1 / 19

The Real Numbers

The Real Numbers. 1.1. Sets. A set is a collection of objects, symbols, or numbers called elements. is a set containing the first three counting numbers. 1, 2, and 3 are elements of the set. Example 1. Example 2. is a set containing the the vowel letters in English language.

katen
Download Presentation

The Real Numbers

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Real Numbers 1.1 Sets A set is a collection of objects, symbols, or numbers called elements. is a set containing the first three counting numbers. 1, 2, and 3 are elements of the set. Example 1 Example 2 is a set containing the the vowel letters in English language Question: What are the elements of this set? Answer: The elements are: a, e, i, o, and u. Let D = { x / x is a day of the week }. What are the elements of D ? Class Exercise

  2. indicate that an object is an element of the set. 3 Set A = { 1, 2, 3, 4 }. i) a B ii) 2 B iii) c B Using the symbols Any object or symbol that is contained in a set is called an element, or a member, of the set. The symbol is used to Example 1 Example 2 January D = { x / x is a day of the week }. Class Excercise Complete each statement with the symbols If B = { 1, 3, 5, a, c }

  3. Equal Sets Two sets are equal if they contain the same elements. Example 1 Let A = { a, b, c, d } and B = { a, d, b, c }. Since A and B have the same elements, then they are equal. We Write A = B Ordered Sets If the elements of a set can be ordered and we wish to indicate that the set continues as described, we use an ellipses, three dots that mean “ and so on”. The set { a, b, c, …, z } represents the entire alphabet Example 1 The set { 1, 2, 3, … , 100 } represents the counting numbers from 1 till 100. Example 2

  4. More Examples Example 3 The set{ 1, 3, 5, … } represents the positive odd numbers Example 4 The set{ 2, 4, 6, … } represents the positive Even numbers Finite or Infinite Sets A set that has a specific number of elements is said to befinite, otherwise, it is infinite. Example 1 The set A = { 1, 2, 3} is finite. Example 2 The set B = { 1, 2, 3,…, 10} is finite. Example 3 The set C= {2, 3,4,…} is infinite.

  5. More Examples Example 4 The Set N= { 1, 2, 3, … } = Set of Natural numbers and it is infinite Example 5 The Set W= { 0, 1, 2, 3, … } = Set of Whole numbers and it is infinite Example 6 The Set Z= { …,-3, -2, -1,0, 1, 2, 3, … } = Set of Integer numbers and it is infinite

  6. v) -1 N vi) -1 W vii) -1 Z viii) 7 Z ix) Z x) W Note : =3.141828…. i) 0 N ii) 0 Z iii) 0 W iv) 1 Z Important Notes Every element in N is in W, and every element in W is in Z. Class Exercise Complete each statement with the symbols

  7. Venn Diagram 1 Set of Integers Z = {…, -3, -2, -1, 0, 1, 2, 3,… } Set of Whole Numbers W = {0,1,2,3,…} Set of Natural Numbers N = { 1,2,3,…}

  8. Numbers as ½ 0.34 5 1.333… 5.2323… -1.5 ¾ -3 Rational Numbers are considered as Rational Numbers Because we cannot list the rational numbers in any meaningful fashion, we define the elements of that set as:

  9. iv) -1 Q Examples of Rational numbers i) 1/2 Q ii) 0 Q iii) 0.34 Q iv) 1.333 Q

  10. Set of Rational Numbers Q Venn Diagram 2 Set of Integers Z = {…, -3, -2, -1, 0, 1, 2, 3,… } Set of Whole Numbers W = {0,1,2,3,…} Set of Natural Numbers N = { 1,2,3,…}

  11. Important Notes About Rational Numbers The numbers 3.5 3.111… 2.6565… 3.141828…. Are decimal numbers Not all Decimal numbers are rational numbers 3.5 Q 3.111… Q 2.6565…… Q Repeating Decimals Terminating Decimal 3.141828… Q

  12. More Notes about Decimal Numbers Repeating Decimal is 1 Repeating Decimal is 65 3.141828…. No Repeating Decimal

  13. Class Participation About Rational Numbers…. Class Exercise Complete the following table with Yes or No

  14. Class Exercise More Class Practice From the set List the elements in N,Z,Q How about the elements

  15. Q` Q Q` Irrational Numbers Q = Set of Rational Numbers • If a number is not rational, then it is irrational Q` = Set of Irrational Numbers Example 1 Q Class Exercise Check whether these numbers are rational Q, or Irrational Q`

  16. Real Numbers The set of real numbers is the union of the sets of rational numbers and irrational numbers. Real Numbers R Irrational Numbers Q` ( Not Q ) Rational Numbers Q All Numbers in N, Z,Q, and Q` are real numbers.

  17. Real Line (Numbered Line ) Numbered Line ( Real Line ) Class Exercise On the number line provided, graph the points named by each set

  18. -7/3 =-2.333 … 5/4 1/2

  19. Home Work Assignment Do all the home work exercises in the syllabus

More Related