Math refresher course Prof. Sacha Varone Haute Ecole de Gestion de Genève Academic year 2011-2012 Contents 1 Review: numbers 3 1.1 Properties . .3 1.2 Notation . .3 2 Fractions 4 2.1 Introduction . .4 2.2 Multiplication and division . .5 2.3 Addition and subtraction . .5 2.4 Percents . .7 2.5 Ratio . .7 3 Powers and roots 8 3.1 Introduction . .8 3.2 Order of operations . .8 3.3 Laws . .9 3.4 Roots . .9 4 Equations 11 4.1 Introduction . 11 4.2 Properties: addition and multiplication . 11 4.3 Linear equation with one variable . 12 5 Systems of Linear Equations 14 5.1 Introduction . 14 5.2 Method of substitution . 14 5.3 Method of elimination . 15 5.4 Graph representation . 17 6 Statistics 18 1 6.1 Measures of central tendency . 18 6.2 Measures of dispersion . 18 7 Exercises 20 7.1 Numbers . 21 7.2 Fractions . 22 7.3 Powers and Roots . 24 7.4 Equations . 25 7.5 Systems . 26 7.6 Statistitics . 28 Remark Syllabus and content may change before or during the course. 2 1 Review: numbers Numbers can be divided into several sets: 1. Natural numbers: 0; 1; 2; 3;::: 2. Integer numbers: :::; −2; −1; 0; 1; 2;::: 2 27 3. Rational numbers: 3 ; − 333 ;::: p those numbers are of the form q where q is different from 0 and p,q are integer numbers p 4. Real numbers: 2; π; −2:333 :::;::: 1.1 Properties Property (a; b real numbers) Illustration (−1) · a = −a (−1)6:3 = −6:3 −(−a) = a −(−2) = 2 (−a)b = −(ab) = a(−b) (−2)5 = −(2 · 5) = 2(−5) (−a)(−b) = ab (−2)(−6) = 2 · 6 Definition The absolute value of a number a is the value of a without its sign. The notation is jaj and its value is a if a > 0 −a if a < 0 1.2 Notation Let's consider two real numbers a and b, with b different from zero 1. Some calculators produce results such as "4.543E13". This means 4:543 · 1013, which is 4543 followed by 10 zeros. −1 1 2. b means b 3. a − b means a + (−b) a 4. a : b means b 3 2 Fractions 2.1 Introduction DefinitionA numeric fraction is a quotient of two numbers. The top number is called the numerator and the bottom number is referred to as the denominator. The denominator cannot equal 0. a where b 6= 0 b 2 10 Example 5 or 100 1 A number could be written in the form 5 4 (an interest rate for example). In order to transform it into a fraction, we simply use the denominator of the fraction, 1 5·4+1 21 and use 5 ∗ 4 + 1 as the numerator. Hence 5 4 = 4 = 4 a bc + a c = b b Definition A fraction is simplified if the numerator and denominator do not have any common factors other than 1. Property a · c a = where b; c 6= 0 b · c b To simplify a fraction, we use a factorisation into prime numbers. A prime number is a whole number that has only two distinct factors, 1 and itself. Examples of prime numbers are 2, 3, 5, 7, 11, ... To simplify a fraction, we can rewrite the denominator and the numerator using a product of prime numbers, and then use the property. 12 Example Simplify 18 . The prime factorization of the numerator 12 would be 12 = (2)(2)(3) . The prime factorization of the denominator 18 would be 18 = (2)(3)(3) Therefore we have 12 2 · 2 · 3 2 = = 18 2 · 3 · 3 3 4 Definition A fraction is irreducible if it can not be further simplified. Example 2 3 is irreducible 2 1 4 is not irreducible since it can be simplified as 2 2.2 Multiplication and division Definition • Multiplying fractions consists in multiplying the numerators together and multiply- ing the denominators together. a c ac = b d bd • Two numbers (and hence fractions) are reciprocal of each other if their product is 1 1 a · = 1 where a 6= 0 a • Dividing a fraction A by another fraction B consists in a multiplying A by the reciprocal of B a c a d ad : = · = b d b c bc 2 4 Example 3 : 5 =? 