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Week 1

Week 1. Given problems with signed real numbers, perform basic arithmetic operations, using a hand help calculator, including conversions of percentages, decimals, fractions, evaluations of mathematics expressions containing exponents and roots, and processing complex orders of operation.

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Week 1

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  1. Week 1 Given problems with signed real numbers, perform basic arithmetic operations, using a hand help calculator, including conversions of percentages, decimals, fractions, evaluations of mathematics expressions containing exponents and roots, and processing complex orders of operation.

  2. Objectives • Real Number System • Sets and Venn diagrams • Signed Numbers • Powers and Roots • Order of Operations

  3. Real Number System • Arithmetic uses only constants like -13, 0, 2/3 which have fixed value • Algebra uses not only constants, but also variables like a,b,x,y, which represent different numbers • The constants and variables we use in Algebra are called the Real numbers.

  4. Real Number System • The origins of number systems date back to the Egyptians, Babylonians, and Chinese. However, these earliest systems were much simpler than the real number system. For example, the number 0 was not widely accepted before the 13th century and the use of negative numbers (-1, -2, -3…) was not generally accepted before the 17th century. • In this first week, you will learn about natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. If you would like to learn more about the historical development of number systems you can visit: • http://mathforum.org/alejandre/numerals.html

  5. The Natural Numbers • Natural numbers or counting numbers are the most fundamental set of numbers • {1,2,3,…} • Braces, { }, are used to indicate a set of numbers • The … after 1,2, and 3, which are read “and so on” mean that the pattern continues without end – there are infinitely many natural numbers

  6. The Whole Numbers • The Natural number together with Zero • {0,1,2,3,…} • Not adequate for indicating losses or debts. • Example: Bank statement • Previous balance: $ 50 • Checks paid: $ 70 • New balance: -$ 20

  7. The Integers • The Whole numbers together with the negatives of the counting numbers form the set of integers • {…-3,-2,-1,0,1,2,3,…}

  8. The Rational Numbers • Any number that can be express as a ratio (or quotient) of two integers • {a/b, where a and b are integers, with b≠0} • Negative and positive fractions • Example: 3/1; 5/4; -77/3

  9. The Irrational Numbers • Real number that is not rational • Can not be written as a ratio of two integers • Example: √2, √3,π • In Computation with irrational numbers we use rational approximation for them • Example: √2≈1.4141; • π≈3.14 : the ratio of the circumference and diameter of every circle • Note: not all square roots are irrational • Example: √9 = 3

  10. The Number Line • Every real number corresponds to one and only one point on the number line. • Every point on the number line corresponds to one and only one real number

  11. Set notation • N = {…-3,-2,-1,0,1,2,3,…} • This is infinite set • A = {0,1,2,3} is finite set • 0 is a member of the set; so 0 ϵ A • Equal sets contain the same numbers • B = {0,1,2,3}; hence A=B • subsets: set C = {1,2,3} is a subset of set A and set B • Set D = {1,2,3,4} is not a subset of A or B • Is C subset of D?

  12. Interval of Real numbers • Interval of real numbers is the set of real numbers that is between two real numbers • Interval notation is representing intervals • (4,6): all real numbers between 4 and 6, but not 4 and 6 • [4,6]: all real numbers between 4 and 6, including 4 and 6 • (-3, ∞): all real numbers between -3 and infinity, but not -3

  13. Applications of SetsVenn Diagrams • Venn Diagrams: useful way of visualizing the possible relationship that can exist between various sets. • The set is represented by A and the universal set is represented by U. U A

  14. Venn Diagrams The set is represented by green color A’ is the complement of set A: includes all elements that does not belong to A. U A’ A

  15. Union of Sets • The union of sets A and B,written as A U Bis the set that contains all the elements of A as well as all the elements of B. • A U Bis represented by the shaded area. In terms of set notation, suppose A = {1, 2, 3} and that B = {d, f, g}. Then A U B = {d, f, g, 1, 2, 3}

