Introduction to Decimals Study Guide (page 4)
Introduction to Introduction to Decimals
The highest form of pure thought is in mathematics.
—PLATO, classical Greek philosopher (427 B.C.E.–347 B.C.E.)
The first decimal lesson is an introduction to the concept of decimals. It explains the relationship between decimals and fractions, teaches you how to compare decimals, and gives you a tool called rounding for estimating decimals.
A decimal is a special kind of fraction. You use decimals every day when you deal with measurements or money. For instance, $10.35 is a decimal that represents 10 dollars and 35 cents. The decimal point separates the dollars from the cents. Because there are 100 cents in one dollar, 1¢ is of a dollar, or $0.01; 10¢ is of a dollar, or $0.10; 25¢ is of a dollar, or $0.25; and so forth. In terms of measurements, a weather report might indicate that 2.7 inches of rain fell in 4 hours, you might drive 5.8 miles to the intersection of the highway, or the population of the United States might be estimated to grow to 374.3 million people by a certain year.
If there are digits on both sides of the decimal point, like 6.17, the number is called a mixed decimal; its value is always greater than 1. In fact, the value of 6.17 is a bit more than 6. If there are digits only to the right of the decimal point, like .17, the number is called a decimal; its value is always less than 1. Sometimes these decimals are written with a zero in front of the decimal point, like 0.17, to make the number easier to read. A whole number, like 6, is understood to have a decimal point at its right (6.).
Each decimal digit to the right of the decimal point has a special name. Here are the first four:
The digits have these names for a very special reason: The names reflect their fraction equivalents.
0.1 = 1 tenth =
0.02 = 2 hundredths =
0.003 = 3 thousandths =
0.0004 = 4 ten thousandths =
As you can see, decimal names are ordered by multiples of 10: 10ths, 100ths, 1,000ths, 10,000ths, 100,000ths, 1,000,000ths, etc. Be careful not to confuse decimal names with whole number names, which are very similar (tens, hundreds, thousands, etc.). The naming difference can be seen in the ths, which are used only for decimal digits.
Reading a Decimal
Here's how to read a mixed decimal, for example, 6.017:
|1.||The number to the left of the decimal point is a whole number. Just read that number as you normally would:||6|
|2.||Say the word "and" for the decimal point:||and|
|3.||The number to the right of the decimal point is the decimal value. Just read it:||17|
|4.||The number of places to the right of the decimal point tells you the decimal's name. In this case, there are three places:||thousandths|
Thus, 6.017 is read as six and seventeen thousandths, and its fraction equivalent is .
Here's how to read a decimal, for example, 0.28:
|1.||Read the number to the right of the decimal point:||28|
|2.||The number of places to the right of the decimal point tells you the decimal's name. In this case, there are two places:||hundredths|
Thus, 0.28 (or .28) is read as twenty-eight hundredths, and its fraction equivalent is . You could also read 0.28 as point two eight, but it doesn't quite have the same intellectual impact as 28 hundredths!
Adding zeroes to the end of the decimal does NOT change its value. For example, 6.017 has the same value as each of these decimals:
6.01700000, and so forth
Remembering that a whole number is assumed to have a decimal point at its right, the whole number 6 has the same value as each of these:
6.000, and so forth
On the other hand, adding zeroes before the first decimal digit does change its value. That is, 617 is NOT the same as 6.017.
Decimals are all around us! They are used in money, measurement, and time, so it's important to read this section carefully and make sure you feel comfortable with them. Using decimals is essential in mastering practical, real-world math skills.
Changing Decimals to Fractions and Fractions to Decimals
Changing Decimals to Fractions
To change a decimal to a fraction:
- Write the digits of the decimal as the top number of a fraction.
- Write the decimal's name as the bottom number of the fraction.
Example: Change 0.018 to a fraction.
|1.||Write 18 as the top of the fraction:||18|
|2.||Since there are three places to the right of the decimal, it's thousandths.|
|3.||Write 1,000 as the bottom number:|
|4.||Reduce by dividing 2 into the top and bottom numbers:|
Changing Fractions to Decimals
To change a fraction to a decimal:
- Set up a long division problem to divide the bottom number (the divisor) into the top number(the dividend)—but don't divide yet!
- Put a decimal point and a few zeros on the right of the divisor.
- Bring the decimal point straight up into the area for the answer (the quotient).
Example: Change to a decimal.
|1.||Set up the division problem:|
|2.||Add a decimal point and 2 zeroes to the divisor (3):|
|3.||Bring the decimal point up into the answer:|
Thus, = 0.75, or 75 hundredths
Some fractions may require you to add more than two or three decimal zeros in order for the division to come out evenly. In fact, when you change a fraction like to a decimal, you'll keep adding decimal zeros until you're blue in the face because the division will never come out evenly! As you divide 3 into 2, you'll keep getting 6s:
A fraction like becomes a repeating decimal. Its decimal value can be written as or , or it can be approximated as 0.66, 0.666, 0.6666, and so forth. Its value can also be approximated by rounding it to 0.67 or 0.667 or 0.6667, and so forth. (Rounding is covered later in this lesson.)
