1. Welcome to our first seminar! We’ll begin shortly

2. The Real number system: Definitions Set: A collection of numbers (elements) represented by enclosing them in brackets Example: { 1, 3, 9, 29} Natural numbers: counting numbers {1,2,3……….} Prime numbers: numbers that are divisible by themselves and 1 only. Examples: 3,5,7,11,13,17,23 etc Composite numbers: Natural numbers that are not prime Whole numbers: Natural numbers plus zero, {0,1,2,3….} Integers: All whole negative and positive numbers and zero, {…. -3,-2,-1,0,1,2,3…..}

3. More definitions Rational numbers: numbers that can be represented by a/b and a and b are integers, b≠0; Example 0/2 = 0, 3/7 Irrational numbers: decimals that do not terminate or repeat. Example π = 3.141592654…… These can be approximated using a calculator. Real numbers: Any number that can be represented on a number line, includes all rational and irrational numbers.

4. Prime numbers Definition: A prime number is any whole number greater than 2 that can only be divided by itself and 1. Other numbers have other factors. I like the factor tree method of factoring. The goal is to keep dividing the number by other numbers until each of them is prime at the bottom. I’ll do an example on the next page.

5. Example: find the factors of 108 108 2 54 2 27 3 9 3 3

6. The greatest common divisor: the largest natural number that divides into each member of a set. To find the GCF: l. Find the prime factorization of each member of the set. 2. List all of the factors giving each the smallest exponent. 3. Multiply all of those numbers in the list.

7. Example Find the GCF for 96 and 108

8. The least common multiple (or denominator: the smallest number that can be divided into each member of a set (often fraction denominators) To find the LCM (LCD) 1. Find the prime factorization of numbers in the set. 2. List all factors giving the largest exponent. 3. Multiply that list of numbers.

9. Example Find the LCD of 96 and 108

10. Signed numbers The easiest way to visualize a signed number is to look at a number line:

11. Think of it like this. If you have a – sign go left and a + sign go right Subtracting Adding Anything to the left is LESS than things to the right For example: -6 is less than – 3 because it is to the left of it.

12. Rules about signed numbers 1.When adding numbers of the same sign, add and give them that sign. Example: 2+4 = +6 -2 + (-4) = -6 To add positive numbers move right to 2 and then 4 more places to 6. To add negative numbers move left to -2 and then 4 more places to -6.

13. 2. To subtract numbers, subtract the smaller one from the larger one and give the answer the sign of the larger number. Example: 2 – 5Subtract: 5 – 2 = 35 is larger and negative so the answer is negative: 2 – 5 = -3 On the number line go right 2 places to +2 and then left 5 places to -3

14. 3.When multiplying or dividing signed numbers if there are an even number of negatives the answer is positive and if there are an odd number of negatives the answer is negative. Examples: 3(-5) 1 number (odd) is negative: the answer is negative: = -15 -2(-9)2 negatives (even): the answer is positive =18 (2)(-4)(-2)(-1) 3 negatives (odd); the answer is negative: =-16 NOTE the importance of parenthesis

17. Orders of operations 1.Parenthesis (innermost out) 2. powers and roots 3. multiply and divide 4.add and subtraction Some people call this PEMDAS (Please excuse my dear aunt sally) Or in a humorous alternative (Please embalm my dead Aunt Sally) Parenthesis,Exponents,Multiply,Divide,Add, Subtract

21. Fractions Book definition: A fraction is any number that can be put in the form a/b where a and b are numbers and b is not equal to zero. the numerator (top term) the denominator (bottom term) We’ll talk about what a fraction means and some of the ways to work with them.

22. Properties 1.To find equivalent fractions (those that represent the same thing) you can use: If a, b, and c are numbers and b and c are NOT zero then: For example:

23. Reducing a fraction to lowest terms: 1. Find the GCM 2. Divide the top and bottom by the GCM Example:

24. Changing improper fractions to mixed numbers 1.Divide the numerator by the denominator 2.Write the quotient as the whole number part and the remainder as the new numerator.

25. To change a decimal to a fraction 1.Write out the name. 2.Put the original numbers on the top of the fraction. 3.Put the place name on the bottom of the fraction

26. To add or subtract fractions: 1. Find the LCD 2. Form equivalent fractions. 3. Add /subtract Example:

27. Irrational numbers: a number which is a non- terminating, non-repeating decimal. These are radicals that have no root.

28. To simplify as square root: 1. Write as any squares 2. Take the things that are squared out as itself

29. The distributive property: a(b + c) = ab + ac Examples:

30. Rules of exponents:

31. Examples

32. Zero exponents Anything to the zero power is ‘1’.

33. Negative exponents A negative exponent means that you move the stuff raised to that exponent to the other side of the fraction bar. For example:

34. Example

35. Thank you for attending!