  16. Union of Sets • A U B • A={0,1, 2, 3} • B={d, f, g} • A U B={d, f, g, 0, 1, 2, 3} U A B

  17. Union of Sets U • A U B • A={0,1, 2, 3} • B={d, f, g,1,2,3} • A U B={d, f, g,0,1, 2, 3} A B

  18. Intersection of Sets U • A ∩ B • A={0,1, 2, 3} • B={d, f, g,1,2,3} • Members of A are members of B: 1, 2, 3 • A ∩B = {1, 2, 3} A B x

  19. Counting formula • Strictly speaking A U B=A + B – A ∩ B • A={0,1, 2, 3} • B={d, f, g,1,2,3} • A U B={d, f, g,0,1, 2, 3} • A ∩ B={1, 2, 3} U A B

  20. Example Venn Diagrams • A total of 1000 students at Concorde Career College were surveyed to determine their course-scheduling preferences. The survey results were as follows: • 620 students liked morning classes • 320 students liked afternoon classes • 230 students liked both morning and afternoon classes • Of those students surveyed, how many: • Like morning classes? • Like afternoon classes • Like morning classes but do not like afternoon classes? • Like afternoon classes but do not like morning classes? • Like neither morning nor afternoon classes?

  21. Using numbers to show opposites • Can you give me an examples of opposite situations in life? • In mathematics, signed numbers are used to represent quantities that are opposites. • Examples: • On a temperature scale, the opposite of 10 above zero is 10 below zero; • A profit of $100 is the opposite of losses of $100 etc. • On the real number line, +4 (read “positive 4”) is the opposite of – 4 (read “negative 4”). Both +4 and – 4 are the same distance from zero.

  22. Using numbers to show opposites • We can say that for every positive number there is a negative number, which is the same distance from zero on the opposite side of the zero. Opposites have the following properties • For any real number a,  ( a) = a • Example: -1 is the opposite of 1 and vice-versa • The sum of a real number and its opposite is zero • Example:-1 + 1 = 0 • Graphical representation

  23. Absolute Value • The absolute value of a number is the distance between 0 and that number on a number line. Another way to put it is that the absolute value of a number is its numerical value, regardless of its sign. • In any pair of opposites, the positive number in the pair is the absolute value of each of the numbers. There is a symbol used to express the absolute value of the number. For any number a,a is how to write an absolute value of a. The absolute value of a (written as a) can be found from the rule: • If a  0, then, |a|=a • If a  0, then, |a|=  a • |5 |= 5 • |-5|= -(-5)=5

  24. Mathematical Operations • The four basic operations used in arithmetic as well as in algebra are: • + addition, usually written as a + b. It means the sum of a and b; a plus b; a added to b. • - subtraction, usually written as a - b. It means the difference of a and b; a minus b; b less than a; • * multiplication, usually written as a * b or ab. It means the product of a and b; a times b. • / or , division,usually written as a/b or a:b. It means the quotient of a and b; a divided by b; a into b.

  25. Fractions • The terms of a fraction are numerator and denominator. Thus, in the fraction a/b (which we read as “a divided by b”), a is the numerator and b is the denominator.

  26. Type of Fractions • Proper fraction a/b: when a<b • Example: 1/2, 3/4, 5/7, etc. • Improper fraction a/b: when a>b • Example: 5/3, 6/5 etc. • Mixed number c a/b: where c is a whole number and a/b is proper fraction • Example: 3 1/2, 5 2/3

  27. Equivalent fractions • Equivalent fractions are fractions that have the same value, but not the same numerator and denominator. There are two ways to find an equivalent fraction.

  28. Equivalent fractions • Multiply the numerator and the denominator by the same number. That number cannot be zero. Why? • Example: 3/4=3*2/4*2=6/8; 3/4=3*3/4*3=9/12. The fractions: 3/4, 6/8, 9/12 are all equivalent. Converting a fractions to an equivalent fractions with a larger denominator is called building upthe fraction.

