Theory behind multiplying decimals

Question

When multiplying two decimal numbers, you first ignore the decimals, find the product, then count the number of decimal places that need to be in the answer by taking the sum of the original decimal places. Why exactly does this work?

Answer 1

$(10^m x)\cdot (10^n y)= 10^{m+n}(x\cdot y)$.

In words, if $x$ has $m$ decimals then $10^m x$ has none. Ditto for $y$. To get $x\cdot y$ you need to take the product of the decimal-free numbers and put the decimal point after $m+n$ digits, counted from the right. This is the same as dividing by $10^{m+n}$.

Answer 2

Suppose we want to multiply $1.2$ with $0.5$. Your method would work this way:

1. "Annihilate" the decimal point of each number, getting $12$ and $5$
2. Multiply those numbers, getting $12\times 5=60$
3. Add many decimal places to this result as the sum of decimal places of each original number, since each one has one decimal position, we have as final result $0.60$.

The reason why this works, is the following.

You can write $1.2$ as $12\times 10^{-1}$ and $0.5$ as $5\times 10^{-1}$ (in general, the number $n\times 10^{-k}$ is obtained by displacing the decimal point of $n$ by $k$ positions to the left), thus, using properties of exponentiation:

$$1.2\times 0.5 = (12\times 10^{-1})\times (5\times 10^{-1})=(12\times 5)\times(10^{-1}\times10^{-1})=60\times 10^{-2}=0.60$$

So note where the sum of the decimal positions appear ($-1-1=-2$).

Can you see how can this be generalized?

Answer 3

Let me run you through an example and hopefully that will clear things up.

Lets do $3.141 *6.28$.

Let's first start by "moving" the decimal place over, and noticing how far over we move it: $$ 3.14 = \frac{31.41}{10}=\frac{314.1}{10^2}=\frac{3141}{10^3}$$ $$ 6.28=\frac{62.8}{10}=\frac{628}{10^2}$$

Now, if we multiply these two numbers, we get: $$ 3.141*6.28=\frac{3141}{10^3}\cdot \frac{628}{10^2}$$

By multiplying the top and bottom, we can see that we get: $$\frac{3141}{10^3}\cdot \frac{628}{10^2} = \frac{3141*628}{10^{2+3}}$$

So from this, we can see that we are multiplying the two original numbers, and then dividing this by $10^{2+3}$ which is the same thing as moving the decimal place backwards twice and then three times, and notice that 2 and 3 are the amount of numbers after the decimal point in $6.28$ and $3.141$. Hopefully this clears things up a bit.