1. Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. Welcome to MM150! Unit 1 Seminar Louis Kaskowitz lkaskowitz@kaplan.edu AIM: xEqualsPi To resize your pods: Place your mouse here. Left mouse click and hold. Drag to the right to enlarge the pod. To maximize chat, minimize roster by clicking here

2. Slide 5 - 2 Copyright © 2009 Pearson Education, Inc. MM150 Unit 1 Seminar Agenda Welcome and Syllabus Review Number Theory The Real Number System Operations with Real Numbers

3. Slide 5 - 3 Copyright © 2009 Pearson Education, Inc. Arrive on time and stay until the end. Seminars are brisk: Be prepared by previewing the topics. Have book, pencil, and paper. You will seldom master a unit’s worth of work in only one seminar. It takes practice, practice, practice, … and patience. All seminars are archived, and the Power Points * are posted in DocSharing after the seminar time. There are several “flex” seminar options for you. Be courteous and encouraging to others. Be positive! Seminar Info

4. Slide 5 - 4 Copyright © 2009 Pearson Education, Inc. Flex seminar cohort for 1101B: Thursday 10 PM ET (this class), Louis Kaskowitz Wednesday 11 AM ET, Kimberly White Sunday 8 PM ET, Mark Johnston Flex Seminars

5. Slide 5 - 5 Copyright © 2009 Pearson Education, Inc. Syllabus can be downloaded from Doc Sharing or found under Course Home: Syllabus. Please read it! (I know it is long) Key things include grading rubrics, attendance requirements, due dates and late policies, info about your Project assignment, and doing your own work. Syllabus

6. Slide 5 - 6 Copyright © 2009 Pearson Education, Inc. Kaplan University Math Center (KUMC) Math Modules Q&A: kumc@kaplan.edukumc@kaplan.edu Math Workshops Live Tutoring Log in to the Kaplan homepage, click “My Studies,” then “Academic Support Center.” Sunday, Wednesday, Thursday: 8 p.m. – 12 a.m. ET Tuesday: 11 a.m. – 12 a.m. ET Monday: 11 a.m. – 5 p.m. ET 8 p.m. – 12 a.m. ET

7. Slide 5 - 7 Copyright © 2009 Pearson Education, Inc. Slide 5 - 7 Copyright © 2009 Pearson Education, Inc. An introduction to number theory Prime numbers Integers, rational numbers, irrational numbers, and real numbers Properties of real numbers Rules of exponents and scientific notation Chapter 1 Number Theory and the Real Number System

8. Slide 5 - 8 Copyright © 2009 Pearson Education, Inc. The study of numbers and their properties. The numbers we use to count are called natural numbers or counting numbers. The Natural Numbers = {1, 2, 3, 4, 5,…} Number Theory

9. Slide 5 - 9 Copyright © 2009 Pearson Education, Inc. Integers The set of integers consists of 0, the natural numbers, and the negative natural numbers. Integers = {…–4, –3, –2, –1, 0, 1, 2, 3 4,…} On a number line, the positive numbers extend to the right from zero; the negative numbers extend to the left from zero.

10. Slide 5 - 10 Copyright © 2009 Pearson Education, Inc. Factors The natural numbers that are multiplied together to equal another natural number are called factors of the product. Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

11. Slide 5 - 11 Copyright © 2009 Pearson Education, Inc. Prime and Composite Numbers A prime number is a natural number greater than 1 that has exactly two factors, itself and 1. A composite number is a natural number that is divisible by a number other than itself and 1. The number 1 is neither prime nor composite, it is called a unit.

12. Slide 5 - 12 Copyright © 2009 Pearson Education, Inc. The Fundamental Theorem of Arithmetic A prime number is a natural number greater than 1 that has exactly two factors, itself and 1. A composite number is a natural number that is divisible by a number other than itself and 1. Every composite number can be expressed as a unique product of prime numbers. This unique product is referred to as the prime factorization of the number.

13. Slide 5 - 13 Copyright © 2009 Pearson Education, Inc. Finding Prime Factorizations Branching Method: – Select any two numbers whose product is the number to be factored. – If the factors are not prime numbers, continue factoring each number until all numbers are prime.

14. Slide 5 - 14 Copyright © 2009 Pearson Education, Inc. Example of branching method Therefore, the prime factorization of 140 = 2 2 5 7.

