Introduction to Decimals Study Guide (page 2)
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
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