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York University (33,498)
Economics (1,498)
ECON 1530 (91)
Lecture

# Topic1.pdf

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School
York University
Department
Economics
Course
ECON 1530
Professor
Gordana Colby
Semester
Fall

Description
I REVIEW OF FUNCTIONS The Real Number System A function from A to B is a rule that assigns to each element of the set A, one and only one element of the set B. In economics we are invariably focused on the way one vari- able eﬀects another: this is a special case of a function. Since these functions are usually represented by numbers, let’s ﬁrst consider what we mean by numbers. Natural Numbers The counting numbers 1,2,3,... Integers Theaeheumes ..., −2,−1,0,1,2,... Rational Numbers: are those that can be represented as the ratio of two integers. It appears as though the rational num- bers should ﬁll up the space between any two numbers on the number line. In fact, they do not. There are always “holes”, an inﬁnite number of which exist between any two rational num- bers. Otherwise put, there exist numbers which are not the ratio of two integers. Irrational Numbers What numbers do the rational numbers not include? Well, we know by Pythagorus’ Theorem that 2 2 2 s =1 +1 =2 √ pherefor√, s = 2, but there does not existdp,aq such that q = s = 2. Thus, we introduce irrational numbers. 1 The rational and irrational numbers make up the real numbers; they “ﬁll up” the real line. We commonly use the “decimal system” to write numbers. This is a good way to understand the diﬀerences between rational and irrational numbers. The decimal system is such that we write a number, say 11as 1.25 1 1 4 such that 1.25 = 1 + 2 × 10 +5 × 102. Rational numbers that can be written using a ﬁnite number of decimal places are ﬁnite decimal fractions. Some rationals, eg. 14 cannot be written 1 3 with a ﬁnite sum but, 1 31 .333... . The decimal fraction is periodic, i.e., the expansion either stops or repeats itself after a ﬁnite number of digits. Irrationals are written x = ±m.α α α ... 1 2 3 where there is not a set of alpha’s that is periodic. Some ex- √ √ √2 2 amples include: 2, − 5,π , 2 ,A = πr We deﬁne a real number as an arbitrary inﬁnite decimal frac- tion. It is not easy to show a number is irrational. √ √ 2 3 Is 2 +3 irrational? We don’t know. a Finally, we say 0 is “undeﬁned” for any real number a. Simi- larily,0is “undeﬁned”. 2 EQUALITIES AND INEQUALITIES Equalities We use equalities in a couple of ways. (i) As an identity: Y = C + I + G by deﬁnition 2 x − 16 = (x − 4)(x +4 ) odsfrall x. (ii) Suppose I have x 2 + x − 12 = 0. This equation does not hold for all xh,i x = 4 does not satisfy the equality but other values of x do. In particular, x =3 ,−4. Often, “≡” is used to distinguish an identity from an equation. Inequalities Inequalities are often used in economics. eg. private expenditures ≤ income. We know if, a ≥ b, then −a ≤− b. 3 EXAMPLES: Given N nonnegative objects 1,x 2...,xn. ArithmeticMean ≥ GeometricMean 1 N N (x1 + x2+ ··· + xN) ≥ (x1× x 2···× x )N Consider the following example with N =. We can see that, if this were not true, then: √ a + b< (a · b) = a · b ⇒ 2 2 12 a + b < (a · b)2 = a · b ⇒ 2 2 a + b − a · b< 0 4 Now, 2 2 2 (a + b) − ab = a + b + 2ab − ab 4 4 4 4 a2+ b +2 ab − 4ab = 4 a2− 2ab + b2 = 4 (a − b) = 4 So, (a + b) (a − b) − ab = < 0 4 4 which is not true. 4 INTERVALS [a,b] − closed (a,b) − open [a,b) − open on the right (a,b] − open on the left [a,∞) ≡ all real numbers greater than or equal to a (−∞,a] ≡ all real numbers less than or equal to a (−∞,∞) ≡ llumes [−∞,∞] ≡ extended real numbersR ∪∞∪− ∞ . ABSOLUTE VALUE |a| = the distance of a from zero +x for x ≥ 0; = −x for x< 0. The distance between a and b is given by, |a − b| = |b − a| Finally, |x|
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