15. Slide 5 - 15 Copyright © 2009 Pearson Education, Inc. Commutative Property Addition a + b = b + a Multiplication a * b = b * a 8 + 12 = 12 + 8 is a true statement. 5 * 9 = 9 * 5 is a true statement. Note: The commutative property does not hold true for subtraction or division. Is putting on your socks commutative? How about putting on shoes and socks?

16. Slide 5 - 16 Copyright © 2009 Pearson Education, Inc. Distributive Property Distributive property of multiplication over addition a * (b + c) = a * b + a * c for any real numbers a, b, and c. Example: 6 * (r + 12) = 6 * r + 6 * 12 = 6r + 72

17. Slide 5 - 17 Copyright © 2009 Pearson Education, Inc. Exponents When a number is written with an exponent, there are two parts to the expression: base exponent The exponent tells how many times the base should be multiplied together.

18. Slide 5 - 18 Copyright © 2009 Pearson Education, Inc. Scientific Notation Many scientific problems deal with very large or very small numbers. 93,000,000,000,000 is a very large number. 0.000000000482 is a very small number. Scientific notation is a shorthand method used to write these numbers. 9.3 x 10 13 and 4.82 x 10 –10 are two examples of numbers in scientific notation.

19. Slide 5 - 19 Copyright © 2009 Pearson Education, Inc. To Write a Number in Scientific Notation 1.Move the decimal point in the original number to the right or left until you obtain a number greater than or equal to 1 and less than 10. 2. Count the number of places you have moved the decimal point to obtain the number in step 1. If the decimal point was moved to the left, the count is to be considered positive. If the decimal point was moved to the right, the count is to be considered negative. 3.Multiply the number obtained in step 1 by 10 raised to the count found in step 2. (The count found in step 2 is the exponent on the base 10.)

20. Slide 5 - 20 Copyright © 2009 Pearson Education, Inc. Example Write each number in scientific notation. a)1,265,000,000. 1.265 x 10 9 b)0.000000000432 4.32 x 10  10

21. Slide 5 - 21 Copyright © 2009 Pearson Education, Inc. Principal Square Root The principal (or positive) square root of a number n, written is the positive number that when multiplied by itself, gives n. For example,

22. Slide 5 - 22 Copyright © 2009 Pearson Education, Inc. Radicals are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand.

23. Slide 5 - 23 Copyright © 2009 Pearson Education, Inc. Product Rule for Radicals Simplify: a) b)

24. Slide 5 - 24 Copyright © 2009 Pearson Education, Inc. Example: Adding or Subtracting Irrational Numbers Simplify:

25. Slide 5 - 25 Copyright © 2009 Pearson Education, Inc. Fractions Fractions are numbers such as: The numerator is the number above the fraction line. The denominator is the number below the fraction line.

26. Slide 5 - 26 Copyright © 2009 Pearson Education, Inc. Reducing Fractions In order to reduce a fraction to its lowest terms, we divide both the numerator and denominator by the greatest common divisor. Example: Reduce to its lowest terms. Solution:

27. Slide 5 - 27 Copyright © 2009 Pearson Education, Inc. Mixed Numbers & Improper Fractions Rational numbers greater than 1 (or less than –1) that are not integers may be written as mixed numbers, or as improper fractions. A mixed number consists of an integer and a fraction. For example, 3 ½ is a mixed number. 3 ½ means “3 + ½”. An improper fraction is a fraction whose numerator is greater than its denominator. An example of an improper fraction is 7/2.

28. Slide 5 - 28 Copyright © 2009 Pearson Education, Inc. Converting a Positive Mixed Number to an Improper Fraction Multiply the denominator of the fraction by the whole part. Add the product obtained in step 1 to the numerator of the fraction. This will be the numerator of the improper fraction. The denominator of the improper fraction is the same as the denominator of the fraction in the mixed number. 3 ½ = 7/2

29. Slide 5 - 29 Copyright © 2009 Pearson Education, Inc. Example Convert to an improper fraction.

30. Slide 5 - 30 Copyright © 2009 Pearson Education, Inc. Terminating or Repeating Decimal Numbers Every rational number when expressed as a decimal number will be either a terminating or a repeating decimal number. Examples of terminating decimal numbers are 0.7, 2.85, 0.000045 Examples of repeating decimal numbers 0.44444… which may be written