Slide 5 - 1Copyright © 2009 Pearson Education, Inc. Slide 5 - 1Copyright © 2009 Pearson Education, Inc.

Welcome to MM150!

Unit 1 SeminarLouis Kaskowitz

lkaskowitz@kaplan.eduAIM: xEqualsPi

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MM150 Unit 1 Seminar Agenda

• Welcome and Syllabus Review

• Number Theory

• The Real Number System

• Operations with Real Numbers

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• 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

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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

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• 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

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Kaplan University Math Center (KUMC)

• Math Modules• Q&A: kumc@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. ETTuesday: 11 a.m. – 12 a.m. ET

Monday:11 a.m. – 5 p.m. ET 8 p.m. – 12 a.m. ET

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• 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

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• 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

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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.

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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.

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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.

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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.

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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.

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Example of branching method

Therefore, the prime factorization of

140 = 2 • 2 • 5 • 7.

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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?

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Distributive Property

• Distributive property of multiplication over additiona * (b + c) = a * b + a * cfor any real numbers a, b, and c.

• Example: 6 * (r + 12) = 6 * r + 6 * 12 = 6r + 72

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Exponents

• When a number is written with an exponent, there are two parts to the expression:

baseexponent

• The exponent tells how many times the base should be multiplied together.

45 44444

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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 1013 and 4.82 x 10–10 are two examples

of numbers in scientific notation.

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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.)

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Example

• Write each number in scientific notation.a) 1,265,000,000.

1.265 x 109

b) 0.0000000004324.32 x 1010

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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,

16 = 4 since 44 =16

49 = 7 since 77 = 49

n

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Radicals

• are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand.

2, 17, 53

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Product Rule for Radicals

• Simplify:a)

b)

ab a b, a 0, b 0

40 410 4 10 2 10 2 10

125 255 25 5 5 5 5 5

40

125

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Example: Adding or Subtracting Irrational Numbers

• Simplify: • Simplify: 4 7 3 7

4 7 3 7

(4 3) 7

7 7

8 5 125

8 5 125

8 5 25 5

8 5 5 5

(8 5) 5

3 5

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Fractions

• Fractions are numbers such as:

• The numerator is the number above the fraction line.

• The denominator is the number below the fraction line.

1

3,

2

9, and

9

53.

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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:

72

81

72 72 9 8

81 81 9 9

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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.

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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

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Example

• Convert to an improper fraction.

5

7

10

(10 5 7)

10

50 7

10

57

10

5

7

10

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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 0.4,

and 0.2323232323... which may be written 0.23.