Tuesday, April 28, 2015

Whole Numbers and Place Values


Objective: Introduce the set of whole numbers and place values.

We begin by defining a set as a collection of objects, typically grouped within braces { }, where each object is called an element.  For example, a set of colors, named C, is expressed as follows:
\[C = \left\{ {{\rm{red, green, blue}}} \right\}\]
When studying mathematics, we focus on special sets of numbers.  The set of natural numbers, or counting numbers, is an infinite set that is denoted using \(\mathbb{N}\).
\[\mathbb{N} = \left\{ {1,\,2,\,3,\,4,\,5,...\} } \right.{\rm{     Natural Numbers}}\]
The three periods (…) is called the ellipsis mark and indicates that the numbers continue without bound.  We define the set of whole numbers, denoted using \(W\), to be the set of natural numbers combined with zero.
\[W = \left\{ {0,1,2,3,4,5,...\} } \right.{\rm{     Whole Numbers}}\]
Developed between the 1st and 5th centuries, the Hindu-Arabic numeral system is the most common positional numeral system. This system allows us to express any whole number using the following set of ten digits.
\[digits = \left\{ {0,1,2,3,4,5,6,7,8,9} \right\}\]
A whole number is expressed using these digits and place values according to the following illustration.
For example, the average distance from the Earth to the Moon is 238,857 miles.
A number is written in expanded form to show the value of each digit. Here is the number 238,857 written in expanded form:
\[200,000 + 30,000 + 8,000 + 800 + 50 + 7\]
Example: Write in expanded form:
a. 375              b. 61,532         c. 5,102,700
Answers:
a. \(375 = 300 + 70 + 5\)
b.\(61,532 = 60,000 + 1,000 + 500 + 30 + 2\)
c. \(5,102,700 = 5,000,000 + 100,000 + 2,000 + 700\)

Try this! Write in expanded form:
a. 8,014,050                b. 71,324                     c. 68,090,004
Answers:
a. \(8,000,000 + 10,000 + 4,000 + 50\)
b.\(70,000 + 1,000 + 300 + 20 + 4\)
c. \(60,000,000 + 8,000,000 + 90,000 + 4\)

Digits are grouped in sets of three, called periods, separated by a comma.   It is sometimes acceptable to leave off the comma. For example, the four digit number 1,561 can also be written as 1561.  From the right, each set of three digits represents the number of ones, thousands, millions, billions, and so on.
A number is read by starting with the leftmost period. With the exception of the ones period, the name of the period is added after verbalizing the corresponding number. For example, 238,857 miles is read, “two hundred thirty-eight thousand, eight hundred fifty-seven miles.” Note that the comma in the written sentence matches the comma in the number.

Example: In 2012, the world population was approximated to be 6,176,534,320 people. Write this number in words.

Solution: Begin by identifying the value of each period.  Use this value for each three digit numerical grouping.

Note: Do not use the word “and” when writing whole numbers in words. As we will see, this word is reserved for the decimal.
Answer: Six billion, one hundred seventy-six million, five hundred thirty-four thousand, three hundred twenty people.

Example: Write the following numbers using words.
a. 371                          b. 71,051                     c. 67,151,205
Answers:
a. Three hundred seventy-one.
b. Seventy-one thousand, fifty one.
c. Sixty-seven million, one hundred fifty-one thousand, two hundred five.

Try this! The speed of light is approximately 186,282 miles per second. Write this speed of light out in words.
Answer: One hundred eighty-six thousand, two hundred eighty-two miles per second.
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Thursday, April 23, 2015

Graph Lines Using Intercepts

The x-intercept is the point where the graph intersects the x-axis and the y-intercept is the point where the graph intersects the y-axis. These points have the form (x, 0) and (0, y) respectively.
x and y-intercepts
To find the x- and y-intercepts algebraically, we use the fact that all x-intercepts have a y-value of zero and all y-intercepts have an x-value of zero.  For example,
Graph:  3x − 5y = 15
Tip 1: To find the y-intercept, set x = 0 and determine the corresponding y-value.  Similarly, to find the x-intercept we set y = 0 and determine the corresponding x-value.
 
Keep in mind that the intercepts are ordered pairs and not numbers.  In other words, the x-intercept is not x = 5 but rather (5, 0).

Two points determine a line. If we find the x- and y-intercepts, then we can use them to graph the line. As you can see, they are fairly easy to find. Plot the points and draw a line through them with a straightedge.
Done. Let’s do another one.
Graph: yx + 9
We begin by finding the x-intercept.
The x-intercept is (3, 0).
The y-intercept is (0, 9). Now graph the two points.
Graph of the Line
Tip 2: Use Desmos.com to check your answer – it’s totally free.  Just type in the equation.

This is a nice and easy method for determining the two points you need for graphing a line.  In fact, we will use this exact technique for finding intercepts when we study the graphs of all the conic sections later in our study of Algebra.

Graph −4x + 3y = 12 using the intercepts.

  

Graph −4x + 2y = −6 using the intercepts.

  

Graph  y = −5x +15 using the intercepts.

   

Graph  y = −3/4 x + 9 using the intercepts.


This brings us to one of the most popular questions in linear graphing.  Do all lines have x- and y-intercepts?  The answer is NO.  Horizontal lines, of the form y = b, do not necessarily have x-intercepts.  Vertical lines, of the form x = a, do not necessarily have y-intercepts.

Graph y = 3.

  

Graph x = −2.

Many students this method, but I will tell you, there is a better way. Even less work... [ Graph Lines using Slope and Intercepts ] Read on!
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