1. Welcome to Survey of Mathematics!
Unit 1: Number Theory and the Real Number System
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2. Agenda Course Introduction: An introduction to number theory
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

3. Here are some Seminar Ground Rules
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:”

4. 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) points You 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 points This 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: points

5. 1.1
Number Theory

6. 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, …}

7. 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.

8. 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.

9. 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.

10. Example of branching method
3190
Therefore, the prime factorization of
3190 = 2 • 5 • 11 • 29.

11. 1.2
The Integers

12. Whole Numbers
The set of whole numbers contains the set of natural numbers and the number 0.
Whole numbers = {0,1,2,3,4,…}

13. 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.

14. 1.3
The Rational Numbers

15. 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:

16. Fractions Fractions are numbers such as:
The numerator is the number above the fraction line.
The denominator is the number below the fraction line.

17. 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:

18. Improper and Mixed Fractions
Improper Fraction:
Mixed Fraction:

19. The Irrational Numbers and the Real Number System
1.4
The Irrational Numbers and the Real Number System

20. Irrational Numbers
An irrational number is a real number whose decimal representation is a non-terminating, non-repeating decimal number.
Examples of irrational numbers:

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

22. Fun Fact:
(pi) π =
Rounded to the first fifty digits following 3
(it keeps going!)

23. Real Numbers and their Properties
1.5
Real Numbers and their Properties

24. 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

25. Relationships Among Sets
Irrational numbers
Rational numbers
Integers
Whole numbers
Natural numbers
Real numbers

26. An Analogy to the set of Real Numbers:
Plants
Animals
Humans
Europeans
Frenchmen
Life on Earth

27. Rules of Exponents and Scientific Notation
1.6
Rules of Exponents and Scientific Notation

28. 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.

29. Scientific Notation
Many scientific problems deal with very large or very small numbers.
93,000,000,000,000 is a very large number.
is a very small number.

30. Scientific Notation continued
Scientific notation is a shorthand method used to write these numbers.
9.3  and 4.82  10–10 are two examples of numbers in scientific notation.

31. 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 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.)

32. Example Write each number in scientific notation. a) 1,265,000,000.
1.265  109 b)
4.32  1010

33. To Change a Number in Scientific Notation to Decimal Notation
1. Observe the exponent on the
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.

34. Example Write each number in decimal notation. a) 4.67  105 467,000

35. Don’t forget to see this presentation in Doc Sharing!
Wrap Up
Don’t forget to see this presentation in Doc Sharing!
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!