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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret.Objectives of this module: be familiar with the real number line and the real numbers, understand the ordering of the real numbers.

Overview

  • The Real Number Line
  • The Real Numbers
  • Ordering the Real Numbers

The real number line

Real number line

In our study of algebra, we will use several collections of numbers. The real number line allows us to visually display the numbers in which we are interested.

A line is composed of infinitely many points. To each point we can associate a unique number, and with each number we can associate a particular point.

Coordinate

The number associated with a point on the number line is called the coordinate of the point.

Graph

The point on a line that is associated with a particular number is called the graph of that number.

We construct the real number line as follows:

    Construction of the real number line

  1. Draw a horizontal line.

    A horizontal line with arrows on both the ends.
  2. Choose any point on the line and label it 0. This point is called the origin .

    A horizontal line with arrows on both the ends,  and a mark labeled as zero.
  3. Choose a convenient length. This length is called "1 unit." Starting at 0, mark this length off in both directions, being careful to have the lengths look like they are about the same.

    A horizontal line with arrows on both the ends, and a mark labeled as zero. There are  equidistant marks on both sides of zero.

    We now define a real number.

Real number

A real number is any number that is the coordinate of a point on the real number line.

Positive and negative real numbers

The collection of these infinitely many numbers is called the collection of real numbers . The real numbers whose graphs are to the right of 0 are called the positive real numbers . The real numbers whose graphs appear to the left of 0 are called the negative real numbers .
The real numbers having graphs on the right side of the origin are positive numbers, and those having graphs on the left side of the origin are negative numbers.

The number 0 is neither positive nor negative.

The real numbers

The collection of real numbers has many subcollections. The subcollections that are of most interest to us are listed below along with their notations and graphs.

Natural numbers

The natural numbers ( N ) :    { 1 , 2 , 3 , }

Graphs of natural numbers one to six plotted on a number line. The numberline has arrows on each sides, and is labeled from zero to six in increments of one. There are three dots after six indicating that the graph continues indefinitely.

Whole numbers

The whole numbers ( W ) :    { 0 , 1 , 2 , 3 , }

Graphs of whole numbers zero to six plotted on a number line. The number line has arrows on each side, and is labeled from zero to six in increments of one. There are three dots after six indicating that the graph continues indefinitely.

Notice that every natural number is a whole number.

Integers

The integers ( Z ) :    { , 3 , 2 , 1 , 0 , 1 , 2 , 3 , }

Graphs of integers negative five to five plotted on a number line. The number line has arrows on each side, and is labeled from negative five to five in increments of one. There are three dots after five indicating that the graph continues indefinitely.

Notice that every whole number is an integer.

Rational numbers

The rational numbers ( Q ) : Rational numbers are real numbers that can be written in the form a / b , where a and b are integers, and b 0 .

Fractions

Rational numbers are commonly called fractions.

Division by 1

Since b can equal 1, every integer is a rational number: a 1 = a .

Division by 0

Recall that 10 / 2 = 5 since 2 5 = 10 . However, if 10 / 0 = x , then 0 x = 10 . But 0 x = 0 , not 10. This suggests that no quotient exists.

Now consider 0 / 0 = x . If 0 / 0 = x , then 0 x = 0 . But this means that x could be any number, that is, 0 / 0 = 4 since 0 4 = 0 , or 0 / 0 = 28 since 0 28 = 0 . This suggests that the quotient is indeterminant.

Questions & Answers

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
hi
Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
how far
Abubakar
cool u
Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
Abegail Reply
hello
BenJay
hi
Method
I am eliacin, I need your help in maths
Rood
how can I help
Sir
hmm can we speak here?
Amoon
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
Arundhati Reply
When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
Jakoiya Reply
how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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