1. Integers, Rational Numbers, and Properties of Real Numbers

2. Integers

3. Definition Integers are the set of real numbers that consists of counting numbers, their inverses, and the number zero. 012345-2-3-4-5

4. Absolute Values Given an integer n, the absolute value of n, represented by |n|, is the distance between n and 0 on the number line. Examples: o |3| = 3 o |-5| = 5 0 123 -2 -3 |3| 0123 4 5-2-3 -4 -5 |-5|

5. Operations on Integers Adding integers: o Positive + Positive: Add, and you’re done. Example: 3 + 2 = 5 o Negative + Negative: Add, then prefix a negative sign to your sum. Example: -2 + (-1) = -3 012345-2-3-4-5 3 2 01 2 3 -2 -3 -2

6. Operations on Integers Adding integers: o Positive + Negative: Subtract, then prefix the sign of the number with the larger absolute value. Example 1: 3 + (-2) = 1 (Because |3| > |-2|) Example 2: -3 + 2 = -1 (Because |-3| > |2|) 0 12 -2 -2-3 3 3 0123-2-3 2

7. Operations on Integers Subtracting Integers: Change the sign of the minuend, and then proceed with integer addition. Examples: o 3 – 2 = 3 + (-2) = 1 o 3 – (-5) = 3 + 5 = 8 o -5 – 2 = -5 + (-2) = -7 o -5 – (-6) = -5 + 6 = 1

8. Operations on Integers Multiplying and Dividing Integers: o When the two numbers have the same sign, just do the required operation and you’re done. 6 * 2 = 12 (-6) / (-3) = 2 o When the two numbers have opposite signs, do the required operation, then prefix a negative sign to the result. 5 * (-4) = -20 (-30) / 6 = -5

9. Operations on Integers Raising an integer to an exponent: o Positive Integers: Just do the operation and you’re done. 2 3 = 8 5 2 = 25 o Negative Integers: If the exponent is even, just do the operation. If the exponent is odd, do the operation, then prefix a negative sign to the result. (-3) 2 = 9 (-5) 3 = -125

10. Word Problems on Integers Height Problems Age Problems Consecutive Integer Problems Digit Problems

11. Problem 1: The highest elevation in North America is Mt. McKinley, which is 20,320 feet above sea level. The lowest elevation is Death Valley, which is 282 feet below sea level. What is the distance from the top of Mt. McKinley to the bottom of Death Valley? (Taken From: http://www.mathgoodies.com/lessons/vol5/intro_integers.html)

12. Problem 1 Solution: We know that: Height of Mt. McKinley = 20,320 feet above sea level = 20,320 feet Height of Death Valley = 282 feet below sea level = -282 feet To get the distance from the bottom of Death Valley to the top of Mt. McKinley, we just have to add their absolute values together: |20,320ft| + |-282ft| = 20,320ft + 282ft = 20,602ft

13. Problem 2: In January of the year 2000, I was one more than eleven times as old as my son William. In January of 2009, I was seven more than three times as old as him. How old was my son in January of 2000? (Taken From: http://purplemath.com/modules/ageprobs.htm)

14. Problem 2 Solution: In January of the year 2000, I was one more than eleven times as old as my son William. In January of 2009, I was seven more than three times as old as him. How old was my son in January of 2000? We are faced with this given: Years elapsed = 2009 – 2000 = 9 January 2000January 2009 Williamxx + 9 Father11x + 1(11x + 1) + 9 = 11x + 10

15. Problem 2 Solution: In January of the year 2000, I was one more than eleven times as old as my son William. In January of 2009, I was seven more than three times as old as him. How old was my son in January of 2000? Also, in 2009, the father’s age is seven more than three times as old as William. 11x + 10 = 3(x + 9) + 7 January 2000January 2009 Williamxx + 9 Father11x + 1(11x + 1) + 9 = 11x + 10

16. Problem 2 Solution: 11x + 10 = 3(x + 9) + 7 11x + 10 = 3x + 27 + 7 11x + 10 = 3x + 34 11x – 3x = 34 – 10 8x = 24 x = 3 (William’s age in January 2000) Final answer: William was 3 years old in January 2000.