If you really have fractionphobia and panic when you have to do fraction arithmetic, just convert each fraction to a decimal and do the arithmetic in decimals. Warning: This should be a means of last resort—fractions are so much a part of daily living that it's important to be able to work with them.
Decimals are easy to compare when they have the same number of digits after the decimal point. Tack zeros onto the end of the shorter decimals—this doesn't change their value—and compare the numbers as if the decimal points weren't there.
Example: Compare 0.08 and 0.1. (Don't be tempted into thinking 0.08 is larger than 0.1 just because the whole number 8 is larger than the whole number 1!)
- Since 0.08 has two decimal digits, tack one zero onto the end of 0.1, making it 0.10
- To compare 0.10 to 0.08, just compare 10 to 8. Ten is larger than 8, so 0.1 is larger than 0.08
Rounding a decimal is a means of estimating its value using fewer digits. To find an answer more quickly, especially if you don't need an exact answer, you can round each decimal to the nearest whole number before doing the arithmetic. For example, you could use rounding to approximate the sum of 3.456789 and 16.738532:
Since 3.456789 is closer to 3 than it is to 4, it can be rounded down to 3, the nearest whole number. Similarly,16.738532 is closer to 17 than it is to 16, so it can be rounded up to 17, the nearest whole number.
Rounding may also be used to simplify a single figure, like the answer to some arithmetic operation. For example, if your investment yielded $14,837,812.98 (wishful thinking!), you could simplify it as approximately $15 million, rounding it to the nearest million dollars.
Rounding is a good way to do a reasonableness check on the answer to a decimal arithmetic problem: Estimate the answer to a decimal arithmetic problem and compare it to the actual answer to be sure it's in the ballpark.
Rounding to the Nearest Whole Number
To round a decimal to the nearest whole number, look at the decimal digit to the right of the whole number, the tenths digit, and follow these guidelines:
- If the digit is less than 5,round down by dropping the decimal point and all the decimal digits. The whole number portion remains the same.
- If the digit is 5 or more, round up to the next larger whole number.
Examples of rounding to the nearest whole number:
- 25.3999 rounds down to 25 because 3 is less than 5.
- 23.5 rounds up to 24 because the tenths digit is 5.
- 2.613 rounds up to 3 because 6 is greater than 5.
Rounding to the Nearest Tenth
Decimals can be rounded to the nearest tenth in a similar fashion. Look at the digit to its right, the hundredths digit, and follow these guidelines:
- If the digit is less than 5, round down by dropping that digit and all the decimal digits following it.
- If the digit is 5 or more, round up by making the tenths digit one greater and dropping all the digits to its right.
Examples of rounding to the nearest tenth:
- 45.32 rounds down to 45.3 because 2 is less than 5.
- 33.15 rounds up to 33.2 because the hundredths digit is 5.
- $14,837,812 rounds down to $14.8 million, the nearest tenth of a million dollars, because 3 is less than 5.
- 2.96 rounds up to 3.0 because 6 is greater than 5. Notice that you cannot simply make the tenths digit, 9, one greater—that would make it 10. Therefore, the 9 becomes a zero and the whole number becomes one greater.
Similarly, decimals can be rounded to the nearest hundredth, thousandth, and so forth by looking at the next decimal digit to the right:
- If it's less than 5, round down.
- If it's 5 or more, round up.
Remember, when rounding to the nearest tenth, you are actually looking at the hundredth place to see if you need to round down or up. It works the same way with other rounding: when rounding to the nearest hundredth, you must look at the thousandth place to determine the rounding. When rounding to the nearest thousandth, you must look at the ten-thousandth place.
As you pay for things throughout the day, take a look at the prices. Are they written in dollars and cents? If so, how would you read the numbers aloud using the terms discussed in this lesson? For a bit of a challenge, insert a zero in the tenths column of the number, thereby pushing the two numbers right of the decimal place one place to the right. Now how would you say the amount out loud? Learning how to correctly express decimals verbally will show others how math-savvy you are!
Introduction to Decimals Sample Questions
- Change the mixed decimal 2.7 to a fraction.
- Change to a decimal
- Put these decimals in order from least to greatest: 0.1, 0.11, 0.101, and 0.011.
Solutions to Sample Questions
- Write 2 as the whole number: 2
- Write 7 as the top of the fraction:
- Since there is only one digit to the right of the decimal, it's tenths.
- Write 10 as the bottom number:
- Set up the division problem: 5√1
- Add a decimal point and a zero to the divisor (1): 5√1.0
- Bring the decimal point up into the answer:
Thus, , or 2 tenths.
|1.||Since 0.0111 has the greatest number of decimal places (4), tack zeroes onto the ends of the other decimals so they all have 4 decimal places:||0.1000, 0.1100, 0.1010, 0.0111|
|2.||Ignore the decimal points and compare the whole numbers:||1,000; 1,100; 1,010; 111|
|3.||The low-to-high sequence of the whole numbers is:||111; 1,000; 1,010; 1,100|
|Thus, the low-to-high sequence of the original decimals is:||0.0111; 0.1, 0.101, 0.11|
Find practice problems and solutions for these concepts at Introduction to Decimals Practice Questions.
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