First of all, you need to know your times tables. You cannot do mental math without those basic facts available.

Single digits.

.. 2 : double the number--add it to itself, or multiply each digit by 2

.. 3 : add the number to its double, or multiply each digit by 3

.. 4 : double the double of the number

.. 5 : multiply the number by 10 and divide the result by 2 (in either order)

.. 6 : multiply by 3 and by 2, or multiply by 5 and add the orginal number

.. 7 : multiply by 6 and add the number; multiply by 5 and add double the number; multiply by 8 and subtract the number; multiply by 10 and subtract 3 times the number

.. 8 : double the result of multiplying by 4, or multiply by 10 and subtract double the number

.. 9 : multiply by 10 and subtract the number

.. 10 : if you don't know how to multiply by 10, study up on place-value number systems.

For multiple-digit products, you need to be aware of the contributors to the number that goes in a particular place in the result. Suppose we multiply 3-digit number XYZ by 3-digit number ABC.

The product that will define the 1s digit of the result is CZ.

The product that will define the 10s digit of the result is BZ+CY (and any carry from CZ).

The product that will define the 100s digit of the result is AZ +BY +CX (and any carry from the previous sum).

The product that will define the 1000s digit of the result is AY +BX (and any carry from the previous sum).

The product that will define the 10,000s digit o the result is AX (and any carry from the previous sum).

I have described the process of developing product digits from right to lefft, but it works the same from left to right. When you do that, you may want to write down the "carry" digits on an additional line (or lines), rather than try to keep track of it all in your head.

Example:

.. 736*529 =

.. digit number (power of 10)

.. 4 .. 3 .. 2 .. 1 .. 0

..35 29. 99. 39. 54 . . . . . . . . dots (.) are for spacing only. Partial sums developed as described above.

so you can write this right to left as

.. 1s digit: (5)4 . . . . . . . . . . . . . . carry to next digit is in parentheses

.. 10s digit: 39+5 = (4)4

.. 100s digit: 99 +4 = (10)3

.. 1000s digit: 29 +10 = (3)9

.. 10,000s digit: 35 +3 = 38

.. end result: 389,344

Working left-to-right, the same set of partial sums would be developed as shown above. They could be written with the 10s digit of the sum on a separate line, arranged to facilitate adding, as

... 3 ... 5 ... 9) .. 9] .. 9} .. 4

... ... ..(2 .. [9 .. {3 ... 5 . . . . . . . . . . . . where ( ), [ ], and { } identify the digits of the intermediate sums 29, 99, 39

Then the sum of all this is

.. 3 ... 8 ... 9 ... 3 ... 4 ... 4 . . . . . . . . . same as right-to-left product

Some folks can carry the necessary intermediate results in their head. We might develop the desired product from the left as

.. 35 . . . . . . . . . first partial sum (10,000s place) is 35

.. 379 . . . . . . . . next partial sum (1000s place) is 29. The 2 adds to the 35.

.. 3889 . . . . . . . next partial sum (100s place) is 99. The 9 (10s digit) adds to the 79

.. 38929 . . . . . . next partial sum (10s place) is 39. The 3 adds to the 89

.. 389344 . . . . . final partial sum (1s place) is 54. The 5 adds to the 29.

Note that every digit but the first and last needed to be amended at least once.