What is long multiplication?

Multiplying two numbers by taking the digits of one apart: each digit of the multiplier produces a line, the lines are shifted left according to the place that digit occupied, and the lines are added. It gives the same product a calculator does, with the intermediate lines kept.

Why are the partial products shifted?

Because a digit's worth depends on where it sits. The 4 in 45 is worth forty, so everything it multiplies has to come out ten times larger than what the 5 produced, and writing the line one place further left is how that is recorded. It is the same rule that makes the columns of an addition line up.

How do I know where the decimal point goes?

Count the decimal places in the two factors and add them. 2.3 has one and 4.5 has one, so the product has two and 1035 becomes 10.35. The digits themselves never see the point — the whole multiplication runs on whole numbers and the point is placed at the end.

Why is a line of zeros kept?

Because the line's position is the information. When the multiplier's ones digit is zero, that line says the ones place contributed nothing, and dropping it would lose the fact that the remaining line is shifted. Keeping it also means the lines always add up to the product, which is the check the page is built to support.

What happens with negative factors?

The product is negative when exactly one of the two factors is, and positive when both are. The partial products are printed without signs either way, matching the way a written calculation keeps the minus outside the working — and it is what keeps them adding up to the product.

Why does an integer product print with a row of zeros?

Because the panel prints the product to a fixed six decimals, so 1035 comes out as 1035.000000. The alternative would be printing the product as text, which would put an ASCII full stop in the answer — wrong in countries where the decimal mark is a comma, and worse than a few trailing zeros.

Multiplication Calculator

Result

1,035.000000

Product

Partial products: 115, 920
Decimal places in the product: 0

Long multiplication multiplies two numbers digit by digit and keeps every partial product in view, along with the shift that puts it in the right place and the decimal point's final position. Multiplying two numbers on this page means multiplying the digits of one by each digit of the other, writing each result on its own line, and adding the lines up — which is the same addition the long addition page performs, one line at a time. The default works out 23 × 45 = 1035 as the partial products 115 and 920, the second shifted one place left. Decimals change nothing about the digit work: 2.3 × 4.5 uses exactly the same two partial products, and only the decimal point moves, two places in from the right because the two factors carry one decimal place each. Negative factors are accepted, and the partial products are still printed unsigned, in the same way a written calculation puts the minus sign outside the working.

23 × 45, one digit of the multiplier at a time

Shift	Multiplier digit	Partial product	Value after shifting
0	5	115	115
1	4	92	920

One row per digit of the multiplier, starting from the ones digit. The shift column is the reason the table exists: on paper the shift is shown by leaving a gap at the start of the line, and a row written into a table has no gaps to leave, so the alignment has to become a number. Reading the last column downwards gives the two lines that are then added, and adding them is an ordinary column addition — the shift of 1 is exactly the zero at the end of 920. Every cell is a whole number, so the table is identical in all ten languages the site serves.

Formula

a × b: write b's digits as b₀ + 10·b₁ + …, take a × b_k on its own line, shift it k places, then add the lines product = (a × b) with the decimal point placed total places from the right

a: The multiplicand, the number being multiplied. The whole of it is multiplied by each digit of the other number in turn, so it stays in one piece while the other number is taken apart.
b: The multiplier, the number whose digits are taken one at a time from the right. Each of its digits produces one line, including a digit of zero, because that line records which place the digit occupied — a place with nothing in it is still information.
shift: How many places left the line is written, equal to the position of the multiplier's digit. This is place value at work: the digit 4 in 45 is worth forty, so its partial product is ten times the size of the ones the ones digit produces. On paper the shift is shown by starting the line further left; in a table it has to become a number.
decimal places: The number of digits after the decimal point, added up across the two factors rather than counted in either one. 2.3 has one and 4.5 has one, so the product carries two, and 1035 becomes 10.35. Nothing in the digit work depends on this: the multiplication runs entirely on whole numbers and the point is placed afterwards.

Use this to multiply two numbers by hand and keep the working: checking a product that was worked out on paper, teaching the shifting step, or handling decimals where the point's position is the part that goes wrong. Repeated multiplication of a number by itself is the exponent page; multiplication of fractions follows its own rule, top by top and bottom by bottom.

