Multiply two whole numbers with exact integer arithmetic instead of a rounded floating-point result. The page traces how the inputs become product before rounding.
Large exact products occur in counting, checksums, integer sequences, and financial minor-unit calculations. Exact text input keeps significant low-order digits intact.
The sample factors produce 121,932,631,137,021,071,359,549,253,925. Reversing their order gives the identical product. This Long Multiplication example can be compared with sum partial products.
For this page, long multiplication is evaluated using the displayed Long Multiplication field order and convention.
This whole-number tool does not place a decimal point for you. Count decimal places separately or use a calculator designed for decimal measurements.
Changing only one field provides a direct Long Multiplication sensitivity check for long multiplication.
Multiply the upper number by each lower digit from right to left. Shift successive partial products one place left, then add those rows with ordinary column addition.
The Long Multiplication meaning depends on First factor. The Long Multiplication meaning also depends on Second factor. Carry those Long Multiplication roles into any later Long Multiplication work.
Long multiplication forms one partial product for each digit of a factor, shifts that row according to place value, and adds the rows. The method is distributive multiplication written by columns. To extend Long Multiplication, open reverse with division.
Before accepting Product, restore the Long Multiplication inputs First factor and Second factor. Estimate Product independently. Then vary Second factor alone and observe the new Long Multiplication output. This isolates the changed part of Long Multiplication.
Count the combined order of magnitude before multiplying. Factors near 10^m and 10^n should produce a result near 10^(m+n). This rough scale check will not verify every digit, but it quickly exposes a missing partial-product shift or a misplaced group of zeros.
No. Whole-number multiplication is commutative.
The purpose is to preserve and display the exact integer.
Yes; the exact product is zero.
Use the integer calculator for signed products.