1. TMAT 103 Chapter 1 Fundamental Concepts

2. TMAT 103 §1.1 The Real Number System

3. §1.1 – The Real Number System Integers –Positive, Negative, Zero Rationals Irrationals Reals –Real number line Complex Numbers Primes

4. §1.1 – The Real Number System Properties of Real Numbers (FYI) –Commutative Property of Addition –Commutative Property of Multiplication –Associative Property of Addition –Associative Property of Multiplication –Distributive Property of Multiplication over Addition –Additive inverse –Multiplicative inverse –Additive identity –Multiplicative identity

5. §1.1 – The Real Number System Signed Numbers –Absolute Value –Adding 2 signed numbers –Subtracting 2 signed numbers –Multiplying 2 signed numbers –Dividing 2 signed numbers

6. §1.1 – The Real Number System Examples – Calculate the following  |–101|  (- 1½) + (- 2¼)  Bill, a diver, is 120 feet below the surface of the Pacific Ocean. Heather is directly above Bill in a balloon that is 260 feet above the Pacific Ocean. Find the distance between Bill and Heather.

7. TMAT 103 §1.2 Zero and Order of Operations

8. §1.2 – Zero and Order of Operations Operations with 0

9. §1.2 – Zero and Order of Operations Examples – Calculate the following  Find values of x that make the following meaningless: 3x – 7 2x + 1  Find values of x that make the following indeterminate: 2 – x. (2x – 7)(x – 2)

10. §1.2 – Zero and Order of Operations Order of Operations – PEMDAS 1.Parenthesis 2.Exponents 3.Multiplications and Divisions in the order they appear left to right 4.Additions and Subtractions in the order they appear left to right

11. §1.2 – Zero and Order of Operations Examples – Calculate the following

12. TMAT 103 §1.3 Scientific Notation and Powers of 10

13. §1.3 – Scientific Notation and Powers of 10 Powers of 10 Laws

14. §1.3 – Scientific Notation and Powers of 10 Scientific Notation –Changing a number from decimal form to scientific notation –Changing a number from scientific notation to decimal form

15. §1.3 – Scientific Notation and Powers of 10 Examples – Calculate the following  Write the following in scientific notation 23700 17070000.00325  Write the following in decimal form 7.23 x 10 6 6.2 x 10 -3

16. TMAT 103 §1.4 Measurement

17. §1.4 – Measurement Measurement –Comparison of a quantity with a standard unit In past, units not standard (1 pace, length of ear of corn, etc.) –Necessity dictated universally standard units Approximate vs. exact –Accuracy (significant digits) –Precision

18. §1.4 – Measurement Accuracy (Significant Digits) Rules 1.All non-zero digits are significant 2.All zeros between significant digits are significant 3.Tagged zeros are significant 4.All numbers to the right of a significant digit AND a decimal point are significant 5.Non-tagged zeros to the right in a whole number are not significant 6.Zeros to the left in a measurement less than one are not significant

19. §1.4 – Measurement Examples – Calculate the following  Find the accuracy (number of significant digits) of the following: 14.7.000000000008 1404040 1404040.00030

20. §1.4 – Measurement Precision –The smallest unit with which a measurement is made. In other words, the position of the rightmost significant digit. –Ex: The precision of 239,000 miles is 1000 miles. –Ex: The precision of 23.55 seconds is.01 seconds

21. §1.4 – Measurement Examples – Calculate the following  Find the precision of each of the following: 1.0 m 360 V 350.000030 V

22. §1.4 – Measurement Precision and accuracy are different!!! –Ex: Determine which of the following measurements are more precise, and which is more accurate: 0.00032 feet 23540000 feet

23. TMAT 103 §1.5 Operations with Measurements

24. §1.5 – Operations with Measurements Adding or subtracting measurements 1.Convert to the same units 2.Add or subtract 3.Round the result to the same precision as the least precise of the original measurements Multiplying or dividing measurements 1.Convert to the same units 2.Multiply or divide 3.Round the result to the same number of significant digits as the original measurement with the least significant digits

25. §1.5 – Operations with Measurements Examples – Calculate the following  Find the sum of: 178m, 33.7m and 100cm  Find the product of: (.065m) and (.9282m)

26. TMAT 103 §1.6 Algebraic Expressions

27. §1.6 – Algebraic Expressions Terminology –Variable –Constant –Term –Numerical coefficient –Monomial, binomial, trinomial, polynomial –Degree of a monomial –Degree of a polynomial

28. §1.6 – Algebraic Expressions Operations on Algebraic expressions –Adding expressions –Subtracting expressions –Evaluating expressions given the values of variables