  29. Equivalent fractions • Divide the numerator and the denominator by the same number- common factor. That number cannot be zero. Why? • Example: 6/8=6÷2/8÷2=3/4. In this case 2 is the common factor of 6 and 8. We have to continue dividing until the numerator and denominator have no common factors greater then 1. • 3/4 is in lowest terms or written in simplest form.

  30. Greatest Common Factor • To find the simplest form of a fraction is to look for the greatest common factor (GCF) of the numerator and denominator. • Example: Let’s find the simplest form of 6/24. The factors of 6 are: 1,2,3,6; the factors of 24 are: 1,2,3,4,6,8,12,24. The GCF of 6 and 24 is 6. So, 6÷6=24÷ 6=1/4; So ¼ is the simplest form of 6/24.

  31. Addition and subtraction of fractions with same denominator • When the fractions have same denominators • Example: 2/5+1/5=3/5; Add the numerators and keep the same denominator • Example: 2/5-1/5=1/5; Subtract the numerators and keep the same denominator

  32. Addition of fractions with different denominator • 5/9 + 3/2=? • Find least common denominator: LCD • Make a list of all multiples of both denominators: • For 9: 9,18; For 2: 2,4,6,8,10,12,14,16,18 • LCD is the lowest number that both of these denominators would be able to divide into evenly ; For 9 and 2 this number is 18; so the common denominator of the resulted fraction should be 18. • Multiply the numerator and denominator of the 5/9 with 2 and numerator and denominator of 3/2 with 9: • 5*2/9*2 + 3*9/2*9 = 10/18+27/18= 37/18; this is improper fraction and should be converted to mixed number ( explanation of how to do this later in the lecture).

  33. Subtraction of fractions with different denominator • 1/3-1/12=? • Find least common denominator: LCD • For 3: 3,6,9,12; For 12: 12, • LCD is the lowest number that both of these denominators would be able to divide into evenly ; For 3 and 12 this number is 12; so the common denominator of the resulted fraction should be 12. • Multiply the numerator and denominator of the 1/3 with 4 and numerator and there is no need to multiply second fraction because it already has denominator is 12. • 1*4/3*4 – 1/12= 4/12 – 1/12 = 3/12 • Reduce to lowest term:1/4 is in its simplest form

  34. Multiplication of fractions • Multiplication of fraction is the easiest of all fraction operations. All you have to do is to multiply straight across- multiply the numerators and the denominators. • Example: 4/5 * 2/3 = 4*2/5*3=8/15

  35. Division of fractions • Let’s consider following example: • 2/5÷1/3=? • Division is a opposite of multiplication; so we change ÷ to * sign and 1/3 with its reciprocal 3/1; • So 2/5÷1/3=2/5*3/1 = 6/5

  36. How to convert mixed number into improper fraction • 3 1/2=? • Multiply denominator 2 by the whole number 3, so 2*3=6; • Add the numerator 1 to the product of the whole number and the denominator-this will be your new numerator: 1+6=7 • Keep the same denominator; So, the new improper fraction is 7/2 • 3 1/2=7/2

  37. How to convert improper fraction into mixed number • Convert 45/4 to a mixed number. • First, do the long division to find the "regular" number part:11 and the remainder:1 • Since the remainder is 1 and you're dividing by 4, the fraction part will be 1/4. • so 45/4 = 11 1/4

  38. How to multiply/divide mixed numbers • (4 1/2) * (3 1/3)=? • Convert mixed numbers to improper fractions • 4 1/2=9/2 • 3 1/3=10/3 • Perform multiplication/division of the two improper fractions following the rules • 9/2*10/3=90/6=15

