<< Chapter < Page Chapter >> Page >
Algebraic and positivity probabilities of real numbers. Ordered fields, isomorphism, bounded below, bounded above, supremum, infimum, and complete are defined.

What are the real numbers? From a geometric point of view (and a historical one as well) real numbers are quantities, i.e., lengths of segments, areas of surfaces,volumes of solids, etc. For example, once we have settled on a unit of length,i.e., a segment whose length we call 1, we can, using a compass and straightedge, construct segments of any rational length k / n . In some obvious sense then, the rational numbers are real numbers. Apparently it was an intellectual shock tothe Pythagoreans to discover that there are some other real numbers, the so-called irrational ones.Indeed, the square root of 2 is a real number, since we can construct a segment the square of whose length is 2by making a right triangle each of whose legs has length 1.(By the Pythagorean Theorem of plane geometry, the square of the hypotenuse of this triangle must equal 2.) And, Pythagoras proved that there is no rational numberwhose square is 2, thereby establishing that there are real numbers tha are not rational. See part (c) of [link] .

Similarly, the area of a circle of radius 1 should be a real number; i.e., π should be a real number. It wasn't until the late 1800's that Hermite showed that π is not a rational number. One difficulty is that to define π as the area of a circle of radius 1we must first define what is meant by the “ area" of a circle, and this turns out to be no easy task.In fact, this naive, geometric approach to the definition of the real numbers turns out to be unsatisfactory in the sense that we are not able to prove or derive from thesefirst principles certain intuitively obvious arithmetic results. For instance, how can we multiply or divide an area by a volume?How can we construct a segment of length the cube root of 2? And, what about negative numbers?

Let us begin by presenting two properties we expect any set that we call the real numbers ought to possess.

Algebraic properties

We should be able to add, multiply, divide, etc., real numbers. In short, we require the set of real numbers to be a field.

Positivity properties

The second aspect of any set we think of as the real numbers is that it has some notion of direction, some notion of positivity.It is this aspect that will allow us to “compare” numbers, e.g., one number is larger than another. The mathematically precise way to discuss this notion is the following.

A field F is called an ordered field if there exists a subset P F that satisfies the following two properties:

  1. If x , y P , then x + y and x y are in P .
  2. If x F , then one and only one of the following three statements is true.
    1. x P ,
    2. - x P , and
    3. x = 0 . (This property is known as the law of tricotomy .)

The elements of the set P are called positive elements of F , and the elements x for which - x belong to P are called negative elements of F .

As a consequence of these properties of P , we may introduce in F a notion of order.

If F is an ordered field, and x and y are elements of F , we say that x < y if y - x P . We say that x y if either x < y or x = y .

We say that x > y if y < x , and x y if y x .

Questions & Answers

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
what is the meaning
Dominic
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Analysis of functions of a single variable' conversation and receive update notifications?

Ask