The formula this page uses
a × (10t + u) = (a × u) + (a × t) × 10, one partial product per digit, each shifted one more place left
Work through long multiplication digit by digit
Use the free long multiplication calculator to work through the number relationship, preserve exact values, and check the result with estimation.
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Arithmetic
Long Multiplication Calculator explained
The short version
- Split the second number into its digits, multiply by each one, then add the rows.
- Each new row shifts one more place to the left, because that digit stands for tens, hundreds and so on.
- Estimate first by rounding, so you notice straight away if a row got shifted wrongly.
What each part means
SymbolWhat it means
a — The top numberThe number being multiplied. It can have any number of digits.
u — The ones digit of the bottom numberIt makes the first row, with no shift.
t — The tens digit of the bottom numberIt makes the second row, shifted one place left.
partial product — One row of the workingThe result of multiplying by a single digit, before anything is added.
Show your work: a full example
- Multiply 347 by 28347 × 28
- Split 28 into 8 and 2028 = 8 + 20
- Multiply by the 8347 × 8 = 2776
- Multiply by the 20347 × 20 = 6940
- Add the two rows2776 + 6940
- Finish the addition2776 + 6940 = 9716
- Check by rounding350 × 30 = 10500, and 9716 is close to that
A second, different case
- A different case: a zero in the middle406 × 35
- Split 35 into 5 and 3035 = 5 + 30
- Multiply by the 5, straight through the zero406 × 5 = 2030
- Multiply by the 30406 × 30 = 12180
- Add the rows2030 + 12180 = 14210
- Check a completely different way406 × 7 = 2842
- Then times 52842 × 5 = 14210, because 35 = 7 × 5
Copy-ready example
347 × 28
Multiply 347 by 28
Questions about the Long Multiplication Calculator
Why does the second row shift one place left?
Because that digit is not really a 2, it is 20. Multiplying by 20 is multiplying by 2 and then by 10, and multiplying by 10 is exactly what shifting one place left does. Some people write a 0 in the empty spot to make this visible.
How does it work for a three-digit by three-digit multiplication?
The same way, with three rows instead of two. The first row has no shift, the second shifts one place, the third shifts two places, and then you add all three. The number of rows always matches the number of digits in the bottom number.
Is the lattice method any different?
Not underneath. The lattice draws a grid, puts each single-digit product in its own box split by a diagonal, then adds along the diagonals. It produces the same partial products in a different arrangement, and it can help if keeping columns straight is the part you find hard.
Why bother estimating first?
Because it catches shift errors instantly. If you expect roughly 350 × 30 = 10500 and your answer comes out as 3470, you know a row went into the wrong column before you hand the work in. The estimate takes five seconds and finds the biggest kind of mistake.
How to use the Long Multiplication Calculator
Use the interactive workspace first, then connect the output to the mathematical definitions and checks below.
What the four-function calculator handles
Use standard arithmetic, parentheses, sign changes, percentages, memory registers, answer recall, and an editable calculation tape. The separate percentage workspace covers percent-of, percent-change, increases, discounts, tax, tips, and split totals.
Order of operations and percentages
Parentheses are evaluated first, followed by multiplication or division and then addition or subtraction at equal priority from left to right. A percent is a ratio per hundred; always identify the base quantity before calculating a discount or change.
How to verify everyday calculations
Estimate before entering values, keep money rounding until the final total, and use an inverse operation when possible. For a split bill, verify that the per-person amount multiplied by the group size returns the calculated total.
Need a complete explanation?
Send the exact question and your current work to MathGPT for a step-by-step method and an independent check.
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Understand arithmetic behind this calculator
A calculator confirms an answer. Working the method yourself is what makes the next problem faster.
GuidesLearn arithmeticConcepts, worked methods, and checks.
PracticePractice arithmetic questionsAnswer review after every attempt.
ReferenceArithmetic formulasConditions and examples for each formula.