Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CHAPTER 1 SECTION 1-1 LANGUAGE OF MATHEMATICS SET– a collection or group of, things, objects, numbers, etc. INFINITE SET – a set whose members cannot be counted. If A= {1, 2, 3, 4, 5,…} then A is infinite FINITE SET – a set whose members can be counted. If A= {e, f, g, h, i, j} then A is finite and contains six elements SUBSET – all members of a set are members of another set If A= {e, f, g, h, i, j} and B = {e, i} , then BA EMPTY SET or NULL SET – a set having no elements. A= { } or B = { } are empty sets or null sets written as The EMPTY SET is a subset of every set Every SET is a subset of itself VARIABLE – represents an unknown number or quantity and is usually denoted by a letter such as a, n, x, y, z VARIABLE EXPRESSION a statement containing a number and/or variables 2x + 4 -10x – 22y – 33z EQUATION – a statement that two numbers or expressions are equal. -6 + 10 = 6 – 2 or 4x + 3 = 19 TRUE/FALSE •9 { -3, 0, 3, 6,…} •{a, b} {a, b, e, i} •The subsets of {b, c} are {b}, {c}, {b, c}, SECTION 1-2 REAL NUMBERS NATURAL NUMBERS set of counting numbers {1, 2, 3, 4, 5, 6, 7, 8…} WHOLE NUMBERS - set of counting numbers plus zero {0, 1, 2, 3, 4, 5, 6, 7, 8…} INTEGERS - set of the whole numbers plus their opposites {…, -3, -2, -1, 0, 1, 2, 3, …} RATIONAL NUMBERS numbers that can be expressed as a ratio of two integers a and b and includes fractions, repeating decimals, and terminating decimals EXAMPLES OF RATIONAL NUMBERS ½, ¾, ¼, - ½, -¾, -¼, .05 .76, .333…, .666…, etc . IRRATIONAL NUMBERS numbers that cannot be expressed as a ratio of two integers a and b and can still be designated on a number line EXAMPLES OF IRRATIONAL NUMBERS , 6, 29, 8.11211121114…, etc . The real number paired with each point is the coordinate of that point. The distance between any two points on the line is equal to the absolute value of the difference of their coordinates. SECTION 1-3 UNION AND INTERSECTION OF SETS UNION OF SETS Two or more sets joining together to form a new set If A = {1,2,3} and B = {-1, -2, -3} then A B = {-3, -2, -1, 1, 2, 3} . INTERSECTION OF SETS Two sets containing elements common to both sets If A = {1, 2, 3} and B = {-1, 0, 1} then A B = {1} . SYMBOLS • - union of sets • - intersection of sets • - complement of a set (not) VENN DIAGRAM Diagram using circles inside a rectangle to represent the union and intersection of sets VENN DIAGRAM A = {4,5,6,7, 8, 9} B = {8, 9, 12, 15, 16} C = {18,20} • Find A B • Find A C • Find B C A = {4,5,6,7, 8, 9} B = {8, 9, 12, 15, 16} C = {18,20} • Find A B • Find A C • Find B C A = {4,5,6,7, 8, 9} B = {8, 9, 12, 15,16} C = {18,20} • Find A • Find (A B) • Find (A C) SECTION 1-4 ADDITION, SUBTRACTION AND ESTIMATION CLOSURE PROPERTY a + b is unique 7 + 5 = 12 COMMUTATIVE PROPERTY a+b =b+a 25 + 60 = 60 + 25 ASSOCIATIVE PROPERTY (a + b) + c = a + (b +c) (5 + 15) + 20 = 5 + (15 +20) IDENTITY PROPERTY a+0=0+a=a -3 + 0 = 0 + -3 = -3 INVERSE PROPERTY a +(-a) = 0 -2 + (2) = 0 For real numbers a and b 1.If a and b are negative numbers, then a + b is negative. • - 16 + (-4) = -20 • (-14) -6 = -20 For real numbers a and b 2.If a is a positive number, b is a negative number, then the sign of the sum of a + b will be the sign of the largest number when the signs are ignored. -9 + 5 = - 4 SECTION 1-5 MULTIPLICATION AND DIVISION CLOSURE PROPERTY ab is unique 7 • 5 = 35 COMMUTATIVE PROPERTY ab = ba 25 • 60 = 60 • 25 ASSOCIATIVE PROPERTY (ab)c = a(bc) (5•15) • 20 = 5(15• 20) IDENTITY PROPERTY a•1=1•a=a -3 • 1 = 1 • -3 = -3 INVERSE PROPERTY a •1/a= 1/a •1 = 1 -2 • (-1/2)= 1 ZERO PROPERTY a •0= 0 •a = 0 5 • 0 = 0 •5 = 0 SECTION 1-7 DISTRIBUTIVE PROPERTY AND PROPERTIES OF EXPONENTS DISTRIBUTIVE PROPERTY a(b + c) = ab + ac 5(12 + 3) = 5•12 + 5 •3 = 75 EXPONENTIAL FORM – number written such that it has a base and an exponent 3 4 = 4 •4 •4 BASE – tells what factor is being multiplied EXPONENT – Tells how many equal factors there are PROPERTY OF EXPONENTS FOR MULTIPLICATION m n m+n a •a = a 2 4 5 •5 = 2+4 5 = 6 5 PROPERTY OF EXPONENTS FOR MULTIPLICATION m)n mn (a = a 2)4 (5 = 2·4 5 = 8 5 PROPERTY OF EXPONENTS FOR MULTIPLICATION m m m (ab) = a b 3 (5·3) = 3 5 3 ·3 PROPERTY OF EXPONENTS FOR DIVISION m n m-n a ÷a = a 5 2 5 5 = 5-2 5 = 3 5 PROPERTY OF NEGATIVE EXPONENTS -m a = 1/a -2 5 = 2 1/5 = 1/25 PROPERTY OF ZERO EXPONENT 0 a = 1 0 5 = 1 SECTION 1-8 EXPONENTS AND SCIENTIFIC NOTATION SCIENTIFIC NOTATION – A number having two factors. The first factor is greater than or equal to 1 and less than 10. The second is a power of 10 STANDARD FORM – the customary way numbers are written 5,283.45 .00789 SCIENTIFIC FORM 496,000 = 4.96 .00059 = 5.9 5 •10 -4 •10 YEAH! THE END