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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Muhammad Reply
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Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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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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