.0 = 100 |
| 0.03 × 0.007 | 3 × 7 = 21 | 2 + 3 = 5 | 0.00021 |
Powers of ten, and what negatives do to the sign
Multiplying by 10, 100 or 1,000 moves the decimal point to the right by the number of zeros. Dividing moves it the same distance to the left. Nothing else about the digits changes.
That last line catches people out. “Add a zero” only works for whole numbers, and it works because appending a zero happens to shift every digit one column left. With decimals, move the point instead.
Signs follow one short rule: count the negatives. An even number of negative factors gives a positive answer, an odd number gives a negative one.
Two negatives make a positive because multiplying by a negative reverses direction on the number line. Reverse twice and you are pointing the way you started.
Mental multiplication shortcuts
Long multiplication always works, but for a lot of everyday numbers there is something faster. Each of these is exact, not an approximation.
| To multiply by | Do this instead | Example |
|---|---|---|
| 5 | Halve, then multiply by 10 | 86 × 5 → 43 × 10 = 430 |
| 9 | Multiply by 10, then subtract the number once | 34 × 9 → 340 − 34 = 306 |
| 15 | Multiply by 10, then add half of that | 24 × 15 → 240 + 120 = 360 |
| 25 | Divide by 4, then multiply by 100 | 36 × 25 → 9 × 100 = 900 |
| 11 (two digits) | Split the digits, put their sum in the middle | 43 × 11 → 4, 4+3, 3 = 473 |
| Anything even | Double one side, halve the other | 14 × 35 → 7 × 70 = 490 |
Doubling and halving is the most useful of the set because it keeps the product identical while making one factor friendlier. 16 × 25 becomes 8 × 50, then 4 × 100, which is 400 with no working at all.
These shortcuts are worth checking against the full working the first few times. When a multiplication sits inside a longer expression, the order of operations calculator will tell you when it should be done at all.
Common mistakes
- Forgetting the shift. The second partial product is the number multiplied by a tens digit, so it needs a placeholder zero. Leave it out and 347 × 28 becomes 2,776 + 694 = 3,470, which is nowhere near right.
- Skipping a zero digit. In 214 × 306 the 0 still owns the tens column. Drop the line and the 3 gets treated as a tens digit instead of a hundreds digit.
- Counting the answer’s decimal places instead of the inputs’. The rule looks at what you started with, before any trailing zeros are removed.
- Losing a carry. Carries belong to the partial product you are working on, not the next one. Cross each one out as you use it.
- Adding a zero to a decimal to multiply by 10. 2.5 × 10 is 25, not 2.50.
- Trusting a long product without a size check. Round both numbers and multiply those in your head first. If the estimate and the answer are a factor of ten apart, a digit has slipped a column.
- Running out of precision. Past about ten digits an ordinary calculator starts rounding. For exact products of very long numbers use the big number calculator.
Frequently asked questions
What is 347 × 28?
347 × 28 = 9,716. Take it in two pieces: 347 × 8 = 2,776 for the units digit, then 347 × 2 × 10 = 6,940 because the 2 sits in the tens column. Adding the two partial products gives 2,776 + 6,940 = 9,716.
What is 0.4 × 0.25?
0.4 × 0.25 = 0.1. Ignore the decimal points and multiply 4 × 25 = 100. The first number has one decimal place and the second has two, so the answer needs three: 100 becomes 0.100, which is 0.1. Both factors are less than 1, so the answer is smaller than either of them.
What is 25 × 16?
25 × 16 = 400. Doubling and halving makes this a mental sum: 25 × 16 becomes 50 × 8, then 100 × 4, which is 400. The product never changes as long as you halve one side each time you double the other.
Why does a negative times a negative give a positive?
Multiplying by a negative flips the direction of the answer on the number line. Doing that twice puts you back where you started. It also keeps the arithmetic consistent: since −12 × 7 = −84, the pattern of adding 12 each time you reduce the second factor forces −12 × −7 to be +84.
Does the order of the two numbers matter?
No. 28 × 347 and 347 × 28 both give 9,716, because multiplication is commutative. The written working does change, though. Putting the shorter number on the bottom gives you fewer partial products to add, so 347 × 28 is less work by hand than 28 × 347.
How do you multiply three or more numbers together?
Multiply two of them, then multiply that result by the next, and keep going. The order makes no difference to the answer, so group whichever pair is easiest first. For 4 × 17 × 25, do 4 × 25 = 100 first, then 100 × 17 = 1,700.
What is the fastest way to multiply by 5?
Halve the number and multiply by 10, because 5 is half of 10. For 86 × 5, halve 86 to get 43, then 43 × 10 = 430. If the number is odd you get a half left over: 37 × 5 is 18.5 × 10 = 185.
Why does multiplying by 10 not just add a zero to a decimal?
Appending a zero only works for whole numbers, and only by accident — it happens to shift every digit one column to the left. What multiplying by 10 really does is move the decimal point one place right. So 2.5 × 10 = 25, and 0.086 × 100 = 8.6.
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