  39. How to subtract/add mixed numbers • 16 1/8 + 5 5/8 = 21 6/8 = 21 3/4 • 129/8 + 45/8 = 174/8 = 21 3/4 • 8 5/8 – 3 1/12 • LCD = 24 • 8 15/24 – 3 2/24 = 5 13/24 • 207/24 – 74/24 = 133/24

  40. Carrying when adding mixed numbers • 9 5/8 + 13 7/8 = 22 12/8 • 12/8 can be written in its lowest terms as 3/2 = 1 ½ • So, 22 12/8 = 22 3/2 = 22 + 3/2 • 22 + 1 ½ = 23 1/2

  41. Borrowing when subtracting mixed numbers • 8 1/3 – 4 3/5 • LCD = 15 • 8 5/15 – 4 9/15 • You can not subtract 9/15 from 5/15 • We borrow from the whole number 8 • 8 5/15 = 7+1+5/15 = 7+15/15+5/15 • 7 + 20/15 = 7 20/15 • 7 20/15 - 4 9/15= 3 11/15

  42. Examples • 7 – 2 5/6 • 14 6/7+15 ½ • 19 2/3 – 11 ¾ • 12 8/15 + 18 3/5

  43. From Fractions to Decimals • Fraction with a denominator of 10, 100, 1000 etc. can be written as decimals numbers: • Examples: 3/10=0.3; 25/100=0.25; 5/1000=0.005 • Fractions with a denominator of 100 are often written as percents • Examples: 25/100=25%; 3/100=3%; 300/100=300%

  44. From Decimals to Fractions • Count the positions to the right of decimals, i.e., tenth’s, hundredth’s, thousandth’s, ten thousandth’s, etc. • Whatever that number ends up being, that number becomes your denominator. • The number in the numerator now loses its decimal, any zeros that remain to the left of any natural number are eliminated. • Example: 0.56=? Fraction; the position to the right of decimal is hundredth’s; hence the denominator of the fraction will be 100 • So, 0.56 = 56/100 • Example: 1.56=?; the denominator is 100, the numerator is 56 and the whole number is 1; the whole number of original decimal now is the whole number of the mixed number • Hence: 1.56 = 1 56/100=1 14/25

  45. From Decimals to Percents • Fractions Decimals Percents • Example: 5.45=?% • Move decimal point two places to the right • 5.45=545% Move decimal point two places From left to right

  46. From Percents to Decimals • Fractions Decimals Percents • Examples: 3.67% = 0.0367 • The process converting decimals to percents and vice-versa calls for a two space movement in either direction. When the spaces are not there, add Zeros (0’s), as many as necessary, to make that 2 position movement and proceed. Move the decimal point two places From right to left

  47. Ratio • Previously, we defined a rational number as the ratio of two integers. • More general definition of a ratio: • If a and b are any two real numbers, with b≠0, then the expression a/b is called the ratio of a and b, or the ratio of a to b. • Example: During soccer game, 240 tickets were sold. There were 500 tickets available. Find the ratio of tickets sold to the total number of tickets. • 240/500 = 12/25; after reducing the initial fraction

  48. Proportion • A proportion is any statement of equality of two ratios. The statement a/b=c/d , (where b, d ≠0) is a proportion; • When two pairs of numbers have the same ratio we say the they are proportional • Example: 4/8=1/2 • Cross-multiplying: the cross products of a proportion are equal • If a/b=c/d then a*d=b*c • Example: 4/8=1/2 , so 4*2=8*1

  49. Proportion • Cross products can be used to find missing numbers in a proportion • 2/3 = n/12 • 2*12 = 3*n • 24 = 3n • n = 8

  50. Operations with signed numbers-Addition • Adding numbers with like signs: add their absolute values; the sum has the same sign as the given numbers • Example: 23+56=79; (-23)+(-56)=-79 • Adding numbers with unlike signs: subtract the absolute value of the smaller number from the absolute value of the larger number; the sum has the sign of the larger number • Example: -23+56=33; -7 + 5 = -2

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