How do you check your answer in multi-digit multiplication?

Answer 1

The best way is to use the method that was used for centuries - until the modern calculators became ubiquitous - estimate the answer.

Consider, for example,2045078 * 9087054 = 18583734220212

Estimate: 2 million * 9 million is 18 million.

So (2 million plus a bit) * (9 million plus a bit) = 18 million plus a bigger bit.

Yes, that does look like it!

A more accurate method is to do carry out a "digital root" check.

A digital root of a multi-digit number is calculated by adding together all the digits. If the answer is greater than 8 then keep repeating with the digits of the answer, until it is not. Equivalently, add together the digits but every time you come to 9 put the sum back to 0.

Then the digital root of the product must equal the product of the digital roots.

Back to the above example:

DR(2045078) = DR(26) = 8 [26 = 9*2 + 8]

DR(9087054) = DR(33) = 6 [33 = 9*3 + 6]

DR(18583734220212) = DR(48) = DR(12) = 3 [48 = 9*5 + 3]

The DR of the left hand side (LHS) of the DR equations is DR(8*6) = DR(48) = 3

and that of the right hand side (RHS) is also 3.

So the multiplication is probably correct. The method is not fool-proof. Errors due to swapping digits are untraceable. Also, replacing 0 by 9 or 9 by 0 cannot be checked.

The latter kind of mistake can be identified if you repeat with DR7, the digital roots in which you discard 7s. So

DR7(2045078) = DR(26) = 5 [26 = 7*3 + 5]

DR7(9087054) = DR(33) = 5 [33 = 7*8 + 5]

DR7(18583734220212) = DR(48) = 6 [48 = 7*7 + 6]

Now DR7(LHS) = DR7(5*5) = DR7(25) = 6 = DR7(RHS).