5.2 Decimal operations  (Page 3/7)

In the next example, we’ll need to add several placeholder zeros to properly place the decimal point .

Multiply: $(0.03)\text{(}0.045\text{).}$

Solution

	$(0.03)(0.045)$
The product is positive.
Write in vertical format, lining up the numbers on the right.
Multiply.
Add zeros as needed to get the 5 places.
The product is positive.	$(0.03)(0.045)=0.00135$

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Multiply by powers of $10$

In many fields, especially in the sciences, it is common to multiply decimal s by powers of $10.$ Let’s see what happens when we multiply $1.9436$ by some powers of $10.$

Look at the results without the final zeros. Do you notice a pattern?

$\begin{array}{ccc}1.9436(10)\hfill & =& 19.436\hfill \\ 1.9436(100)\hfill & =& 194.36\hfill \\ 1.9436(1000)\hfill & =& 1943.6\hfill \end{array}$

The number of places that the decimal point moved is the same as the number of zeros in the power of ten. [link] summarizes the results.

We can use this pattern as a shortcut to multiply by powers of ten instead of multiplying using the vertical format. We can count the zeros in the power of $10$ and then move the decimal point that same of places to the right.

So, for example, to multiply $45.86$ by $100,$ move the decimal point $2$ places to the right.

Sometimes when we need to move the decimal point, there are not enough decimal places. In that case, we use zeros as placeholders. For example, let’s multiply $2.4$ by $100.$ We need to move the decimal point $2$ places to the right. Since there is only one digit to the right of the decimal point, we must write a $0$ in the hundredths place.

Multiply $5.63$ by factors of ⓐ $10$ ⓑ $100$ ⓒ $1000.$

Solution

By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.

ⓐ
	$56.3\left(10\right)$
There is 1 zero in 10, so move the decimal point 1 place to the right.
	$56.3$

ⓑ
	$5.63\left(100\right)$
There are 2 zeros in 100, so move the decimal point 2 places to the right.
	$563$

ⓒ
	$5.63\left(1000\right)$
There are 3 zeros in 1000, so move the decimal point 3 places to the right.
A zero must be added at the end.	$\mathrm{5,630}$

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Divide decimals

Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.

To understand decimal division, let’s consider the multiplication problem

Remember, a multiplication problem can be rephrased as a division problem. So we can write

We can think of this as “If we divide 8 tenths into four groups, how many are in each group?” [link] shows that there are four groups of two-tenths in eight-tenths. So $0.8÷4=0.2.$

Using long division notation, we would write