Worked examples

23 × 45
1. Multiply by the ones digit: 23 × 5 = 115, written with no shift
2. Multiply by the tens digit: 23 × 4 = 92
3. The tens digit is worth ten times as much, so that line shifts one place left and reads 920
4. Add the lines: 115 + 920 = 1035
5. Neither factor has a decimal place, so the product has none
6. The product is 1035
The default case. The 0 at the end of 920 is the shift made visible — on paper that line simply starts one column further left, and in the reading it becomes a digit that has to be written. The two lines added together are an ordinary column addition, which is the next page over.
The same digits with decimal points
1. Ignore the points and multiply the digits: 23 × 45 is the previous example
2. The partial products are 115 and 920, unchanged
3. Count the decimal places: 2.3 has one and 4.5 has one, so the product has two
4. Place the point two digits in from the right of 1035
5. The product is 10.35
The clearest statement of how decimals work here: the digit work is identical to multiplying 23 by 45, and the whole of the difference is where the point lands. That is also why the decimal places are reported as a sum rather than counted off either factor.
A zero digit in the multiplier
1. Multiply by the ones digit: 23 × 0 = 0, written with no shift
2. Multiply by the tens digit: 23 × 4 = 92, shifted one place left, so 920
3. Add the lines: 0 + 920 = 920
4. The product is 920
The zero line is kept rather than skipped. Written by hand it is usually left out, but its position is the thing being recorded — it says the ones digit of 40 contributed nothing — and the check that the lines add up to the product only holds if the line is there. This is the opposite choice from the FOIL page, where a term that works out to zero is dropped: there the empty term does not exist, while here a line really is written down.

Limitations

The two factors may each carry at most six decimal places between them, and the sum is what is limited rather than either number. That is a display limit, not an arithmetic one: the panel prints the product to six decimals, so a product needing eight would be printed as 0.000000 — a wrong answer wearing the clothes of a right one. Each factor is limited to a magnitude of one million, and the whole-number form of the product to the largest exactly representable integer, because the digits are read one place at a time and beyond that limit the partial products stop adding up to the answer. Negative factors are accepted, and the partial products are printed without signs; the sign appears only on the final product. Zero is a legitimate factor, and the product of a negative and a zero is printed as 0 rather than as negative zero. This page multiplies; division is a separate procedure with its own column rules.

Frequently asked questions

What is long multiplication?: Multiplying two numbers by taking the digits of one apart: each digit of the multiplier produces a line, the lines are shifted left according to the place that digit occupied, and the lines are added. It gives the same product a calculator does, with the intermediate lines kept.
Why are the partial products shifted?: Because a digit's worth depends on where it sits. The 4 in 45 is worth forty, so everything it multiplies has to come out ten times larger than what the 5 produced, and writing the line one place further left is how that is recorded. It is the same rule that makes the columns of an addition line up.
How do I know where the decimal point goes?: Count the decimal places in the two factors and add them. 2.3 has one and 4.5 has one, so the product has two and 1035 becomes 10.35. The digits themselves never see the point — the whole multiplication runs on whole numbers and the point is placed at the end.
Why is a line of zeros kept?: Because the line's position is the information. When the multiplier's ones digit is zero, that line says the ones place contributed nothing, and dropping it would lose the fact that the remaining line is shifted. Keeping it also means the lines always add up to the product, which is the check the page is built to support.
What happens with negative factors?: The product is negative when exactly one of the two factors is, and positive when both are. The partial products are printed without signs either way, matching the way a written calculation keeps the minus outside the working — and it is what keeps them adding up to the product.
Why does an integer product print with a row of zeros?: Because the panel prints the product to a fixed six decimals, so 1035 comes out as 1035.000000. The alternative would be printing the product as text, which would put an ASCII full stop in the answer — wrong in countries where the decimal mark is a comma, and worse than a few trailing zeros.

References

Base — why shifting a partial product one place left multiplies it by ten, which is what the shift column records — Wolfram MathWorld (United States)
Exponentiation — multiplication applied to a number and itself, the case where both factors are the same — Wolfram MathWorld (United States)
Division — the inverse operation, and the reason it needs a different column procedure rather than a variation on this one — Wolfram MathWorld (United States)

Related calculators

Long Addition CalculatorAdd whole numbers column by column, with carries shown.Exponent CalculatorRaise a number to any power, including negative exponents.Multiplying Fractions CalculatorMultiply fractions and simplify the result.