## Operations With Decimals and Powers of Ten: Overview

In mathematics the digits 0 to 9 and place-value position are used to represent numbers. The place a digit occupies in a written number gives its value in the number. For example, the 2 in 245 means “two hundred,” while the 2 in 524 means “twenty.”

Students are expected to read, write, and understand large numbers to 100 billion and small numbers to millionths. This is a review of material learned in Grade 5, except for the inclusion of the decimal places ten thousandths, hundred thousandths, and millionths. See the place-value chart below for a visual representation of these whole-number and decimal positions.

The large whole number in the chart above—one billion, six hundred seventy-nine million, nine hundred thirty-five thousand, five hundred—is the number of quarters, placed end to end, it would take to circle Earth at its equator.

The small decimal number in the chart represents the time in seconds it takes for a computer chip to send a signal. Many computer operations are measured in millionths of a second. The number 0.000049 means forty-nine millionths.

You can compare whole or decimal numbers by lining them up and comparing the place-value positions. For example, suppose two library books have the call numbers 811.542 and 811.526. To determine which book comes first on the shelf, compare the numbers by first lining up the decimal points. Compare the digits from left to right until you find digits in the same place that are not equal. The decimal with the smaller digit is the smaller number and so will come first on the shelf.

811.542

811.526

The tenths-place digits in each number, 4 and 2, are not equal. Since 2 < 4, the book with the call number 811.526 comes first.

When you round a number to a particular place-value position, you are really finding which of two numbers is closer to the original number. To round 92.8 to the nearest whole number, think *“Is 92.8 closer to 92 or 93?”*

The number line shows it is closer to 93. You can round any decimal to any place-value position. For example, to round 0.001347 to the nearest ten thousandth, find the digit in the ten thousandths place (in this case, the 3). The 3 remains unchanged if the digit to the right is 0, 1, 2, 3, or 4. The 3 rounds to 4 (increases by 1) if the digit to the right is 5, 6, 7, 8, or 9. Since the digit to the right of the 3 is 4, the 3 remains unchanged. So, 0.001347 rounded to the nearest ten-thousandth is 0.0013.

When 10 is multiplied by itself several times, you can use an **exponent** to make the notation simpler. For example, you can write 10 x 10 x 10 x 10 as 10^{4}. The number 10^{4} is read "ten to the fourth power" and is equal to 10,000. The exponent 4 tells us how many times 10, the **base,** is used as a factor.

The powers of ten are displayed in the table below.

For the numbers greater than 1, the exponents are positive and correspond to the number of zeros in the standard form of the number. For the numbers less than 1, the exponents are negative and tell us the number of decimal places in the standard form.

Adding and subtracting with decimals is similar to adding and subtracting with whole numbers. Numbers must be aligned according to their place value.

Remember to line up the decimal points when you add or subtract decimals. You can use zeros as placeholders.

Students multiplied and divided whole numbers and decimals in Grade 5. You may need to remind them that, when they multiply decimals, the sum of the number of decimal places in the factors should equal the number of decimal places in the product. Below are some examples involving multiplication with decimals.

If a music CD is on sale for $11.98, 3 CDs would cost $35.94.

You can check this multiplication by using repeated addition.

Ginger root costs $1.99 per pound. If a piece of ginger root weighs 0.39 pounds, you can determine its cost by multiplying.

Sometimes when you multiply two decimals there aren't enough digits in the result to place the decimal point. In such cases you can add as many zeros as needed on the left to place the decimal point.

Since the factors have a total of six decimal places, you must write the product with six decimal places: 0.423 x 0.003 = 0.001269.

Dividing decimals is similar to dividing whole numbers, which students learned in Grade 5. The following example illustrates how to divide a decimal by a whole number.

Kevin skates six times around the lake on a paved bike path. If he skates a total of 14.4 km, what is the distance around the lake?

Divide, disregarding the decimal point.

Place the decimal point in the quotient directly above the decimal point in the dividend.

When you divide a decimal by another decimal, there is one more step to follow. Change the divisor to a whole number by multiplying both the divisor and dividend by a power of 10. Here is an example.

Jose has exactly $4.83 in his pocket. If containers of yogurt cost $0.69 each, how many can Jose buy?

Multiply 0.69 and 4.83 by 100.

You can place zeros after the decimal point in the dividend without changing the value of the decimal.

In this example, the quotient has no remainder after placing two zeros in the dividend.

There are many problem-solving situations that require operations with decimals. In some situations, only an estimated answer is needed.

Three baseballs cost $4.79 each. Is $15 enough to buy 3 baseballs?

The total cost is less than $15, so $15 is enough.

At other times we need an exact answer. How much change will you receive if we pay for the baseballs with a $20 bill? Assume there is no sales tax.

You will receive $5.63 in change.