1. MM150 Unit 1 Seminar Agenda Welcome and Syllabus Review –Brief Syllabus Review –Contact Information for Instructor –Seminar Rules –Discussion Topics –Whole Numbers, Integers, Rational, Irrational –Product, Quotient rules for Exponents, Radicals –Commutative, Associative Distributive Properties –Zero, Negative exponent, Power Rules

2. Syllabus Highlights Due Dates All learning activities for a unit are due by Tuesday 11:59 pm ET. Instructor Contact Email: kbaalman@kaplan.edu AIM name: kathrynbaalman Download AIM from http://www.aim.com/get_aim/win/other_win.adp

3. Seminar Rules, Structure Usual format –Discussion of a concept –Responses to questions I post on the concepts Posting a wrong answer will not negatively affect your participation grade. It is important that you try to participate rather than simply watch Do not interrupt if you enter seminar late Social posts (It is raining, Hi, Bye, My car broke down) are not appropriate and will not help your participation grade.

4. Replying to Instructor When I want each of you to reply to a question or solve a problem, I will say: EVERYONE: followed by my question. If no specific answer is requested, your response will be made by either typing Yes or No, followed by the Enter key, or clicking Send.

5. Sets of Numbers Natural Numbers: {1, 2, 3, 4, …} (… called an ellipsis) –Prime Numbers: {2, 3, 5, 7, 11, 13, 17 …} –Composite Numbers: Greater than 1, has divisors Whole Numbers: {0, 1, 2,3, …} Integers: {…-3, -2, -1, 0, 1, 2, 3, …} –Even Numbers: {2, 4, 6, 8, …} –Odd Numbers: {1, 3, 5, 7, …} Rational Numbers: ½, 0.5, -6 –Terminating: 0.5, 0.75 –Repeating Decimals: 1.33…, Irrational Numbers: √[2], √[3] (Non repeating, non terminating) Real Numbers: all rational and irrational numbers

6. Mathematical Operators Grouping symbols: ( ) Parentheses [ ] Brackets { } Braces

7. Mathematical Operators Symbols Used in Discussion: + Add - Subtract * Multiply / Divide ^ Raising to a power = Equation, such as x = 2 + y

8. Rules of Divisibility 285The number ends in 0 or 5. 5 844 since 44  4 The number formed by the last two digits of the number is divisible by 4. 4 846 since 8 + 4 + 6 = 18 The sum of the digits of the number is divisible by 3. 3 846The number is even.2 ExampleTestDivisible by

9. Divisibility Rules (continued) 730The number ends in 0.10 846 since 8 + 4 + 6 = 18 The sum of the digits of the number is divisible by 9. 9 3848 since 848  8 The number formed by the last three digits of the number is divisible by 8. 8 846The number is divisible by both 2 and 3. 6 ExampleTestDivisible by

10. 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.

11. Find Prime Factors, using a Factor Tree 72 2 36 2 18 2 9 3

12. YOU TRY, using a Factor Tree 48 2

13. SOLUTION, using a Factor Tree 48 2 24 2 12 2 6 2 3

14. Find Prime Factors, using division 2 | 72 2 |36 2 |18 3 |9 3 So 72 = 2*2*2*3*3 or 2 3 * 3 2

15. YOU TRY, using division 2 | 56

16. SOLUTION, using division 2 | 56 2 |28 2 |14 7 So 56 = 2*2*2*7 or 2 3 * 7

17. Finding the Greatest Common Divisor (GCD) Largest natural number that divides all of the numbers Consider this example: 12 = 2 * 2 * 3 72 = 2 * 2 * 2 * 3 * 3 GCD = 2 * 2 *3 or 2 2 * 3

18. Another Example (GCD) Find the GCD of 63 and 105. 63 = 3 2 7 105 = 3 5 7 Smallest exponent of each factor: 3 and 7 So, the GCD is 3 7 = 21.

19. Least Common Multiple (LCM) Smallest natural number divisible by each of the given numbers Find the LCM of 63 and 105. 63 = 3 2 7 105 = 3 5 7 Greatest exponent of each factor: 3 2, 5 and 7 So, the LCM is 3 2 5 7 = 315.

20. Operations with Integers Addition Rules Same sign: ADD and take that sign -3 + -5 = -8 or 2 + 4 = 6 Different sign: SUBTRACT and take the sign of the larger 3 + -5 = -2 or -7 + 9 = 2 Subtraction Rule Change the operation to addition, adding inverse: -5 - 2 = -5 + (-2) = -7

21. Whole Number – Addition Properties Addition Property of 0 0 + any number = that numberExample: 0 + 8 = 8 Commutative Property Changing order of addends gives the same sumExample: 2+3 = 3+2 Associative Property Changing grouping of addends gives the same sumExample: (2+3)+ 4 = 2+(3+4) 5 + 4 = 2 + 7

22. Sign Rules for Multiplication/Division Same sign: positive answer EXAMPLES: -3 * -5 = +15 2 * 4 = 8 Different sign: negative answer EXAMPLES: -3 * +5 = -15 2 * -4 = -8

23. Whole Number – Multiplication Properties Multiplication property of 0 0 times any number is 0 Example: 0 * 3 = 0 Multiplication property of 1 1 times any number is that number Example: 1*3 = 3 “0” in Division: IF 0 is in the numerator, the answer is 0 0/3 = 0 IF 0 is in the denominator, the answer is UNDEFINED 3/0 = undefined

24. 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 numerator is the number above the fraction line. The denominator is the number below the fraction line.

25. 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:

26. Mixed Numbers & Improper Fractions A mixed number consists of an integer and a fraction. For example, 3 ½ is a mixed number. An improper fraction is a fraction whose numerator is greater than its denominator.

27. Converting Mixed - Improper Convert to an improper fraction.

28. Converting Improper - Mixed Convert to a mixed number. The mixed number is

29. Multiplying & Dividing Fractions Evaluate the following. a) b)

30. Add or Subtract Fractions Add: Subtract:

31. Finding an LCD Evaluate: Solution:

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

33. Simplifying Radicals Simplify: a) b)

34. Adding or Subtracting Irrational Numbers Simplify:

35. MULTIPLY Radicals 3√[6] (2√[3]) GIVEN 6√[18] Multiply inside, multiply outside 6 √[2*3*3] Prime factor radicand 6*3 √[2] Remove perfect squares 18 √[2] Multiply

36. Rationalizing Denominators Rationalize the denominator of Solution:

37. Commutative Property Addition a + b = b + a 8 + 12 = 12 + 8 Multiplication a b = b a 5  9 = 9  5

38. Associative Property Addition (a + b) + c = a + (b + c), (3 + 5) + 6 = 3 + (5 + 6) Multiplication (a b) c = a (b c), (4  6)  2 = 4  (6  2)

39. 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

40. Practice Exercises