2 4 2 5 First rewrite the division with a multiplication by the reciprocal: 3 : 5 = 3 · 4 2 5 1 5 Then try to simplify the product: 3 · 2·2 = 3 · 2 1·5 5 Finally do the product: 3·2 = 6 2.3 Addition and subtraction If both fractions have the same denominator, then a b a + b a b a − b + = − = c c c c c c 5 3 5+3 8 Example 7 + 7 = 7 = 7 5 If the fractions to be added or subtracted do not have a common denominator, then 1. multiply each numerator by the denominator of the other fraction 2. add the two results. This is the new numerator. 3. multiply the denominators together. This is the new denominator 4. simplify the obtained fraction. a c ad + bc + = b d bd 2 4 Example 3 + 5 =? 1.2 · 5 = 10 and 4 · 3 = 12 2. the new numerator is 10 + 12 = 22 3. the new denominator is 3 · 5 = 15 22 2·11 22 4. the fraction is 15 = 3·5 = 15 6 2.4 Percents Percent = "per hundred". Notation: x% 10 Example 10% = 100 = :10 How to write a percent as a decimal number? Drop the percent sign and move your decimal two places to the left. Example 75% (=75:00%) becomes .75 163% becomes 1.63 0:2% ! 0:002 How to write a decimal number as a percent? Move your decimal two places to the right and then put the % sign at the end of the number. Example :62 ! 62% 8 ! 800% 0:123 ! 12:3% 2.5 Ratio DefinitionA ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. It is often written as two numbers separated by a colon (:). Example • Most movie theater screens have an aspect ratio of 16:9, which means that the screen is 16/9 as wide as it is high. • In probability, the ratio of the probability of something happening to the probability of it not happening is called the odds of the thing happening. 7 3 Powers and roots 3.1 Introduction Definition The power of a number, noted an is the product of a by itself, n times. a is called the base and n is called the exponent. We refer to an as a to the nth power (or simply a to the n) an = a · a · a · ::: · a | {z } n times Example • 24 = 2 · 2 · 2 · 2 = 16 • (−3)2 = (−3) · (−3) = 9 Property Property (a; b real numbers) Illustration a1 = a a 6= 0 (−2:3)1 = (−2:3) 1a = 1 1123 = 1 a0 = 1 a 6= 0 20 = 1 0k = 0 k 6= 0 025 = 0 −k 1 −2 1 1 a = ak a 6= 0 3 = 32 = 9 1 p p a q = q a 160:5 = 16 = 4 3.2 Order of operations Please Parenthesis or grouping symbols Excuse Exponents (and radicals) My Dear Multiplication/Division left to right Aunt Sally Addition/Subtraction left to right Example −32 + 23 · (−3)2 = −3 · 3 + 2 · 2 · 2 · (−3) · (−3) = −9 + 8 · 9 = 63 8 3.3 Laws The followings laws apply for real numbers a and b, and integer numbers m and n Law Illustration aman = am+n 3231 = 33 = 27 (am)n = amn (23)2 = 26 = 64 (ab)n = anbn (10)2 = (2 · 5)2 = 2252 = 4 · 25 = 100 a n an 3 2 32 9 ( b ) = bn where b 6= 0 ( 2 ) = 22 = 4 am m−n 24 1 an = a 23 = 2 = 2 3.4 Roots 1 p Roots can be considered as special cases of powers: a q = q a Definition Let n be a positivep integer greater than 1, and let a be a real number. The principal nth root n a of a real number a is defined as follows: p 1. If a = 0, then n a = 0. p 2. If a > 0, then n a is the positive real number b such that bn = a. 3. If a < 0, then p (a) if n is odd, then n a is the negative real number b such that bn = a. p (b) if n is even, then n a is not a real number. Warning: if a 6= 0 and b 6= 0 p p p • a + b 6= pa + b p p p Example: 7 + 2 = 9 6= 7 + 2 p p p 2 2 2 2 • a + b 6=p a + b p p p Example: 22 + 32 = 13 6= 4 + 9 = 5 p 2 • x = jxj p Example: 9 = +3 9 Remark It is a good idea to rationalize the denominator of a fraction: p p 1 1 a a p = p · p = a a a a 10 4 Equations 4.1 Introduction Definition An equation is a statement that two quantities or expressions are equal. Example A car is moving during t = 2 hours at the speed of r=50 km per hour. What is the distance x traveled after t = 2 hours? x = t · r that is to say x = 2 · 50 = 100 DefinitionA solution or also called root of an equation in x is a number b that yields a true statement when substituted for x.