Monday, February 15, 2016

Decimals

I. Intro
Decimals are numbers that are written with a decimal point. The number to the left of a decimal point is a whole number (or integer) and the number to the right is a fractional number (number between 0 and 1). This fractional part uses base 10 place value.

Hence, writing numbers with decimals (decimal notation) is another way of writing fractions and mixed numbers.

Reading Decimal Numbers
There are two ways to read a decimal number. 

1. Digit-by-Digit 
The first way is to simply read the whole number followed by "point", then to read the digits in the fractional part separately. It is a more casual way but more common way to read decimals. For example, we read 85.64 as eighty-five point six-four. 

2. Place Value Reading
The second way is to read the whole number part followed by "and", then to read the fractional part in the same way as we read whole numbers but followed by the place value of the last digit. For example, we can also read 85.64 as eighty-five and sixty-four hundredths. 

This method can be confusing so let's work through it.
12.1 - Twelve and one tenth
12.12 - Twelve and twelve hundredths
12.123 - Twelve and one hundred twenty three thousandths
12.1234 - Twelve and one thousand two hundred thirty four ten thousandths
12.12345 - Twelve and twelve thousand three hundred forty five hundred thousandths
12.123456 - Twelve and one hundred twenty-three thousand four hundred fifty-six millionths


II. Conceptual Models
Number Line
We can use the number line to help conceptually make sense of decimals. Here we are asked to find 0.6 on the number line. To do so, we divide 0 to 1 into ten parts. Next, we move from 0 to the right counting six places till we get to 0.6.


Area Model
Decimals can also be graphically represented with the use of an area model. The whole block is divided into 100 parts with the 35 blue colored blocks representing the number. 

III. Place Value
Place value with decimal numbers can be thought of as an expansion of the place value explanation used with whole numbers. Numbers are written using digits 0 through 9 with the position of the digit in the number determining the value of the digit. Each place has a value 10 times the place to its right. For example, the number 364.314 in expanded form is:

3 x 100 + 6 x 10 + 4 x 1 + 3 x 1/10 + 1 x 1/100 + 4 x 1/1,000

or

3 Hundreds + 6 Tens + 4 Ones + 3 Tenths + 1 Hundredths + 4 Thousandths

Here is an image showing place values of both whole and decimal number parts.


Lets focus on the digits to the right of the decimal. Each of these place units refer to a whole divided into equal parts. So the tenths place is a whole divided into ten equal parts, the hundredths is a whole divided into a hundred equal parts, the thousandths is a whole divided into a thousand equal parts, and so on.

Here are examples of equivalent forms of decimals and fractions for the place value units to the right of the decimal point in the chart above:

Tenths: 0.1 = 1/10
Hundredths: 0.01 = 1/100
Thousandths: 0.001 = 1/1,000
Ten Thousandths: 0.0001 = 1/10,000
Hundred Thousandths: 0.00001 = 1/100,000
Millionths: 0.000001 = 1/1,000,000

10 Times Relationship
The following graphic helps drive home the point that the relationship between any two adjacent places is that the place on the left is worth ten times as much as the place on the right, and the place on the right is worth one tenth as much as the place on the left:
Face Value x Place Value
Another important idea in our base ten system that extends to decimal numbers is that the quantity represented by a digit is the product of its face value and its place value. The face value is the value of the digit without regard to its position. In the number 73.6, the quantity represented by the 7, for example, is its face value 7 multiplied by its place value 10, which is 7 x 10 = 70. The face value 3 is multiplied by it's place value 1, which is 3 x 1 = 3  The face value of the 6 is multiplied by it's place value of 0.1 (or 1/10), which is 6 x 0.1 = 0.6. 

The result is 7 x 10 + 3 x 1 + 6 x 0.1 = 73.6

Regrouping With Decimals

IV. Types of Decimals
Decimals can be divided into different categories.

Terminating decimals: Terminating decimals mean it does not reoccur and end after a finite number of decimal places. For example: 543.534234

Non-terminating decimals: It means that the decimal numbers have infinite digits after the decimal point. For example, 54543.23774632439473747... 
The non-terminating decimal numbers can be further divided into 2 types:

1) Recurring (Repeating) decimal numbers: In recurring decimal numbers, digits repeat after a fixed interval. For example, 94346.374374374...

2) Non-recurring decimal numbers: In non-recurring decimal numbers, digits never repeat after a fixed interval. For example 743.872367346...

V. Conversions 
A. Converting Terminating Decimals to Fractions
Place the number to the right of the decimal point in the numerator. Next, place the number 1 in the denominator and then add as many zeroes as the numerator has digits to the right of the decimal. Reduce if necessary

Example:

5.025 = 5 .025/1 = 5 25/1000 = 5 1/40

B. Converting Repeating Decimals to Fractions
See:
Cliffnotes: Changing Infinite Repeating Decimals to Fractions
Note: Non-recurring, non-terminating decimals can not be converted to a fraction.

C. Converting Fractions to Decimals
Convert fractions to decimals by dividing the numerator by the denominator.

Example:
                                            .5 
1/2 = 2) 1    = 2) 1.0   = 2) 1.0
                                           1.0
                                           0 0

D. Converting Decimals to Percentages & Converting Percentages to Decimals
See Percentages


Practice:
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