Welcome to Survey of

Mathematics!Unit 1: Number Theory and the Real Number System

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

Click on “Fit to Screen” below

Agenda

• Course Introduction:– Seminar Rules– Syllabus Notes

• 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

• To gain my attention, please “raise your hand” by using the symbol:

//• Please respect classmates’ right to speak

by not using this session as a “chat session” – It is still a class. Thanks!

• When I pose a question to everyone that I’d like you to answer, I will precede it by saying “ALL:”

Here are some Seminar Ground Rules

Notes concerning the Syllabus

• Grades based on…– Discussion (9 at 30 points each) 270 points

Be careful to ensure you use your best writing skills here; if you need help, you should get help from the writing center.

– Seminar (9 at 5 points each) 45 pointsYou can either attend one of the three flex seminar sessions (Weds. 8 PM, Thurs. 11AM or Thurs. 10 PM) or complete the seminar quiz by the Tuesday following the seminar

– MML Graded Assignment (9 at 60 points) 540 pointsThis option allows you to see examples, retry problems of the same type and contact the instructor.

– Final Project (1 at 186 points) 145 points– Total: 1000 points

1.1

Number Theory

Number Theory

• The study of numbers and their properties.

• The numbers we use to count are called natural numbers, N , or counting numbers.N = {1, 2, 3, 4, 5,

…}

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.

Prime and Composite Numbers

• A prime number is a natural number greater than 1 that has exactly two factors (or divisors), 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.

The Fundamental Theorem of Arithmetic

• 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.

• Find the Prime Factorization using the 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.

Example of branching method

Therefore, the prime factorization of

3190 = 2 • 5 • 11 • 29.

3190

319 10

11 29 5 2

1.2

The Integers

Whole Numbers

• The set of whole numbers contains the set of natural numbers and the number 0.

• Whole numbers = {0,1,2,3,4,…}

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.

1.3The Rational Numbers

The Rational Numbers

• The set of rational numbers, denoted by Q, is the set of all numbers of the form p/q, where p and q are integers and q 0.

• The following are examples of rational numbers:

1

3,

3

4,

7

8, 1

2

3, 2, 0,

15

7

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.

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

Improper and Mixed Fractions

Improper Fraction:

Mixed Fraction:

1.4The Irrational Numbers and the

Real Number System

Irrational Numbers

• An irrational number is a real number whose decimal representation is a non-terminating, non-repeating decimal number.

• Examples of irrational numbers:

5.12639573...

6.1011011101111...

0.525225222...

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

Fun Fact:

(pi) π = 3.14159265358979323846264338327950288419716939937510...Rounded to the first fifty digits following 3 (it keeps going!)

1.5Real Numbers and their

Properties

Real Numbers

• The set of real numbers is formed by the union of the rational and irrational numbers.

• The symbol for the set of real numbers is

Relationships Among Sets

Irrational numbers

Rational numbers

Integers

Whole numbersNatural numbers

Real numbers

An Analogy to the set of Real Numbers:

Plants

Animals

Humans

EuropeansFrenchmen

Life on Earth

1.6

Rules of Exponents and Scientific Notation

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

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 continued

• Scientific notation is a shorthand method used to write these numbers.

• 9.3 1013 and 4.82 10–10 are two examples of numbers in scientific notation.

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.)

Example

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

1.265 109

b) 0.0000000004324.32 1010

To Change a Number in Scientific Notation to

Decimal Notation1. Observe the exponent on the 10.2. a) If the exponent is positive, move the

decimal point in the number to the right the same number of places as the exponent. Adding zeros to the number might be necessary.

b) If the exponent is negative, move the decimal point in the number to the left the same number of places as the exponent. Adding zeros might be necessary.

Example

• Write each number in decimal notation.a) 4.67 105

467,000

b) 1.45 10–7

0.000000145

Wrap Up

• My office hour is 7 – 8 PM Sunday Nights and 6:30 – 7:30 PM Wednesdays (before the seminar) via AOL Instant Messenger. My AIM Name is “peeplesKCO”

• You can access notes via Doc Sharing• The topics we’ve covered:

• Natural and Prime Numbers• Rational and Irrational Numbers• The Real Number System• Properties of Real Numbers

• See you on the Discussion Board!

Don’t forget to see this

presentation in Doc Sharing!