Why do you not line up decimals when you multiply?

Answer 1

You could line up the decimals if you prefer; it's just that when multiplying, unlike adding, lining them up doesn't make things easier. Also, your teacher is likely to misunderstand your creative technique and mark you down even if you get the right answer.

When adding, it makes things easier to line up the digits of the same order of magnitude because you only want to add ones to ones and tens to tens. But when you multiply, you end up multiplying every digit in one number by every digit in the other number anyway, so you don't gain anything by lining them up. It just makes it more confusing to figure out the position of the decimal point in the answer.

Another way to look at it, is to think of a decimal number as a fraction. 3.14 is the same as 314/100. So when you multiply, say, 3.14 * 2.2, that's the same as (314/100) * (22/10). When you multiply fractions you can multiply the numerators, over the denominators: (314*22)/(100*10). If you write out the integer numerators to multiply them, it's natural to right-align them so that there's a ones column and a tens column, etc. Putting the decimal points in is just a way to keep track of how many digits total were behind the decimal point originally.

To think of it yet a third way, consider the problem 3.14+2.2. Clearly this is not the same as 31.4+.22. But 3.14*2.2 isthe same as 31.4*.22. When multiplying, you can move the decimal points around and get the same result, provided that when you move it forward in one number, you move it backward in the other number. That makes it meaningless which digit lines up with which other digit when you write it out on two lines.