Decimal Multiplication

Multiplying with decimals is almost exactly like multiplying whole numbers -- you just have to pay attention to the decimal point.  And, when you multiply numbers with decimals, after you multiply, you count the number of digits behind the decimal point in each factor, add those numbers together, and move the decimal point in the answer that many digits to the left. 

The process is very simple -- the reasons behind it are pretty simple too, although we don't always think through the process while we're doing it:

When you multiply numbers together, they grow... and the larger the numbers you multiply, the more they grow.  When you multiply factors of ten, it's easy to see how place value is involved in multiplication:

$100 \times 100 = 10000$

$1000 \times 1000000= 10000000000$

When you multiply factors of 10, your answer will contain the same number of zeros the zeros in your factors added together. 

The process is easy to see when you multiply factors of ten, but in truth, whenever you multiply whole numbers with different place values, the number of place values in your answer grows:

$345 \times 10 = 3450$

$345\times 12 = 4140$

When you multiply by 12, you can't just add a zero to the end of your number, but because you multiply by something with a tens place, your answer is a place value bigger than your first factor.  Likewise, if you multiply by somethihng with a hundreds place, your answer will be two place values bigger than your first factor:

$12 \times 542=6504$

See? The two-digit number 12 becomes a 4-digit number when multiplied by something with a hundreds place. 

The reverse happens when you multiply with decimals. Think about it: when you multiply by a whole number, your answer is bigger than the original factors.  When you multiply by a decimal (or fraction) the answer is smaller than the first factor. If you multiply two decimals together, the answer is smaller than either of the decimals!

How much smaller?

When, just as you can predict that answers will grow by the place value of the number you multiply it by, numbers also get smaller by the number of place values of the decimal you multiply it by.

So, the rule of multiplying with decimals is that you figure out the place value of the decimals in all factors and you reduce the place value of your answer by that amount -- how do you do it?  You do it by moving the decimal point to the left. 

Example:

$4.5 \times 3.12=$

Because there is a commutative property of multiplication, it doesn't matter what order you multiply the numbers in.  When multiplying multi-digit numbers it is usually most efficient to put the longer number (the number with more digits) on top.  Because you will deal with the place value after you complete the multiplication, you do not have to line up the decimals when you multiply with decimals. 

$$\eqalign{3.12&\\\times4.5&\\\hline\text{ }}$$

Then you multiply, following all of the rules of multiplying multi-digit numbers:

$$\eqalign{3.\!^112&\\\times4.5&\\\hline160}$$

$$\eqalign{3.12&\\\times4.5&\\\hline160&\\1248\emptyset&\\\hline\text{ }}$$

$$\eqalign{3.12&\\\times4.5&\\\hline160&\\1248\emptyset&\\\hline1404}$$

Once you have completed the multiplication problem, you just have to place the decimal point.Count the number of digits behind the decimal point in each of the factors:

$$\left \lbrace \eqalign{3.\underline{1}\underline{2}&\\\times4.\underline{5}&\\\hline160&\\1248\emptyset&\\\hline1404&}\right\rbrace\text{There are 3 digits behind the decimal point in the factors}$$

So, place the decimal in the answer, that many digits from the right.

$$\eqalign{3.12&\\\times4.5&\\\hline160&\\1248\emptyset&\\\hline1.\underset{\mathbf{\hookleftarrow}}{4}\underset{\mathbf{\hookleftarrow}}{0}\underset{\mathbf{\hookleftarrow}}{4}&}$$