5.1+Decimals+and+Fractions

Section 5.1 - Decimals and Fractions
Decimals are related to fractions and mixed numbers. Although we will talk about converting between decimals and fractions later in the chapter, it helps to understand the structure of decimal numbers by comparing them to mixed numbers.

In the example of the mixed number above, the 2 represents the whole number, while the fraction part is less than 1. In the decimal number example, anything to the left of the decimal point is the whole number, while anything to the right of it represents less than 1.

If a mixed number is negative, the minus sign will be in front of the whole number part. The same goes for a decimal number -- if it is negative, the minus sign will be in front of the number and will affect all of the decimal number.


 * Place Value**

Decimal numbers are usually easier to do math with than fractions because they rely on multiples of ten for place value rather than whatever the denominator of a fraction happens to be. Think of a pizza cut into 16 pieces. If you were counting pieces, you would be counting by 16ths. It takes longer to add, subtract, multiply, and divide fractions because you are forced to find common denominators, or multiply by "odd" factors such as 16ths. However, decimal numbers always use powers of 10, so it is almost as easy to do math with them as it is with whole numbers.

Just as with whole numbers, the places after a decimal have names to match their value, and they almost exactly mirror the whole number place values.



There is no such thing as "oneths"! The first place value following the decimal point is tenths. If you look at the pattern from left to right, each place value is the same as the one before it, divided by 10. That pattern continues following the decimal point. Tenths are ones divided by 10. Hundredths are tenths divided by 10.


 * Which Zeroes Matter?**

Which number is bigger: 456 or 00,000,456? They are the same. The zeroes in front of the 456 do not affect the value, just the way the number looks. However, if there was a digit to the left of the zeroes, they would be important as place holders (100,000,456).

Zeroes to the left of whole numbers do not matter. Zeroes to the right of a decimal number do not matter. We call them insignificant. They do not change the value of the number, but might be helpful in doing some math. 6.73 has the same value as 6.73000000

But, if there was another digit at the end, the zeroes would be important as place holders in order to have the correct place value: 6.730000004


 * From Decimal to Fraction or Mixed Number**

It is easy to translate decimal numbers into fractions or mixed numbers, as long as you pay attention to the place value of the decimal part. (The real question is why you would want to.)

The easiest is if there is a single non-zero digit in the decimal part. In that case, the digit becomes the numerator of the fraction part, and the place value is the denominator.



If the decimal part contains a multiple-digit part, the key digit is the right-most one. This is the one that specifies the place value for the fraction (denominator).



Yikes! You would need to reduce the fraction to lowest terms, but I left them this way to show you how they translate.

Hopefully you can see that it would also be easy to translate from fractions to decimals -- as long as the denominator is a multiple of 10. If it isn't, it takes a bit more work, and will be covered in a later section.




 * Terminating, Repeating, Other**

A repeating decimal is one in which a pattern of digits repeats itself, forever, meaning the decimal part would never really terminate, or end. The ellipses (dots) at the end of the number indicate that it continues on forever in that direction. A repeating bar is used above a digit, or set of digits, to indicate that it repeats. The digits, or pattern of digits below the repeating bar are repeated forever.



A terminating decimal does not go on forever. Examples would be 5.8809 or 0.43 Terminating and repeating decimals are part of a set of numbers called "Rational Numbers," but rational is better defined as an integer divided by a non-zero integer.

It is possible to have a decimal number that is not terminating and also not repeating. These numbers belong to a special set of numbers called "irrational". An example might be: 5.3041294003. . . It is possible for that number, or part of it, to repeat later on, but it is not apparent that it does, so it is called irrational.The most important irrational number is **Pi**, which is usually shortened to **3.14**, though in reality the decimal goes on forever in no discernible pattern.


 * Rounding Decimals**

Round decimal numbers the same way you round whole numbers, except that you can get rid of insignificant zeroes.



If you need to round a decimal number in order to estimate, use front-end rounding, which means rounding to the first __non-zero__ digit.

If you are asked to round to the nearest penny, that means the hundredths place (a penny is 1 hundredth of a dollar).


 * Comparing Decimal Numbers**

With whole numbers, it is easy to see which number is larger by counting the digits -- the number with more digits is larger. If they have the same number of digits, you start at the left side and compare digits until you decide which one is larger.

The number of digits following a decimal point does not have anything to do with the value of the number and cannot be used to compare two numbers. You must compare individual digits, starting at the left, where the place value is greater. If you always start at the decimal point, you will always be comparing the same place value for decimal numbers.



As you look from left to right after the decimal point, you see the digits in green are the same. But when you get to the digits in red, they are different and you can see that the bottom number is greater than the top number. It does not matter how many digits, or which digits, follow that red digit -- it will not change the fact that the bottom number is larger.

Watch out for negative numbers, because just like with whole numbers, if the number is larger, but negative, it is really less than a smaller negative number.

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 * Chapter Five Practice Problems**

**Where to from here?**
5.1 Decimals and Fractions 5.2 Adding and Subtracting Decimals 5.3 Multiplying Decimals 5.4 Dividing Decimals 5.5 Application Problems with Decimals Chapter Five Summary Chapter Five Homework