29. §1.6 – Algebraic Expressions Examples – Calculate the following  Find the degree of x 2 y  Find the degree of x 2 y + w 4 + a 3 b 2  (4y + 11) + (11y – 2)  (x 2 + x + 17) – (3x – 4)

30. TMAT 103 §1.7 Exponents and Radicals

31. §1.7 – Exponents and Radicals Laws of Exponents

32. §1.7 – Exponents and Radicals Examples – Simplify the following

33. §1.7 – Exponents and Radicals Radicals –Simplifying simple radicals Ex: –Simplifying radicals with the following property: Ex:

34. TMAT 103 §1.8 Multiplication of Algebraic Expressions

35. §1.8 – Multiplication of Algebraic Expressions Distributive Property FOIL Vertical multiplication Multiplication of general polynomials

36. §1.8 – Multiplication of Algebraic Expressions Examples – Calculate the following  x 2 (y 3 + z – 2)  (x + 2)(x – 2)  (3x 2 + 4x – 1)(2y – 3z + 7)

37. TMAT 103 §1.9 Division of Algebraic Expressions

38. §1.9 – Division of Algebraic Expressions Division by a monomial Division by a polynomial

39. §1.9 – Division of Algebraic Expressions Examples – Calculate the following  14x 2 – 10x 2x  6x 4 + 4x 3 + 2x 2 – 11x + 1 (x – 2)  4y 3 + 11y – 3 (2y + 1)

40. TMAT 103 §1.10 Linear Equations

41. §1.10 – Linear Equations Four properties of equations 1.The same value can be added to both sides 2.The same value can be subtracted from both sides 3.The same non-zero value can be multiplied on both sides of the equation 4.The same non-zero value can divided on both sides of the equation

42. §1.10 – Linear Equations Examples – Calculate the following  x – 4 = 12  4(2y – 3) – (3y + 7) = 6  ¼(½x + 8) = ½(x – 16) + 11

43. TMAT 103 §1.11 Formulas

44. §1.11 – Formulas Formula – equation, usually expressed in letters, that show the relationship between quantities Solving a formula for a given letter

45. §1.11 – Formulas Examples – Calculate the following  Solve f = ma for a  Solve e  = ƒx +  for x  Solve for R 3 :

46. TMAT 103 §1.12 Substitution of Data into Formulas

47. §1.12 – Substitution of Data into Formulas Using a formula to solve a problem where all but the unknown quantity is given 1.Solve for the unknown 2.Substitute all values with units 3.Solve

48. §1.12 – Substitution of Data into Formulas Examples – Calculate the following  Solve f = ma for a when f = 3 and m = 17  Solve e  = ƒx +  for x when e  = 11, ƒ = 3.5 and  =.01  Solve for R 3 when R B, R 1, and R 2 are all 11

49. TMAT 103 §1.13 Applications involving Linear Equations

50. §1.13 – Applications involving Linear Equations Solving application problems 1.Read the problem carefully 2.If applicable, draw a picture 3.Use a symbol to label the unknown quantity 4.Write the equation that represents the problem 5.Solve 6.Check

51. §1.13 – Applications involving Linear Equations Examples – Calculate the following  The difference between two numbers is 6. Their sum is 30. Find the 2 numbers.  The perimeter of an isosceles triangle is 122cm. Its base is 4cm shorter than one of its equal sides. Find the lengths of the sides of the triangle.

52. TMAT 103 §1.14 Ratio and Proportion

53. §1.14 – Ratio and Proportion Ratio: Quotient of 2 numbers or quantities Proportion: Statement that 2 ratios are equal

54. §1.14 – Ratio and Proportion Examples – Calculate the following  Find x: 25 = 75 96 x  The ratio of the length and the width of a rectangular field is 5:6. Find the dimensions of the field if its perimeter is 4400m.

55. TMAT 103 §1.15 Variation

56. §1.15 – Variation Direct Variation –If 2 quantities, y and x, change and their ratio remains constant (y/x = k), the quantities vary directly, or y is directly proportional to x. In general, this relationship is written in the form y = kx, where k is the proportionality constant. –Example: m varies directly with n; m = 198 when n = 22. Find m when n = 35.

57. §1.15 – Variation Inverse Variation –If two quantities, y and x, change and their product remains constant (yx = k), the quantities vary inversely, or y is inversely proportional to x. In general, this relation is written y = k/x, where k is called the proportionality constant. –Example: d varies inversely with e; d = 4/5 when e = 9/16. Find d when e = 5/3.

58. §1.15 – Variation Joint Variation –One quantity varies jointly, with 2 or more quantities when it varies directly with the product of these quantities. In general, this relation is written y = kxz, where k is called the proportionality constant. –Example: y varies jointly with x and the square of z; y = 150 when x = 3 and z = 5. Find y when x = 12 and z = 8.