* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 1.1-1.3
History of logarithms wikipedia , lookup
Law of large numbers wikipedia , lookup
History of mathematical notation wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Large numbers wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Location arithmetic wikipedia , lookup
Hyperreal number wikipedia , lookup
Surreal number wikipedia , lookup
Elementary algebra wikipedia , lookup
Factorization wikipedia , lookup
Positional notation wikipedia , lookup
System of polynomial equations wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Real number wikipedia , lookup
P-adic number wikipedia , lookup
1.1 Introduction to Algebra Arithmetic vs. Algebra ____________________________________ _________________________________ A variable is a ___________________________________________________________ An algebraic expression is a _______________________________________________ _______________________________________________________________________ ex. Let x represent my yearly salary in dollars. If my boss makes three times as much as me, we can say that her salary is ________. If I make 50,000 dollars per year, then x = ___________, and my boss then makes ___________________________ dollars. We evaluate an expression by substituting a number for the variable as in the example above. ex. Evaluate the algebraic expression 3(5 + 2 x − y ) − 1 for x = 2 x and y = 6 . Order of Operation 1. Perform all operations within parentheses or other grouping symbols. 2. Multiplication & Division in the order which they occur from left to right. 3. Addition & Subtraction in the order which they occur from left to right. An equation is a __________________________________________________________ ________________________________________________________________________ Solution(s) of an equation are _______________________________________________ ________________________________________________________________________ ex. Is 10 a solution to the equation 2 x − 8 = 7 + x ? (Substitute 10 in for x everywhere and determine if both sides are equal.) ex. Is 15 a solution to the equation 2 x − 8 = 7 + x ? Translating from English to Math (see p.4, table 1.1) Five more than a number : __________________________________________________ Three less than a number : __________________________________________________ Nine decreased by twice a number : ___________________________________________ The quotient of a number and 5 is 4 : __________________________________________ Mathematical Models are formulas that describe relationships between real-world variables. 1.2 Fractions An improper fraction has a numerator that is___________ than its denominator. ex. A mixed number consists of the addition of an ________ and ________, but the addition sign is invisible. ex. To rewrite an improper fraction into a mixed number, we _____________ the denominator into the numerator and write in the form quotient ex. 14 = 5 remainder denominator ex. 228 = 7 To rewrite a mixed number as an improper fraction we multiply the ____________ and the __________ and add the ___________, and place result over ___________. ex. 1 2 = 4 A prime number is a natural number greater than 1, which only has ________ and ____ as factors. ex. A composite number is a natural number greater than 1, which is not a _____________. Every composite number can be expressed as the product of ____________________. ex. 6= 84 = 90 = 108 = Fundamental Principle of Fractions: a⋅c a = b⋅c b Reducing a fraction to its lowest terms: 1. Rewrite numerator and denominator as products of prime numbers. 2. Divide out common factors. ex. 8 = 10 32 = 48 Multiplying Fractions: ex. a c a⋅c ⋅ = b d b⋅d 7 1 ⋅ = 9 2 Dividing Fractions: 7 ⋅2 = 9 a c a d ÷ = ⋅ b d b c change to and multiplication ex. 2 1 6 ⋅2 = 3 4 7 3 ÷ = 8 4 invert to get reciprocal 10 9 ÷ = 3 2 When Adding and Subtracting fractions they need to have the same denominators. a b a+b + = d d d and a b a −b − = d d d If the denominators are different, first rewrite the fractions as equivalent fractions with the least common denominator. (Find the LCD using prime factorization.) ex. 2 10 + = 9 9 1 1 − = 24 45 3 1 + = 4 6 1.3 The Real Numbers A _______ is a collection of objects. The objects in a set are called the _____________. ex. {2, 4, 6, 8, ...} = The Number Line 0 The sets that make up the real numbers Natural numbers = Whole numbers = Integers = Rational Numbers = Irrational Numbers = ex. Natural number: 3 5, 0, - , 2π, -5, 0.45, 0.16 . Determine which is a 4 Rational number: Whole number: Irrational number: Integer: Real number: Given the numbers 18, -3.5, The Absolute Value of a number describes the ______________ from ____ on the number line. ex. 8= 0= −5 = The symbols < and > are called __________________. a < b “a is ________ than b” a ≤ b “a is _______________________ b” a > b “a is ________ than b” a ≥ b “a is _______________________ b” ex. 8 2 −8 2 −8 −2 −8 2