1. 1.2 SKM & PP 1 Types of Numbers There are many “types” of numbers. Each type can be grouped into a collection called a SET.

2. 1.2 SKM & PP 2 Sets In general, any collection of objects is called a SET. A set can be defined in several ways:  English: A description in words  Set Builder: A mathematical rule  Roster: A list of the objects or numbers inside braces

3. 1.2 SKM & PP 3 Sets: Example 1 Consider the set of even numbers: 0,2,4,6,…  English: “The Even Numbers”  Set Builder: {x| x is divisible by 2}  Roster: {0, 2, 4, 6, 8, … }

4. 1.2 SKM & PP 4 Sets: Example 2 Consider the set of digits: 0,1,2,3,4,5,6,7,8,9  English: “Digits”  Set Builder: {x| x is a digit}  Roster: {0,1,2,3,4,5,6,7,8,9}

5. 1.2 SKM & PP 5 C The Number Line We use a Number Line to graph sets of Real Numbers. Zero is in the center. Positive numbers are on the right. Negative numbers are on the left.

6. 1.2 SKM & PP 6 The Natural Numbers Natural numbers are usually the first set that we learn. They are also called Counting numbers. {1, 2, 3, 4, 5, …}

7. 1.2 SKM & PP 7 The Natural Numbers Here are the Natural numbers graphed on the number line: {1, 2, 3, 4, 5, …} …

8. 1.2 SKM & PP 8 The Whole Numbers The set of Whole numbers is the set of Natural numbers along with zero. {0, 1, 2, 3, 4, 5, …} …

9. 1.2 SKM & PP 9 The Opposite Each Natural number to the right of zero has an Opposite to the left of zero. -1 and 1 are Opposites. -2 and 2 are Opposites. -3 and 3 are Opposites. and so on...

10. 1.2 SKM & PP 10 Opposite Numbers Opposite numbers are the same distance from zero, but they are on opposite sides of zero. -a and a are opposites.

11. 1.2 SKM & PP 11 What about Zero? Two numbers are opposite if their sum is zero. -1 + 1 = 0 Zero is it’s own opposite. -2 + 2 = 0 -3 + 3 = 0 Since 0 + 0 = 0

12. 1.2 SKM & PP 12 The Integers The Integers are the Whole numbers together with their Opposites. {…,-3,-2,-1,0,1,2,3,…} … …

13. 1.2 SKM & PP 13 The Rational Numbers The set of Rational Numbers consists of all quotients of Integers with non-zero denominators. a and b are integers,

14. 1.2 SKM & PP 14 Convert: Rational Number to Decimal To convert a Rational Number into Decimal form, divide the numerator by the denominator. A Rational number can always be converted to a Terminating Decimal or a Repeating Decimal.

15. 1.2 SKM & PP 15 Conversion Example 1 Terminating Decimal

16. 1.2 SKM & PP 16 Conversion Example 2 Repeating Decimal Ellipsis show the 3 repeats.

17. 1.2 SKM & PP 17 Conversion Example 3 Terminating Decimal

18. 1.2 SKM & PP 18 Conversion Example 4 Repeating Decimal Ellipsis show the 571428 repeats.

19. 1.2 SKM & PP 19 Conversion Example 5 Terminating Decimal

20. 1.2 SKM & PP 20 Conversion Example 6 The denominator can never equal zero!

21. 1.2 SKM & PP 21 Conversion Example 7 Terminating Decimal

22. 1.2 SKM & PP 22 Conversion Example 4 Repeating Decimal Ellipsis show the 6 repeats.

23. 1.2 SKM & PP 23 What about Negatives? The negative sign can be in front of the ratio or in the numerator or in the denominator. Usually, it is best to place it in the front.

24. 1.2 SKM & PP 24 What about Negatives? Example 1 “Negative three-fourths”

25. 1.2 SKM & PP 25 What about Negatives? Example 2 “Negative two and one-half”

26. 1.2 SKM & PP 26 Irrational Numbers Any Real number that is not a rational number is called Irrational. Irrational numbers cannot be written as the ratio of integers. The decimal approximation for an irrational number will not terminate or repeat.

27. 1.2 SKM & PP 27 Irrational Numbers Here are a few examples of numbers that are Irrational.   3.14159… e  2.71828…  1.41421…  3.6055512…

28. 1.2 SKM & PP 28 The REAL Numbers REAL NUMBERS The set of numbers that correspond to points on the number line. The REAL NUMBERS include the following: Natural, Whole, Integers, Rational, and Irrational

29. 1.2 SKM & PP 29 REAL NUMBERS Rational Numbers: a/b with b  0 Integers: …-2,-1,0,1,2,… Whole Numbers: 0,1,2,3,… Natural Numbers: 1,2,3,… A Map of the Number Sets Irrationals: pi,e,  3,…

30. 1.2 SKM & PP 30 Order: Small to Large The Real Numbers are named on the number line from small to large. If we choose any two numbers on the number line, the number on the left is smaller and the number on the right is larger.

31. 1.2 SKM & PP 31 Order: Small to Large The Real Numbers are named on the number line from small to large. If we choose any two numbers on the number line, the number on the left is smaller and the number on the right is larger.

32. 1.2 SKM & PP 32 An Example: “Negative three is less than one” -3 < 1 “One is greater than negative three” 1 > -3

33. 1.2 SKM & PP 33 > or < How do these numbers compare? -52< 11-13> 06< -50<

34. 1.2 SKM & PP 34 Absolute Value The ABSOLUTE VALUE of a number, |x|, is its distance from zero on the number line. |-5|= 5 |5|= 5

35. 1.2 SKM & PP 35 |x| Examples |-9|= 9 |20|= 20 |0|= 0 -|-9| = -1  |-9|= -1  9= -9

36. 1.2 SKM & PP 36 That’s All for Now!