While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals.

Sterntaler baby abs krabbelsöckchen mit vollplüsch esel emmi girlIn fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers.

It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century.

This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. John Stillwell is a professor of mathematics at the University of San Francisco.

Lower-division undergraduates. Turner, Choice, Vol. There are extensive historical notes about the evolution of real analysis and our understanding of real numbers. I think this is very successful, and his book … is much more informative and enjoyable. It is intended for senior undergraduates who have already studied calculus, but a wide range of readers will find something interesting, new, or instructive in it.

It is full of interesting examples, very clear explanations, historical background, applications. Each new idea comes after proper motivation.

JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser. Mathematics Analysis. Undergraduate Texts in Mathematics Free Preview. Fills a gap in the standard curriculum by linking analysis to set theory Contains background, history, examples, and explanatory remarks Includes almost two courses for the price of one by providing a unified treatment of analysis and set theory see more benefits. Buy eBook. Buy Hardcover. Buy Softcover.

Rent the eBook. FAQ Policy. About this Textbook While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light.

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Infinite Sets Pages Stillwell, John. Functions and Limits Pages Stillwell, John. Ordinals Pages Stillwell, John. Borel Sets Pages Stillwell, John.We will use this as a spring board for the day's lesson. This lesson is a foundation topic meant to prepare students to enter the complex number system.

The real number system was addressed in prior years but I feel that it is important enough to look at again to ensure that all students will be coming into the next couple of lessons from the right place. My hook for this activity is a video from teacher tube on the history of numbers. It is about 7 minutes and provides a nice survey of the development of numbers into our current number systems.

As we build this diagram, we relate the numbers back to the historical perspectives in the video as well as the students' experiences learning numbers as they grew up. Each pair receive a set of cards with the names of each number system: natural, whole, integer, rational, irrational, and real; each of which have been printed on separate colored paper. As I pull up the numbers on the PowerPoint, the pairs will hold up the number systems that apply to that number Math Practice 2.

I will then call on a student to explain a portion of their solution. An alternate method is to make this a race where the first 3 or 5 groups get a point for holding up the correct systems. The first problem gives them several numbers to categorize within the number systems. The remainder of the problems asks the students to come up with numbers that fit into certain systems but not others Math Practice 1. Empty Layer.

## Introduction to Sets

Home Professional Learning. BetterLesson reimagines professional learning by personalizing support for educators to support student-centered learning.

See what we offer. Sign Up Log In. Algebra II Amelia Jamison. Students will be able to classify numbers in the real number system. Big Idea This lesson relates the real number system to its historical perspective. Lesson Author. Grade Level. MP3 Construct viable arguments and critique the reasoning of others.

Warm up and Homework Review 5 minutes. Introduction to Number Systems 17 minutes. Classifying Numbers Activity 20 minutes. Competition- Number Systems. Exit Ticket 3 minutes. Today's lesson asks students to determine a number that is a rational number but not an integer. Student Assignment- Number Systems. Previous Lesson. Next Lesson. Related Lessons. The Origins of Imaginary Numbers. Roots of Polynomial Functions - Day 2 of 2.In this section we will explore sets of numbers, perform calculations with different kinds of numbers, and begin to learn about the use of numbers in algebraic expressions.

The numbers we use for counting, or enumerating items, are the natural numbers : 1, 2, 3, 4, 5, and so on. The natural numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers.

In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers. Notice from the definition that rational numbers are fractions or quotients containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:. We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

At some point in the ancient past, someone discovered that not all numbers are rational numbers. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers.

Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown. Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

Simplify and divide. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number. Try It. Given any number nwe know that n is either rational or irrational.

It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers.

Zero is considered neither positive nor negative. The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer or other basic value on either side of 0. Any real number corresponds to a unique position on the number line.

The converse is also true: each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence.

We refer to this as the real number line. Classify each number as either positive or negative and as either rational or irrational.

Does the number lie to the left or the right of 0 on the number line? Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far.First we specify a common property among "things" we define this word later and then we gather up all the "things" that have this common property. There is a fairly simple notation for sets.

We simply list each element or "member" separated by a comma, and then put some curly brackets around the whole thing:. The three dots OK, there isn't really an infinite amount of things you could wear, but I'm not entirely sure about that! After an hour of thinking of different things, I'm still not sure. So let's just say it is infinite for this example.

So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic.

Who says we can't do so with numbers? And we can have sets of numbers that have no common property, they are just defined that way. For example:. Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are.

Math can get amazingly complicated quite fast. But there is one thing that all of these share in common: Sets. We call this the universal set. It's a set that contains everything. Well, not exactly everything. Everything that is relevant to our question.

In Number Theory the universal set is all the integersas Number Theory is simply the study of integers.Tipsters must explain their arguments for each tip, set the odds, set the stake and select the bookmaker they will place their bet on. Community members can then check and rate the quality of that tip. All these steps help you to personally decide whether a tip is reliable or not.

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**The Map of Mathematics**

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### The Real Numbers

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### The Real Number System

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