17. Problem 3: The sum of three consecutive integers is 36. What are those integers?

18. Problem 3 Solution: We can write this problem as: x + (x + 1) + (x + 2) = 36 3x + 3 = 36 3x = 36 – 3 3x = 33 x = 11 x + 1 = 12 x + 2 = 13

19. Problem 4: The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number. (Taken from: http://www.purplemath.com/modules/systprob.htm)

20. Problem 4 Solution: The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number. First Condition: The sum of the digits of a two-digit number is 7. Equation 1: x + y = 7  x = 7 - y Second Condition: When the digits are reversed, the number is increased by 27 Equation 2: 10y + x = (10x + y) + 27  10y – y + x – 10x = 27  9y – 9x = 27  9(y – x) = 27  y – x = 3

21. Problem 4 Solution: The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number. y – (7 – y) = 3 y – 7 + y = 3 y + y = 3 + 7 2y = 10 y = 5 x = 7 – y = 7 – 5 = 2 Answer: The number is 25.

22. Rational Numbers

23. A number can be called rational if it can be written as a quotient of two integers. Examples: 2 = 2 / 1 0.5 = 1 / 2 0.3333… = 1 / 3

24. Irrational Numbers Any number that does not satisfy the said condition is called irrational. Two Types: o Irrational Constants (Examples: Pi and e) o Radicals that cannot be written as a whole number (Examples: √2 and 3 √10)

25. Properties of Irrational Numbers n √xy = n √x * n √y n √(x / y) = n √x / n √y ( n √x) n = x n √( m √x) = nm √x a n √x (+ or -) b n √x = [a (+ or -) b] n √x

26. Simplifying Irrational Numbers Simplify √12.

27. Simplifying Irrational Numbers Simplify √12. √12 = √(4 * 3) = √4 * √3 = 2√3

28. Rationalizing Irrational Numbers Simplify 3 / √7.

29. Rationalizing Irrational Numbers Simplify 3 / √7. (3 / √7) * (√7 / √7) 3√7 / 7

30. Rationalizing Irrational Numbers Simplify (√2 + √8) / (√5 – 8).

31. Rationalizing Irrational Numbers Simplify (√2 + √8) / (√5 – 8). (√2 + √(4 * 2)) / (√5 – 8) (√2 + √4 * √2) / (√5 – 8) (√2 + 2√2) / (√5 – 8) (3√2) / (√5 – 8) [(3√2) * (√5 + 8)] / [(√5 – 8) * (√5 + 8)] (3√(2 * 5) + (3√2 * 8)) / (5 – 64) (3√10 + 24√2) / (-59)

32. Properties of Real Numbers

33. (All info after this slide are courtesy of: http://www.math.com/school/subject2/lessons/S2U2L1DP.html)

34. Properties of Real Numbers 1) Commutative Property Addition: 5a + 4 = 4 + 5a Multiplication: 3 x 8 x 5b = 5b x 3 x 8

35. Properties of Real Numbers 2) Associative Property Addition: (4x + 2x) + 7x = 4x + (2x + 7x) Multiplication: 2x 2 (3y) = 3y(2x 2 )

36. Properties of Real Numbers 3) Distributive Property 2x(5 + y) = 10x + 2xy 4) Density property - It simply states that given two real numbers, we can find infinitely other real numbers that lie between them.

37. Properties of Real Numbers 5) Identity property Addition: 5y + 0 = 5y Multiplication: 2c × 1 = 2c

38. Imaginary Numbers An imaginary number is any number that is not a real number. Famous example: i = √-1

39. Imaginary Numbers We notice a pattern when we raise i to an exponent n: i 1 = i i 2 = -1 i 3 = -i i 4 = 1 And so on…

40. Imaginary Numbers In general, the value of the imaginary number i raised to an exponent n, where n is an integer and n >= 0, is given by: i n = i (n mod 4) Note: n mod 4 means that you should take the remainder when n is divided by 4.

41. Problem 1: Evaluate i 8.

42. Problem 1 Solution: Evaluate i 8. 8 mod 4 = 0 Thus, i 0 = i 4 = i 8 = 1.

43. Problem 2: Evaluate: (i + 2) 2.

44. Problem 2 Solution: Evaluate: (i + 2) 2. (i + 2) 2 = i 2 + 2i + 4 (-1) + 2i + 4 2i + 3

45. Sources: