2.17: Concepts - Mathematics

  1. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If for some reason the election had to be held again and C decided to drop out of the election, which caused B to become the winner, which is the primary fairness criterion violated in this election?
  1. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If for some reason the election had to be held again and many people who had voted for C switched their preferences to favor A, which caused B to become the winner, which is the primary fairness criterion violated in this election?
  1. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If in a head-to-head comparison a majority of people prefer B to A or C, which is the primary fairness criterion violated in this election?
  1. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If B had received a majority of first place votes, which is the primary fairness criterion violated in this election?

2.17: Concepts - Mathematics

Math 300: Fundamental concepts of mathematics Math 300: Fundamental concepts of mathematics
Spring 2017

Instructor: Paul Hacking, LGRT 1235H, [email protected]
TA: Shelby Cox, [email protected]

Classes: Tuesdays and Thursdays, 1:00PM--2:15PM, in LGRT 143.
Coseminar: Tuesdays 4:00PM-4:50PM, LGRT 1234 Wednesdays 4:00PM--4:50PM, LGRT 1114, and Wednesdays 5:00PM--5:50PM, LGRT 1114.
Office hours: Tuesdays 2:30PM--3:30PM and Wednesdays 2:00PM--3:00PM in LGRT 1235H (Paul Hacking) Wednesdays 3:00PM--4:00PM in LGRT 1117 (Shelby Cox).

Course text: Mathematical Reasoning: Writing and Proof, by T. Sundstrom, available for free download at the author's website here.

Prerequisites: Math 132.

The goal is that you learn to read, understand, and construct coherent, logically correct proofs, so that you may more easily make the transition from calculus to the more theoretical junior-senior courses, especially abstract algebra and modern analysis. Starting with explicit axioms and precisely stated definitions, you will systematically develop basic propositions about integers and modular arithmetic, induction and recursion, real numbers, infinite sets, and such other topics as time may allow. You will be provided with the needed background about logic, sets, and functions. For nearly every class you will create written mathematical proofs. You are expected to participate actively in class, including at the co-seminar.

We will cover Chapters 1--9 of the textbook.

There will be weekly homework, due at the beginning of Thursday's class. (First homework due Thursday 2/2/17.)

There will be two midterm exams and one final exam as follows:

Midterm 1: Tuesday 2/28/17, 7:00PM--8:30PM, in LGRT 206.
The syllabus for Midterm 1 is Sections 2.1, 2.2, 2.3, 2.4, 3.1, 3.2, and 3.3 of Sundstrom.
There will be a review session for Midterm 1 on Monday 2/27/17, 7:00PM--8:30PM, in LGRT 202.
Please try the review problems here before the review session.

Midterm 2: Tuesday 4/11/17, 7:00PM--8:30PM, in LGRT 206.
The syllabus for Midterm 2 is Sections 3.1, 3.2, 3.3, 3.4, 3.5, 4.1, 4.2, 4.3, 8.1, 8.2, and 8.3 of Sundstrom.
There will be a review session for Midterm 2 on Monday 4/10/17, 7:00PM--8:30PM, in LGRT 204.
Please try the review problems here before the review session.

Final exam: Wednesday 5/10/17, 10:30AM--12:30PM, in LGRT 143.
The syllabus for the final exam is Sections 2.1, 2.2, 2.3, 2.4, 3.1, 3.2, 3.3, 3.4, 3.5, 4.1, 4.2, 4.3, 5.1, 5.2, 5.3, 6.1, 6.3, 6.4, 6.5, 7.1, 7.2, 7.3, 8.1, 8.2, and 8.3 of Sundstrom. (See also the class log here.)
There will be a review session for the final exam on Tuesday 5/9/17, 7:00PM--8:30PM, in LGRT 204.
Please try the review problems here before the review session.

You are allowed one letter-size sheet of notes (both sides) for each exam. Calculators, additional notes, and the textbook are not allowed on exams and quizzes. You should bring your student ID (UCard) to each exam.

Your course grade will be computed as follows: Homeworks and quizzes 30%, Midterm exams 20% each, Final exam 30%.

2.17: Concepts - Mathematics


6 unit subject -- Spring 2007

This year, the subject 2.035 will cover selected topics from Linear Algebra and the Calculus of Variations. It will be aimed mainly (but not exclusively) at students aiming to study mechanics (solid mechanics, fluid mechanics, energy methods etc.), and the course will introduce some of the mathematical tools used in these subjects. Applications will be related mostly (but not exclusively) to the microstructures of crystalline solids.

  • Instructor: Rohan Abeyaratne, 3-173, x3-2201, [email protected]
  • Term: Spring Term 2007
  • Prerequisites: Matrices and Multivariable Calculus
  • Time: Tuesdays and/or Thursdays 11:00-12:30 (6 unit subject 14 classes)
  • Place: Room 1-134
  1. J.K. Knowles, Linear Vector Spaces and Cartesian Tensors, Oxford, 1998
  2. Rohan Abeyaratne, Lecture Notes on The Mechanics of Elastic Solids: Volume 1: A Brief Review of Some Mathematical Preliminaries, free e-book, 2006. Download from

Tentative Course Outline:

  • Linear vector spaces
  • Euclidean vector spaces
  • Linear transformations
  • Cartesian tensors

Calculus of Variations: (6 classes)

  • First variation
  • Second variation.
  • Variational principles in mechanics.
  • Approximate solutions.

Grading: Midterm = 40%, Final Exam = 60%

Further references: (On reserve at Barker Library)

  1. I.M. Gelfand, Lectures on Linear Algebra, Dover, 1989.
  2. P. R. Halmos, Finite Dimensional Vector Spaces, Van Nostrand-Reinhold, 1958.
  3. I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice Hall, 1963.
  4. M. Giaquinta and S. Hilderbrandt, Calculus of Variations I, Springer, 1996.
  5. J.L. Troutman, Variational Calculus with Elementary Convexity, Springer-Verlag, 1983.

Tentative Course Schedule

Assigned Readings and Problem Sets:

  • Read pages 1-8 (Knowles)
  • Key concepts: Vector Space, Linear Independence, Dimension of a Vector Space, Basis for Vector Space, Components of a Vector
  • Work problems 1.1 to 1.11 (Knowles)
  • Read pages 9-17 (Knowles)
  • Key concepts: Scalar Product, Length of Vector, Distance between Vectors, Angle between Vectors, Orthonormal basis
  • Work problems 1.12 to 1.20 (Knowles)
  • Read pages 18-20, 23-26 (Knowles)
  • Key concepts: Linear Transformations, Invariant Subspace, Eigenvalue problem
  • Work problems 2.1, 2.3, 2.6, 2.17 except questions about singular/non-singular/inverse transforms (Knowles)
  • Read Chapter 2 (Knowles)
  • Key concepts: Null Space, Singular/Non-Singular Linear Transformations, Inverse, Components of a Linear Transformation
  • Work problems 2.1-2.5, 2.8, 2.9, 2.11, 2.15-2.17
  • Read pages 27-32, 42-46 (Knowles)
  • Key concepts: Components of a Linear Transformation, Components in Different Bases, Scalar Invariants, Cartesian Tensors, Symmetric Tensors, Skew-Symmetric Tensors
  • Work problems 3.1-3.12 (Knowles)
  • Read pages 42-52 (except Tensor Products), 56-57 (Knowles)
  • Key concepts: Eigenvalues of a symmetric tensor, Principal basis, Positive-Definite Tensor, Orthogonal Tensor, Proper/Improper Orthogonal Tensor
  • Work problems 3.13-3.18, 3.20, 3.24-3.26 (Knowles)
  • Read pages 44, 57-59 (Knowles), Chapters 2 and 3 (Abeyaratne)
  • Key concepts: Tensor Product of 2 Vectors, Polar Decomposition of a Non-Singular Tensor
  • Work problems 3.3-3.7, 3.11-3.16, 3.22, 3.23 (Knowles)

Midterm Exam: (April 3, and April 3-5)

  • Part-1: In-Class. Tuesday April 3, 11:00 AM - 12:30 PM. You can use your own handwritten class notes only.
  • Part-2: Take home. Due Thursday April 5 at 11 AM. You can use Knowles and your own handwritten class notes only.
  • Old Exam: Mid-Term-Exam-Part-1 , Mid-Term-Exam-Part-2

Final Exam: (Distributed May 8, due May 15)

  • The final exam will be entirely take home.
  • The exam will be given out in class on May 8.
  • Your solutions should be turned in no later than 11 AM on May 15.
  • All problems will be on Calculus of Variations.
  • There will be 6 problems of which you can work any 5. You should not devote more than 2 hrs on any one problem.
  • The final exam will count for 50% of your grade.

Math Extended Essay Topics & Editing Assistance at

In some math courses, especially in college or university, the teacher may assign an essay paper project on math extended essay topics. At first, it may seem really daunting. Solving equations differ from writing about them, and initially, you will feel unfamiliar about the topics. But we are here to help you out with any challenging mathematical home assignment. It is a completely doable thing for our experts.

Properties of Division of Integers – Importance

Knowing the properties of the division of integers is important because most of the candidates often get confused between the properties of multiplication and division. Both the operations have different properties and are mandatory to follow all the rules. The properties help to solve many problems in an easy manner. To recall, integer numbers are positive or negative numbers, including zero. All the properties for multiplication, subtraction, division, and addition are applicable to integers. Integers are denoted by the letter “Z”.

Rules for Division of Integers

Rule 1: The quotient of 2 positive(+) integers is positive(+).

Rule 2: The quotient of a positive(+) integer and a negative(-) integer is negative(-).

Rule 3: The quotient of 2 negative(-) integers is positive(+).

In brief, if the signs of the two integers are the same, then the result will be positive. If the signs of the two integers are different, then the result will be negative.

Integers Division Properties

The division is the inverse operation of multiplication.

Let us take the example of whole numbers,

24/4 which means dividing 24 by 4 is nothing but finding an integer when multiplied with 4 that gives us 24, such integer is 6.

Hence, for each whole-number multiplication statement, there are two division statements.

1. Division of negative integer by positive integer

Whenever a negative number is divided by a positive number, the result will always be negative.

  1. Divide the given number as the whole number first.
  2. Then add, minus symbol before the quotient. Thus we get the final result as a negative integer.

2. Division of positive integer by negative integer

Whenever a positive integer is divided by a negative integer, the result will always be negative.

  1. Divide the given number as the whole number first.
  2. Then add, minus symbol before the quotient. Thus we get the final result as a negative integer.

3. Divide a negative integer by a negative integer

Whenever a negative integer is divided by a negative integer, then the result will always be positive.

  1. Divide the given number as the whole number first.
  2. Then add, plus symbol before the quotient. Thus we get the final result as a positive integer.

4. Closure Under Division Property

Generally, the closure property is, if there are 2 integers, then the addition or subtraction of those integer results in an integer. But integers division does not follow closure property.

Let us consider the pair of integers.

(-12)/(-6) = 2 (Result is an integer)

(-5)/(-10) = -1/2 (Result is not an integer)

From the above examples, we conclude that integers are not closed under division.

5. Commutative Property of Division

The commutative property states that swapping or changing the order of integers does not affect the final result. Integers division does not follow commutative property also.

Let us consider the pairs of integers.

From the above examples, we conclude that integers are not commutative for integers.

6. Division of Integer by Zero

Any integer divided by zero gives no result or meaningless result.

When zero is divided by an integer other than zero it results in zero.

7. Division of Integer by 1

When an integer is divided by 1, it gives the result as 1.

The above example shows that when a negative integer is divided by 1, it gives the same negative integer.

Dividing Integers Properties Examples

Verify that a/(b+c)≠(a/b)+(a/c) when a = 8, b = – 2, c = 4.

Hence the above equation is verified.

(– 80)/(4) is not the same as 80/(–4). True/False

As (– 80)/(4) = 80/(–4), therefore the above statement is false.

Solution: [(– 8)+4)]/[(–5)+1]

From the question, (-15625)/(-125)

Find the value of [32+2*17+(-6)]/15

How to Divide Whole Numbers?

  1. After dividing the first digit of the dividend by divisor, if the divisor is a larger number than the first digit of the dividend, then divide the first 2 digits of the dividend by divisor and so on.
  2. Always, write the quotient above the dividend.
  3. Multiply the quotient value by the divisor and write the product value under the dividend.
  4. Subtract the product value from the dividend and bring down the next digit of that dividend.
  5. Repeat solving from Step 1 until there are no digits left in the dividend.
  6. Finally, verify the solution by multiplying the quotient times the divisor.

Hope you can now get the complete information on the Properties of the Division of Integers. Get the latest updates on all types of mathematical concepts like Integers, Time and Work, Pipes & Cisterns, Ratios and Proportions, Variations, etc. Follow all the articles to get complete clarity on the Integers topic. Stay tuned to our site to get the complete data or information regarding mathematical concepts.


To complete this section, you must know how to write inequalities in interval notation. See the section titled &ldquoIntervals,&rdquo pages 337-338 of the textbook.

Calculus is the area of mathematics that specializes in solving problems by establishing the function or functions that represent the quantities involved. In this introductory course, you will learn to apply different types of functions to solve problems, and it is our hope that, in the process, you will come to appreciate how powerful the concept of function is for problem solving. But before we discuss functions, we must consider variables and relations.

In mathematics, quantities are called &ldquovariables.&rdquo Quantities that do not change are called constant variables or constants those that do change are called changing variables or simply variables.

Definition 2.1. A variable is a quantity that may or may not change (vary) according to what it represents.

You may think that this definition contains a contradiction, but what we intend is to give ourselves the flexibility of changing a constant into a variable if we need to. In this course, all variables are real numbers.

Example 2.1. Your age is a changing variable. It changes (varies) every year while you age, but it becomes a constant on your death.

Example 2.2. Income is a changing variable (increasing, one hopes). It changes according to cost of living adjustments, promotions, etc.

Example 2.3. The distance of a traveling object from its origin is a changing variable.

Example 2.4. The volume of a wooden box is a constant variable. [Why?]

Example 2.5. The volume of a melting ice cube is a changing variable. [Agree?]

In general, it is easy to decide when a variable is constant or changing. Consider the following exercises.

  1. Is the number of children in a family a changing or a constant variable?
  2. Is the acceleration of Earth&rsquos gravity a constant variable?
  3. Is the number of days in a year a constant or a changing variable?

When working with variables, it is important to know which quantities are changing, and which are constant.

When we identify one or more variables, we name them for easy reference therefore, different variables must have different names. Once a variable is named, we represent it as a symbol this is the first step in abstract thinking&mdashwhat mathematicians call &ldquomathematical modeling.&rdquo The type of symbol is not important, but to facilitate the manipulation of the variables, we tend to give them symbols that are related to what they represent, using as few letters or other characters as possible. Since each person could choose a different symbol for each variable, once we have assigned a symbol, we must let others know what the symbol represents. That is, we assign names and their corresponding symbols to the variables in order to define them.

Always define the variables you are using and give different names to different variables.

It is also convenient to reflect on the possible range of values of the variables we define.

Example 2.6. We could assign the symbols I and $ to the variables that refer to two different types of income. We can define them by stating: &ldquoWe denote by I the income earned by an employee before the year 2000, and by $ the income earned by the same employee after the year 2000.&rdquo What are the possible values for I and $ ? We expect that I and $ are positive nonzero real numbers that is, I > 0 and $ > 0 .

Example 2.7. To the variable &ldquoage&rdquo of a person we could assign the letter A and state: &ldquoThe age (in years) of a person is denoted by A . &rdquo The variable A is positive or zero, and we are probably safe to assume that it is less than 120 hence, 0 ≤ A < 120 that is, the range of values for A is the interval [0,120).

Example 2.8. The variable &ldquodistance&rdquo can be represented by s , and we can say: &ldquo . . . the distance $s$ of a traveling object . . . . The variable $s$ is positive that is, s ≥ 0 .

  1. What symbol would you give to the variable &ldquovolume of an ice cube&rdquo? How would you define it?
  2. How would you define the variable &ldquothe body temperature of a living human being&rdquo? What is the possible range of values for this variable?

Having defined the variables, we must try to understand the possible relationships among them.

Example 2.9. The &ldquoincome&rdquo of an employee is related to the &ldquonumber the employee&rsquos working hours.&rdquo So we have a relation between the changing variables &ldquoincome&rdquo and &ldquonumber of worked hours.&rdquo

Example 2.10. The cost of mailing a letter depends on its weight, so we have a relation between the changing variables &ldquoweight of a letter&rdquo and &ldquocost of postage.&rdquo

Example 2.11. The &ldquoarea of any triangle&rdquo depends on the &ldquolength of its base and its height.&rdquo So we have a relation between the variables &ldquoarea of a triangle&rdquo and &ldquolength of base and height.&rdquo

Example 2.12. The &ldquodistance&rdquo of a traveling object depends on the &ldquotime&rdquo traveled.

Observe that in the examples give above, we have related variables because we found a relation of dependency between them. But we can also relate variables based on other criteria.

Example 2.13. We can relate the &ldquoSocial Insurance Number&rdquo of a Canadian citizen with his or her &ldquoage in the year 1998.&rdquo

Example 2.14. Your &ldquoage&rdquo can be related to your &ldquoweight.&rdquo

Definition 2.2. A relation is an association between two variables or among several variables.

The criterion used to associate the variables must be established&mdashthat is, defined&mdashand to establish it, we must introduce a mathematical model.

Example 2.15. In Example 2.9, we have the relation between the variable &ldquoincome&rdquo and the variable &ldquonumber of worked hours.&rdquo We start by defining these variables. Let I be the income and w be the number of worked hours. There are two ways to define the relation between these two variables:

by ordered pairs. The pair ( w , I ) indicates that I and w are two related variables. To establish the relationship between them, we say that the relation is the set of all pairs ( w , I ) where I is the income earned for w worked hours, and we write { ( w , I ) | I is the income earned for w worked hours } . We can also give a name to this relation, say T , and we write

The brackets $<>$ are read as &ldquothe set of,&rdquo and the symbol | is read as &ldquosuch that&rdquo or &ldquowhere.&rdquo So the expression is, &ldquo T is the set of all pairs of the form ( w , I ) such that I is the income for w worked hours.&rdquo

by explicit definition of the relation. We decide on the name of the relation first, say  T and then we write w T I to indicate that w is related to I by T . We define this relation explicitly, as follows:

$wTI$ if and only if $I$ is the income earned for $w$ worked hours.

In calculus, we prefer the notation that uses ordered pairs.

Example 2.16. In Example 2.10, we have the variables cost of postage and weight, which we define as P and w , respectively. If we name the relation C , we either write

to be read as &ldquo C is the set of all pairs ( w , P ) such that P is the cost of postage of a piece of mail of weight w ,&rdquo or we write w C P iff [1] P is the cost of posting a piece of mail of weight w .

We read &ldquo P is related to w by the relation C if and only if P is the cost of posting a piece of mail of weight w .&rdquo

Example 2.17. In Example 2.11, we define A as the area of a triangle, and b and h as the base and height of the triangle, respectively. We name the relation T , and we define it as

( b , h ) T A iff A is the area of a triangle with base b and height h .

Example 2.18. Refer to Example 2.13. We will let SIN be the social insurance number of a person, and we let A be that person&rsquos age in the year 1998. We name the relation K , and we have

$ K = (mbox,A) | A $ is the age, in the year 1998, of the person associated to $mbox$

$ mbox K,A $ iff $ A $ is the age in the year 1998 of the person associated to $mbox$.

Example 2.19. The relation S between a positive integer n and its square root q is written as

Since we already have a symbol to denote the square root of a positive integer n , we can also write S = { ( n , n ) | n   is   a   positive   integer } .

  1. How would you state the relation in Example 2.12?
  2. How would you read the relation below?
  3. T = { ( ( b , h ) , A ) | A   is   the   area   of   a   triangle   of   base   b   and   height   h }
  4. How would you read the relation below?
  5. ( b , h )   T A iff A is the area of a triangle with base b and height h
  6. Define the variables &ldquoage&rdquo and &ldquoweight&rdquo and establish their relation as given in Example 2.14.
  7. How would you read the relation below?
  8. T = { ( A , y ) | A   is   your   age   in   the   year   y }

Once a relation is established (defined), we can decide which particular pairs belong to it.

Definition 2.3. We say that a pair ( a , b ) belongs to a relation T if a is related to b by T , and we write ( a , b ) ∈ T . If a is not related to b by T , then we write ( a , b ) ∉ T .

A relation T is well defined if we can determine whether any given pair ( a , b ) belongs to the relation T or not.

From Example 2.15, we can see that if the pair ( 1 2 , 5 0 0 ) ∈ T , then for 1 2 hours of work the income is $$500mbox<.>00$, we also see that the pair ( 1 2 , 0 ) ∉ T , since the income cannot be $mbox<.>00$ for $12$ hours of work. It is also clear that ( − 4 , 100 ) ∉ T .

In Example 2.16, we write ( 2 0 , 0 . 5 0 ) to indicate that, for a piece of mail weighing 2 0 grams, the cost of postage is $ 0 . 5 0 .

In Example 2.17, we have ( ( 4 , 3 ) , 6 ) ∈ T and ( ( 5 , 2 ) , 7 ) ∉ T .

In Example 2.19, we have ( 9 , 3 ) ∈ S , ( 9 , - 3 ) ∈ S and ( 9 , 4 ) ∉ S . [Why?]

Observe that the order in which the pair is presented matters, the pairs ( 1 2 , 5 0 0 ) and ( 5 0 0 , 1 2 ) in the relation T of Example 2.15 are not the same: as we said ( 1 2 , 5 0 0 ) means that the income for 1 2 hours of work is $ 5 0 0 . 0 0 , the pair ( 5 0 0 , 1 2 ) says that for 5 0 0 hours of work the income is $ 1 2 . 0 0 .

  1. Consider the relation
  2. M = { ( s , A ) | A   is   the   area   of   a   square   of   side   s } .
  3. Identify a pair that is in M and a pair that is not in M .
  4. Indicate whether the pair ( 3 4 , 2 0 0 0 ) belongs to the relation
  5. T = { ( A , y ) | A   is   your   age   in   the   year   y } .

We use a special notation when there is a relation of dependency between variables. In Example 2.15, the income I depends on the number of worked hours w , and we express this fact by writing I ( w ) . In Example 2.16, we write P ( w ) to indicate that the cost P of posting a piece of mail depends on its weight w . In Example 2.17, we are told that A ( b , h ) &mdashthe area of a triangle depends on its base b and height h . In Example 2.12, the distance traveled $s$ depends on the time traveled t hence, s ( t ) . However, in Example 2.18, the variable SIN does not depend on the variable A . This is not a relation of dependency, so it is not correct to write A (SIN).

Definition 2.4. If a variable F depends on the variable m , we write F ( m ) . In this case, we refer to F as the dependent variable, and m as the independent variable.

For convenience, the relation of dependency takes the name of the dependent variable F . Moreover, the pair ( a , b ) belongs to the relation F iff F ( a ) = b .

Observe that if the pair ( a , b ) belongs to the relation F , then a is the independent variable and b is the dependent variable. When we write F ( a ) = b , we indicate two things: F depends on a , and the value F that corresponds to a is b . So we write

The relation in Example 2.15 is a relation of dependency therefore, it is no longer referred to as T instead, it is called I . Note that I ( 1 2 ) = 5 0 0 because ( 1 2 , 5 0 0 ) belongs to the relation. The statement I ( 1 0 ) = 7 0 indicates that for 1 0 worked hours, the income is $ 7 0 . 0 0 that is, the pair ( 1 0 , 7 0 ) belongs to the relation I . We also recognize that ( - 4 , 1 0 0 ) does not belong to the relation hence, I ( - 4 ) is undefined.

In Example 2.16, P ( 2 0 ) = 0 . 5 0 in Example 2.17, A ( 3 , 4 ) = 6 and in Example 2.19, S ( 9 ) = 3 , S ( 9 ) = - 3 and S ( 9 ) ≠ 4 .

  1. Is the relation in Example 2.11 a relation of dependency? How would you write it?
  2. If $s$ is the distance traveled (in metres) and t is the time (in seconds), then s ( t ) is the relation of dependency between $s$ and t . How would you write the statement, &ldquothe object travels 5 0 m in 3 0 seconds,&rdquo in symbolic form?
  3. Establish the relation of dependency between the following pairs of variables.
    1. temperature T , and wind chill factor c
    2. cost of living L , and taxes t
    3. amount of interest paid r , and principal amount P

    The problems that calculus investigates are those that can be represented by a relation of dependency, where the dependent variable is uniquely determined by the value of the independent variable or variables. We call these special relations &ldquofunctions.&rdquo Some functions have to do with variables that change with respect to time, such as distance, velocity, area, volume, population, etc. [Note that, for a given time, there is only one distance, one velocity, one area, one volume or one population.]

    In this course, we consider only functions of one independent variable functions of more than one independent variable, such as that shown in Example 2.17, are studied in more advanced courses.

    Definition 2.5. A function is a relation of dependency F between two variables, such that if the pairs ( a , b ) and ( a , c ) belong to the relation [i.e., if F ( a ) = b and F ( a ) = c ], then b = c . That is, the dependent variable F associated to the independent variable a is uniquely determined.

    If two pairs ( a , b ) and ( a , c ) belong to a relation and b ≠ c , then the relation is not a function.

    Example 2.20. In Example 2.19, the relation $S$ is a relation of dependency, but it is not a function, because $S(9) = 3$ and $S(9) = -3$. That is, it is not a function because both of the pairs $(9,3)$ and $(9,-3)$ belong to $S$.

    Example 2.21. The relation between the area of a square and the length of its side is a function because each length of a square&rsquos side is associated with only one area.

    Example 2.22. The GST we pay for a product of cost $C$ is a relation of dependency, GST($C$), and it is a function. [Why?]

    The relations in Examples 2.15 and 2.16 are functions. [Why?]

    Later, we will need to solve problems by establishing the function or functions that represent the problem. The key to doing so is understanding the relation of dependency between the variables of the problem.

    We have been paying attention to the range of values of the dependent and independent variables. These ranges of values are important for identifying the variables we are working with. We have special names for them.

    Definition 2.6. The domain $D_F$ of a function $F(s)$ the largest set of acceptable values of the independent variable $s$.

    The range or rank R F of a function F is the largest set of values of the dependent variable F .

    Thus a pair ( s , F ) belong to the function F , if $s$ is in $D_F$ and F is in R F .

    To find the domain of a function, ask yourself, &ldquoWhat are the possible values for the independent variable?&rdquo

    If we have a function F with independent variable v &mdashthat is, F ( v ) &mdashwhat we try to do next is to find a mathematical expression that gives the value of F for each value of v . This mathematical expression is the mathematical representation (model) of the function F . This model is also referred as a &ldquoformula&rdquo for F . In this course, the formula we want must involve constants and the variable v only. That is, we want to express F only in terms of the independent variable v .

    Example 2.23. If A ( s ) is the function that gives the area A of a square of side $s$, then the mathematical representation of A is s 2 . [Why?] So, we write A ( s ) = s 2 . In this case, A is expressed in terms of s . The domain and range of the function A is the interval D A = R A = [ 0 , ∞ ) . [Why?]

    The formula A ( s ) = s 2 can be used to solve any problem that involves the area of a square. Observe that A ( 1 0 ) = 1 0 0 , this means that the area of a square of side 1 0 is 1 0 0 . We say that &ldquothe value of A at 1 0 is 1 0 0 , &rdquo or &ldquothe area A is 1 0 0 if $s$ is 1 0 .&rdquo

    Example 2.24. If $< ext>(C)$ is the function of Example 2.22, above, then $< ext>(C) = 0.05C$. [Agree?] What is the formula for the total cost $T$ (including the GST) that we would pay for a product of cost $C$? That is, what is the mathematical model (formula) in terms of $C$ of $T$? What is the value of $T$ for an item priced at $$165.45$?

    Example 2.25. If income paid I is at a rate of $ 1 0 . 5 0 per hour, then the mathematical model of the function I ( w ) of Example 2.15 is I ( w ) = 1 0 . 5 0 w . The income for 1 0 worked hours is I ( 1 0 ) = 1 0 5 . 0 0 . Although we can multiply 1 0 . 5 0 by a negative number, it does not make sense to say that the value of I at - 2 is - 2 1 that is, it is true that I ( - 2 ) = 1 0 . 5 0 ( - 2 ) = - 2 1 , but the meanings of I and w are lost here. The domain and range of this function are the interval D I = R I = [ 0 , ∞ ) , and we say that we cannot evaluate the function I at negative numbers, not because we cannot multiply 1 0 . 5 0 by a negative number, but because the variable w represents worked hours, and this variable takes only positive values.

    We now have the concepts of mathematical domain, as defined in Definition 2.6, above, and of physical domain, as the set of acceptable values of the independent variable, according to what the independent and dependent variables involved represent.

    Example 2.26. If $A$ is the area of an equilateral triangle with side $s$, then the formula of the function $A(s)$ in terms of $s$ is $(sqrt 3 )/4$ that is,

    To arrive at this relation, we must study the area of equilateral triangles. The area of any triangle is half of the product of the base and the height. The base of the equilateral triangle is $s$. To find the height h (height is a variable) we use Pythagoras&rsquo Theorem.

    Figure 2.1. Equilateral triangle with side equal to $s$

    According to Pythagoras&rsquo Theorem,

    since h and $s$ are positive. [Why?] Therefore,

    and the area is as indicated.

    An equilateral triangle with side 5 6 has an area of

    A ( 5 6 ) = 5 6 2 3 4 ≈ 1 3 5 7 . 9 3 .

    In words, the area of an equilateral triangle with side 5 6 is approximately equal to 1 3 5 7 . 9 3 . What are the domain and range of A ? What is the value of A at 9 ? What is the interpretation of A ( 4 ) = 4 3 ?

    Note: We are ignoring the units for the time being. Be aware, however, that you will be expected to use the correct units in assignments or examinations.

    Example 2.27. If the area of a rectangle is 1 6 m 2 , what is the formula for P ( l ) , where P is the perimeter of the rectangle and l is the length of one of its sides? In other words, how can we express P in terms of l ?

    Step 1

    We start with a picture of the problem:

    Figure 2.2. Rectangle with length equal to l , and width equal to w

    Step 2

    $l$ is the length of the base
    $w$ is the height of the rectangle

    $P$ is the perimeter of the rectangle
    $A$ is the area

    Step 3

    We relate the variables to what we already know about the perimeter and area of the rectangle hence, P = 2 l + 2 w and A = l w = 1 6 .

    Since we want to find a formula for the perimeter that depends only on l , we must substitute for the variable w an expression with only l s. Hence,

    w = 1 6 l and P = 2 l + 2 1 6 l = 2 l + 3 2 l ,

    and we conclude that P ( l ) = 2 l + 3 2 l .

    We have found the formula that gives the perimeter P of a rectangle of side l and area 1 6  m 2 . We can find the value of P for any positive value of l &mdashall we have to do is to apply the formula. For example,

    P ( 6 ) = 2 ( 6 ) + 3 2 6 = 5 2 3 .

    We call this process &ldquoevaluating P at 6 ,&rdquo and the answer is expressed as, &ldquothe value of P at 6 is $52/3$.&rdquo

    The domain and range of P are the interval ( 0 , ∞ ) . [Agree?]

    What is the perimeter of a rectangle of area 1 6 m 2 if one of its sides is 5 m?

    1. A rectangle has perimeter of $20$ m. Express the area of the rectangle as a function of the length of one of its sides.
    2. Express the surface area of a cube as a function of its volume.
    3. An open rectangular box with volume 2 m 2 has a square base. Express the surface area of the box as a function of the length of a side of the base.
    4. The cost of renting a car is $ 5 0 . 0 0 plus 4 3 cents per kilometre traveled.
      1. Define the variables that correspond to this problem, and establish the relation of dependency between them.
      2. Find the formula that gives the rental cost in terms of the kilometres traveled.
      3. Give the domain and range of this function.
      4. What is the cost for renting a car in Edmonton to travel to Calgary? Include the appropriate taxes.

      The formulas or mathematical models of functions can take different forms. We cannot always give a single formula for a function. For instance, in Example 2.16, the cost of postage P is fixed depending on a certain range of values of w . That is, P is $ 0 . 5 0 if 0 < w ≤ 3 0 and P is $ 1 . 0 0 if 3 0 < w ≤ 1 0 0 . In this case, we write

      As you can see, P ( 1 0 2 ) = 1 . 7 0 and P ( 1 0 0 ) = 1 . 0 0 . The range of P , written as a set, is < 0 . 5 0 , 1 . 0 0 , 1 . 7 0 , 2 . 4 5 >, and the domain of P is the interval ( 0 , 5 0 0 ] . See Example 2.10.

      We may also be able to give different formulas for different ranges of the independent variable. Consider a function defined as follows:

      First, we notice that the values for $s$ are in the intervals ( 0 , 5 0 ) and ( 5 0 , 6 0 ] , that is ( 0 , 5 0 ) ∪ ( 5 0 , 6 0 ] is the domain of g . Then 5 0 and 0 are not in the domain, in other words there is no value for g at 0 or 5 0 . And we say that g is undefined at 0 and 5 0 . Observe also that g is not defined at any number bigger than 6 0 (e.g., 6 0 . 0 0 1 or 1 9 8 ), nor is it defined for negative numbers.

      James Stewart, the author of your textbook, calls the functions defined in this fashion piecewise functions, and we will adopt this name. See Examples 7 and 8 on pages 14 and 15 of the textbook.

      1. For each of the piecewise functions (i)-(iv), below,
        1. ( f(x) = <eginx & mbox x + 1 & mbox end> )
        2. ( f(x) = <egin2x + 3 & mbox 3 − x & mbox end> )
        3. ( f(x) = <eginx + 2 & mbox x^2 & mbox end> )
        4. ( f(x) = <egin-1 & mbox 3x + 2 & mbox 7 - 2x & mbox end> )
        1. give the domain of the function f .
          Hint: For function (iv), see Box 6 on page 341 of the textbook.
        2. give the values of f ( 1 ) and f ( - 1 ) .

        Finally, we may not be able to find one given formula for a function, but we may be able to give a table of pairs that belong to the function (see Example 4 on page 12 of the textbook). This is the case when we make a finite number of observations, such as temperature, population, number of consumers of a product, etc.

        Example 2.28. The table below shows the population of a certain city recorded every two years for 10 years.

        Table showing the population, P, of a certain city recorded on a particular year, Y
        P Y
        56000 1990
        56800 1992
        57000 1994
        57500 1996
        57850 1998
        58000 2000

        The variables are the population P and the year Y when the population was recorded. The function is P ( Y ) . This table indicates that the pairs

        $(1990,56000)$, $(1992,56800)$, $(1994,57000)$, $(1996,57500)$, $(1998,57850)$, $(2000,58000)$

        are in the function P . So P ( 1990 ) = 56000 , P ( 1992 ) = 56800 , and so on.

        If the data show a certain pattern or uniformity, one would hope to be able to associate a formula to the function. Different techniques are available.

        1. If a metal rod is being heated, its volume depends on the temperature. If T denotes the temperature (in °C) and V the volume (in m 3 ), then V ( T ) , and the domain of the function V is all possible values of the temperature T . We have the following recorded volumes: Table showing how the volume of metal rod, V, varies with the temperature, T
          1. What do you observe about the volume of the rod with respect to the temperature?
          2. What is the value of V for T = 80 ?
          3. Estimate the volume of the rod when the temperature is 73 °C.

          So far we have been working with functions for which the variables have a very specific interpretation however, to get to the problems we want to solve, we must learn to work with functions in a general sense. That is, we must learn to work with functions and their corresponding formulas when the variables do not have any specific meaning. We must learn about the general properties of functions, and we must learn to do different operations with them. Once we can perform these operations, we will be able to solve problems using functions. This learning approach is the same as the one you used when learning to read: you learned the basic alphabet first.

          For example, in general terms, we defined the domain of a function as the set of possible values of the independent variable. In Example 2.23, the variables of the function A ( s ) have specific meaning. Therefore, the domain of the function A ( s ) is the interval [ 0 , ∞ ) because s represents a positive quantity: the length of one side of a square. However, if we ignore the meaning of s as a length, and consider the function A ( s ) = s 2 in general terms, then we see that we can evaluate the function A at any value of s , including negative numbers:

          Hence, the domain of the function A ( s ) = s 2 in general is all of the real numbers.

          Working in general terms has its advantages, because different practical problems can be solved with functions that share the same formula.

          Example 2.29. If we consider the function

          T ( t ) = t 2 + 5 t − 1 2 t − 3 ,

          in general terms, then we can evaluate the function T for any value of t except t = 3 ∕ 2 . For this value of t , the denominator of this expression is zero that is, the expression is undefined. So, for any value of t other than t = 3 ∕ 2 , it is possible to apply the formula of T .

          You can check that T ( 3 ∕ 4 ) = - 2 . 0 8 3 3 3 , and that T ( - 5 . 6 7 ) = - 0 . 1 9 5 1 8 .

          Since 3 ∕ 2 is the only value for which T is not defined, the domain of T is all real numbers except 3 ∕ 2 . In interval notation, we would write

          If T represents the temperature of an object at time t , then the possible values of t are all positive real numbers except t = 3 ∕ 2 . In interval notation, the domain is

          Example 2.30. The domain of the function

          consists of all numbers such that c - 6 ≥ 0 (since only positive numbers have real square roots). So, the domain of F is all real numbers greater than or equal to 6 that is, c ≥ 6 , or in interval notation, [ 6 , ∞ ) .

          We can also write, F ( c ) = c - 6 only for c ≥ 6 .

          Example 2.31. What is the domain of

          To find the domain, we look for those values a for which we may have problems evaluating H ( a ) , either because the denominator is 0 for this particular value, or because the square root is negative. We see that what we need is 2 a - 8 > 0 , so a > 4 . So the domain of H is all values a that are strictly greater than 4&mdashin interval notation ( 4 , ∞ ) .

          Example 2.32. For the function

          we see that s - 1 ≥ 0 (the square root must be non-negative), and the divisor must be nonzero.

          If 2 cos  s - 1 = 0 then cos  s = 1 ∕ 2 , and therefore,

          s = π 3 + 2 k π and s = - π 3 + 2 k π , for any integer k .

          Since s ≥ 1 , the function is undefined for

          s = π 3 , and for all s = ( 6 k ± 1 ) π 3 , for any k > 0 .

          We conclude that the domain is all numbers s ≥ 1 , except

          s = π 3 , and s = ( 6 k ± 1 ) π 3 , for any integer k > 1 .

          If we have a function with its corresponding formula, we need to understand what we mean by &ldquoevaluating the function.&rdquo

          at 9 , we replace the independent variable a by 9 in the formula:

          H ( 9 ) = 9 - 6 2 ( 9 ) - 8 = 3 1 0 .

          We can also evaluate the function at any other expression in the same way. For example, we can evaluate this function at x + h by replacing the independent variable a by x + h :


          f ( x ) = 5 x + 4 x 2 - 3 x + 2

          Express the domain of each of the functions below in interval notation.


          [1] It is customary to write iff instead of &ldquoif and only if&rdquo.

          Share this knowledge with your friends!

          Last reply by: Professor Selhorst-Jones
          Thu Jul 13, 2017 1:33 PM

          Post by John Stedge on July 13, 2017

          For solving equations of unusual things i got y^2=27/z-x^2 would this also be considered correct?

          Last reply by: Professor Selhorst-Jones
          Wed Oct 7, 2015 11:14 AM

          Post by Peter Ke on October 4, 2015

          For the percent example at 11:00. I thought 30% off means multiplying 70 by .30 and same for 20% off from the coupon. But, instead you did 100% - 30% = 70% and multiply 70 by .70, why is that? Can you explain it pls?

          Last reply by: Professor Selhorst-Jones
          Sun Apr 20, 2014 8:54 PM

          Post by Tommy Lunceford on April 15, 2014

          During the Domain and Range example, does the +3 change to a negative -3 just because its under the square root sign or because of the negative in front of the square root symbol?

          If that negative sign in front of the square root symbol was not there, would it still turn into a -3 or remain positive as it is under the square root symbol?

          Math Concept Petting Zoo: Part 1

          • If you want to do well on the Math section of the SAT, you need to be familiar with a wide variety of concepts. This is the first of two lessons covering the vast majority of the concepts on the SAT.
          • In the first lesson, we'll generally cover topics relating to the interaction of numbers, equations, and functions. The second lesson will focus more on geometry, shapes, and a few miscellaneous topics.
          • In the Zoo, each concept will be presented by showing a question based on the concept. While there will be some explanation of the questions, the point of the Zoo is to quickly review concepts, not teach them.
          • As you watch this lesson, make sure you have a pencil and paper. Make sure you could solve each problem on your own before the answer is revealed. If you're not absolutely certain you could, pause the video and try to work it out yourself before you continue watching.
          • Whenever you find a concept you don't know well, write it down and note how much help you need.
          • Later, after you've finished watching the lesson, go study the concepts you wrote down. Begin with the ones you find most difficult. Watch the corresponding SAT Math-specific lessons, and if you still need more help, browse, search the web, or talk to someone who is good at math.
          • Here's a list of all the concepts in this half of the Zoo:
            • Intersection and Union,
            • Sequences / Patterns,
            • Even/Odd Properties,
            • Primes,
            • Percent,
            • Percent Change,
            • Average: Mean, Median, Mode,
            • Absolute Value,
            • Solving Equations for Unusual Things,
            • Distance = Speed · Time,
            • Radicals,
            • Exponents,
            • Concept of a Function,
            • Domain and Range,
            • Function Transformations,
            • Expanding Factors (aka: FOIL),
            • Factoring,
            • Solving Polynomials.

            Math Concept Petting Zoo: Part 1

            Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

            2.17: Concepts - Mathematics

            Department Chairperson: Rohan Attele

            Faculty: Victor K. Akatsa, Kapila Rohan Attele, Jan-Jo Chen, Johng-Chern Chern, Raymond

            H. Y. Chu, John F. Erickson, Dawit Getachew, Lun-Pin Ho, Daniel J. Hrozencik, Lixing (Adam) Jia, Paul M. Musial, Sharon O’Donnell, Howard A. Silver (Emeritus), Richard J. Solakiewicz, Marjorie M. Stinespring (Emeritus), Luis Vidal-Ascon, Guang-Nay Wang, Jesse Y. Wang, Asmamaw Yimer, George I. Zazi

            The Department of Mathematics and Computer Science offers a Bachelor of Science degree in Mathematics with two options: (a) Mathematics (b) Secondary Teaching. Within the Mathematics option, a student may take an actuarial science concentration. The built-in flexibility of the Mathematics option will prepare students for careers in banks, insurance, industry, government, or to pursue advanced degrees in mathematics.

            Completion of the Secondary Teaching option in Mathematics qualifies students for an Initial Type 09 Illinois High School Certificate with a high school endorsement in mathematics for grades 9–12, and a middle grade endorsement in mathematics for grades 6–8.

            Certification requires the successful completion of the Illinois Certification Tests of Basic Skills, Mathematics, and Assessment of Professional Teaching (Secondary 6–12). The secondary teaching program is accredited by the National Council of Teachers of Mathematics (NCTM), and meets Illinois State Board of Education (ISBE) standards in mathematics education.

            The department offers a minor sequence in mathematics. It will provide students majoring in other disciplines access to more potent professional tools, and help them to gain a deeper understanding of their own fields.

            All entering freshmen and transfer students are required to take the university placement examination in mathematics. These students may not register for any mathematics or computer science course until they have taken the examinations. These examinations are designed to place students into the appropriate mathematics course from Applied Intermediate Algebra to Calculus. Students may not use MATH 0880/088, 0900/090, 0950/095, 0980/098, 0990/099, 160 or 161 toward satisfying general education mathematics or university graduation requirements. Credit will not be given for any mathematics course which is a prerequisite for a course in which a grade of C or better has already been earned.

            Mathematics Option (with concentrations in Mathematics and Actuarial Science)

            Admission to the program is contingent upon completion of MATH 1210/163 or MATH 1250/171 with a grade of C or higher, cumulative grade point average of 2.0 or higher, and acceptance by the department.

            The department will not accept D grades in any required major courses or required supportive courses, either as transfer credit or completed at Chicago State University.

            Requirements include completion of 120 semester hours of work: 39 hours in general education 44 hours in mathematics 15 hours of supportive courses 22 hours in electives selected with the departmental advisor’s approval and passing the examination on the state and federal constitutions.

            By demonstrating proficiency, a student may be able to obtain credit for certain mathematics and computer science courses at the recommendation of the department.

            Specific Requirements (Mathematics Concentration)

            The 9 credit hours in Physical and Life Sciences must be selected from BIOL/CHEM/PHYS/PHSCI and include at least one laboratory course.

            Required Courses (44 credit hours)

            MATH 1900/180, 2200/201, 2300/230, 1410/261 or 1415, 1420/262, 2430/263, 2550/271, 4110/342, and 4940/392 one of the following - MATH 4210/327, 4230/308, or 4250/361 one of the following – MATH 4410/358 or 4450/356 one of the following in applied mathematics – MATH 3820/325, 4510/354, 4520/355, 4600/315, 4650/318, 4800/350, or 4840/326 two additional courses selected from the above or MATH 2800/283, 3210/329, 3800/313, 4710/345, 4900/370.

            Required Supportive Courses (15 credit hours)

            Physical and Life Sciences that must be selected from BIOL/CHEM/PHYS/PHSCI: three additional credit hours CPTR 1100/141, nine additional credit hours selected from accounting, biology, botany, chemistry (1550/155 or above) computer science (above 1100/141), economics, mathematics (2810/251 only), physics, zoology. At least two courses must be from the same discipline.

            Elective Courses (22 credit hours)

            22 credit hours of electives selected with the department advisor’s approval.

            Specific Requirements (Actuarial Science Concentration)

            Required Courses (44 credit hours)

            MATH 1900/180, 1410/261 or 1415, 1420/262, 2200/201, 2300/230, 2430/263, 2550/271, 3630/274, 3800/313, 4110/342, 4600/315, 4650/318, and 4940/392 one of the following:

            MATH 2800/283, 3820/325, 4230/308, 4840/326, 4800/350 or 4900/370.

            Required Supportive Courses (15 credit hours)

            Physical and Life Sciences that must be selected from BIOL/CHEM/PHYS/PHSCI: three additional credit hours complete one of the following two sets of courses: CPTR 1100/141, ACCT 2110/110 and 2111/111, and FIN 2660/266 or CPTR 1100/141, ACCT 2110/110, FIN 2660/266 and 3680/368. ECON 1010/101 and 1020/102 are strongly recommended to fulfill the General Education Social Science requirement.

            Elective Courses (22 credit hours)

            22 credit hours of electives selected with the department advisor’s approval.

            Secondary Teaching Option in Mathematics

            To be considered for recommendation for admission to the College of Education, students must have:

            • completed MATH 1420/262 and two 4000/300-level mathematics courses with a grade of C or higher
            • completed with a grade of C or higher, or be concurrently enrolled in MATH 2430/263 and MATH 4110/342
            • passed the Illinois State Basic Skills Examination, and a 2.5 GPA or higher in 1000-level and above mathematics courses.
            • Pass the examination on the state and federal constitutions.
            • Complete 120 credit hours in: General Education 39 credit hours, Area of Specialization 50 credit hours, Professional Education 27 credit hours, Elective Courses 4 credit hours.

            Specific General Education Requirements

            General Education 39 credit hours: the 3 hours in mathematics is satisfied by the major.

            In addition, the nine credit hours in Physical and Life Sciences must include a two-course science laboratory sequence. Also, the nine hours in Social Sciences must be chosen from the following:

            HIST 1300/130 or 1310/131 or POL 1010/101 PSYC 1100/141 and 2040/204.

            Area of Specialization: 50 credit hours

            Required Mathematics Courses (47 credit hours)

            MATH 1900/180, 1410/261 or 1415, 1420/262, 2200/201, 2300/230, 2430/263, 4010/347, 4020/348, 4250/361, 4110/342, 4450/356 or 4410/358, 4600/315, and 4710/345 at least six

            additional hours selected from MATH 2550/271, 2800/283, 3210/329, 3820/325, 4210/327, 4230/308, 4410/358, 4450/356, 4510/354, 4650/318, 4800/350, 4840/326, 4900/370 or 4940/392.

            Required Supportive Courses: (3 credit hours)

            MATH 0920/092 or pass state teacher certification subject matter test in mathematics

            MATH 2810/251 or three additional hours in the Physical and Life Sciences (BIOL/CHEM/PHYS/PH SCI).

            Professional Education (27 credit hours)

            ELCF 1520/152 and 2000/200 PSYC 2020/206 S ED 4301/301 and 4303/303* ELCF 5500/353* READ 4100/306* CAS 3630/363*, MATH 4000/363* and 4005/375*.

            Course must be passed with at least a grade of C.

            * Restricted to students admitted to the College of Education.

            Elective Courses (4 credit hours)

            Sample Curriculum Pattern (Mathematics Option, Mathematics Concentration)

            BümoBrain Live Class Program Refunds

            Cancellations for live classes must be made at least 7 calendar days prior to the first class date of said cancelled program. For instance, if Client cancels a live class program that is scheduled to start on March 20, 2021, then the program should be cancelled no later than March 13, 2021 by 11:59 PM Pacific Time. For such cancellations, Client will receive a full refund within 3 business days. Any cancellation within 7 days of the first class of said program will not be eligible for a refund. Client cannot transfer a credit for such classes for another class for any reason. Should Client want to switch classes, Client must cancel original classes/program and book new classes. Should Client cancel and be eligible for a refund and has not received a refund, please email [email protected]

            Have Questions?

            2.17: Concepts - Mathematics

            Since 1986 the Mathematics section at ICTP has played an important role in fostering mathematics research and education in developing countries. Research is carried out in various fields of Mathematics by the permanent staff, postdocs, and graduate students, as well as by scientific visitors from all over the world.

            Typically, the section organizes from 5 to 10 focused activities a year involving an average of 100 participants. These activities are the core of the section's activities and are crucial for disseminating current mathematics knowledge of the highest level as widely as possible.

            In addition the Mathematics section, like all the other sections at ICTP, participates in the Diploma program. Since 2011 Diploma students can apply to stay on to work on a PhD in Mathematics in a joint program with SISSA.

            The Mathematics section also offers opportunities for postdocs and research fellows click here for latest announcements.

            Once a month, the section organises The Basic Notions Seminar Series to broaden the understanding of some mathematical concepts.

            Mathematics Section is Hiring!

            Deadline extension: Senior Research Scientist to lead Algebraic Geometry research

            Remembering M.S. Narasimhan

            Colleagues to host online memorial meetings on 4 & 7 June

            Celebrating Physics Excellence

            ICTP Prize ceremony will honour 2019 recipients

            In Memoriam

            The Future is in Data

            ICTP among academic and business partners of new data science institute

            ICTP's 2021 Dirac Medal

            Nomination deadline extended to 25 June

            Alicia Dickenstein Wins L'Oreal-UNESCO International Award

            Argentinian mathematician and ICTP Simons Associate honored

            Deadline 15 February

            ICTP Postgraduate Diploma Programme

            Mathematics Role Model

            Carolina Araujo accepts 2020 Ramanujan Prize

            2021 Ramanujan Prize

            Nomination deadline 1 March 2021

            Seminars view all

            Europe/Rome 2021-06-18 14:00:00 2021-07-16 16:00:00 IGAP/MPIM lecture course “From 3-manifold invariants to number theory” Research Group: Geometry and Mathematical Physics Course Type: PhD Course Questions from topology have led to interesting number theory for many years, a famous example being the occurrence of Bernoulli numbers in connection with stable homotopy groups and exotic spheres, but some developments from the last few years have led to much deeper relationships and to highly non-trivial ideas in number theory. The course will attempt to describe some of these new interrelationships, which arise from the study of quantum invariants of knot complements and other 3-dimensional manifolds. [Joint work with Stavros Garoufalidis] Topics to be studied include: * The dilogarithm function, the 5-term relation, and triangulations of 3-manifolds * Quantum invariants of 3-folds (Witten-Reshetikhin-Turaev and Kashaev invariant) definitions and first properties * The Habiro ring (this is a really beautiful algebraic object that should be much better known and in which both of the above-named quantum invariants live) * Perturbative series (formal power series in h) associated to knots * Turning divergent power series into actual functions (this has connections with resurgence theory and involves some quite fun analytic considerations) * Numerical methods (the ones needed are surprisingly subtle) * Holomorphic functions in the upper half-plane (q-series) associated to knots * Modular properties of both the Habiro-like and of the holomorphic invariants These topics are all interconnected in a very beautiful way, formally summarized at the end by a single matrix invariant having different realizations in the Habiro world, the formal power series world, and the q-series world. Although some quite advanced topics will be reached or touched upon, the course assumes no prerequisites beyond standard basic definitions from either topology, number theory, or analysis. To access the course use the following Zoom coordinates: Zoom meeting ID 969 5251 6566 Password 307018 Zoom Link: Online/Presence ICTP [email protected] 18 Jun 2021 - 16 Jul 2021

            » IGAP/MPIM lecture course “From 3-manifold invariants to number theory”

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            Europe/Rome 2021-07-19 08:00:00 2021-07-23 22:00:00 Trieste Algebraic Geometry Summer School (TAGSS) 2021 - Hyperkähler and Prym varieties: classical and new results | (smr 3609) An ICTP Virtual Meeting Hyperkähler manifolds are a class of manifolds with vanishing first Chern class, constituting an active area of current research.  Another fundamental problem concerns moduli space of polarized abelian varieties, studied via Prym varieties and the Prym map to the moduli of abelian varieties.   Hyperkähler manifolds, an overview and some open problems   Hyperkähler manifolds are mainly characterized by their second cohomology. The period maps from the moduli spaces of hyperkähler manifolds to the period domains of their second cohomology are surjective, which is a rare phenomenon happening almost exclusively in weight 1 and weight 2. The course will give an introduction to hyperkähler manifolds and their known examples, survey some of the known results, and present some open problems. In the examples, interesting connections between hyperkähler manifolds, Fano manifolds and abelian varieties will be shown.   Prym varieties   The course we will review the classical theory and recent advances on Prym varieties and the Prym map, with special focus on the low genera cases which display beautiful geometry. The moduli aspect and the appearances of Prym varieties in other mathematical contexts will also be discussed.   Topics: Characterization of hyperkähler manifolds via their cohomology Period maps from the moduli spaces of hyperkähler manifolds to the period domains of their second cohomology Connections between hyperkähler manifolds, Fano manifolds and abelian varieties Fano manifolds Abelian varieties and Polarized abelian varieties Prym varieties and the Prym map: classical theory and applications Speakers: E. IZADI, University of California, San Diego, USA A. ORTEGA, Humboldt-Universität zu Berlin, Germany Additional: Contributed talks: Participants interested in giving a short communication are invited to submit an abstract. Registration: There is no registration fee. Online - ICTP [email protected] 19 Jul 2021 - 23 Jul 2021

            » Trieste Algebraic Geometry Summer School (TAGSS) 2021 - Hyperkähler and Prym varieties: classical and new results | (smr 3609)

            Europe/Rome 2021-08-12 08:00:00 2021-08-20 22:00:00 EAUMP-ICTP School: Topics in Concrete Mathematics | (smr 3604) This is an EMA (Ecole Mathematique Africaine) School of CIMPA. This hybrid School is nominally based at the ICTP-East African Institute for Fundamental Research [EAIFR] in Kigali, Rwanda it will offer students concrete and effective mathematical tools from algebra, group theory and geometry that can be applied to any scientific field. Consisting of 8 working days 12-15 August and 17-20 August, with one rest day the  2+2 courses shall be running concurrently. Lectures will be given remotely by video link. Description of Courses: WEEK 1 [12-15 August] R. K. Ramakrishna: Class numbers of number rings A. Taribi: Group representation theory and combinatorics WEEK 2 [17-20 August]: C. Kurujyibwami / B. Szendroi: Lie algebras K. Wendland: Modular forms Online - ICTP [email protected] 12 Aug 2021 - 20 Aug 2021

            » EAUMP-ICTP School: Topics in Concrete Mathematics | (smr 3604)

            Europe/Rome 2021-08-16 08:00:00 2021-08-20 22:00:00 AGRA IV (Arithmetic, Groups and Analysis) - Part I (smr 3617) An IMPA - ICTP Online School -  held in Spanish and Portuguese The AGRA series of summer schools (Santiago de Chile, 2012 Cusco, 2015 Córdoba, 2018) has as its goal to foster the development of number theory in Latin America: it provides a formative experience for young researchers, and it also brings together senior and mid-career mathematicians working in the field. The primary languages for this event are going to be Spanish and Portugese. Description: The target audience of AGRA includes undergraduate students, graduate students and young researchers with a serious interest in number theory and neighboring fields. The AGRA IV has been divided into two parts: part 1 is this online school in August 2021, while part 2 is preliminarily scheduled to take place in the first semester of 2022 in Cabo Frio, Brazil. The adjoined list of topics is representative of the two parts. The AGRA series has traditionally been held in Spanish but, this time, Spanish and Portuguese will share a role as primary languages of the school. Part 1 will consist of 4 minicourses delivered by a pool of distinguished mathematicians, followed by tutorials and problem sessions.   Topics: • Arithmetic geometry • Group approximations • Arithmetic combinatorics • Mahler measure • Model theory • Square-tiled surfaces • Analytic methods in number theory Programme: Minicourse 1: Introducción al análisis de Fourier de orden superior (Introduction to higher order Fourier analysis) • Julia Wolf (Cambridge, UK) • Pablo Candela (UAM / ICMAT, Spain)   Minicourse 2: Aritmética de variedades de alta dimensión (Arithmetic of high dimensional varieties) * • Tony Varilly-Alvarado (Rice, USA) • Damiano Testa (Warwick, UK) Minicourse 3: Combinatória de superfícies quadriculadas e geometria de espaços de módulos (Combinatorics of lattice surfaces and geometry of moduli spaces) • Carlos Matheus (École Polytechnique / CNRS, France) • Vincent Delecroix (Bordeaux / CNRS, France) Minicourse 4: Ecuaciones diofantinas con pocas soluciones (Diophantine equations with few solutions) • Hector Pasten (PUC, Chile) *preliminary title. Registration:  There is no registration fee   Online - ICTP [email protected] 16 Aug 2021 - 20 Aug 2021

            » AGRA IV (Arithmetic, Groups and Analysis) - Part I (smr 3617)

            Europe/Rome 2021-11-15 07:00:00 2021-12-10 21:00:00 Markov Partitions and Young Towers in Hyperbolic Dynamics | (smr 3642) In the 1970s, Sinai, Ruelle, and Bowen, developed groundbreaking new ideas and techniques which made it possible to apply the powerful results of Ergodic Theory to concrete, and sometimes quite explicit, differentiable dynamical systems. In particular they showed that smooth Uniformly Hyperbolic systems admit Markov Partitions, from which one can obtain a Symbolic Coding with a finite number of symbols. This Symbolic Coding makes it possible to apply methods from statistical mechanics to describe the statistical properties of the system through the construction of a particular class of invariant measures which are now called Sinai-Ruelle-Bowen (SRB) measures. Over the last 20 years there has been a huge progress in extending the results of Sinai, Ruelle, and Bowen, to the much larger class of more general (Nonuniformly) Hyperbolic systems, including systems with discontinuities/singularities. The geometry of these systems is much more complicated and one cannot expect to be able to code them with a symbolic dynamics with a finite number of symbols, making them much more challenging to study. Two inter-related but distinct approaches have emerged. At the end of the 1990s, Lai-Sang Young introduced a construction which is now generally refereed to as a Young Tower, based on constructing an induced uniformly hyperbolic system within the given system. More recently, around 2013, Sarig generalised the original Sinai-Ruelle-Bowen approach to construct infinite Markov Partitions. Both approaches have proved quite powerful and have been used to construct SRB measures and to study their statistical properties in a number of classes of dynamical systems of great interest. On the other hand, both approaches are also technically non-trivial and as a consequence, notwithstanding the applications of both to similar systems and their inevitable underlying connections, most researchers have developed an expertise in either one or the other. The main purpose of this event is to bring together experts in both areas in order to create opportunities to understand better the similarities and differences between them, and the advantages and disadvantages of the two approaches. The entire event will be held online with 2 mini courses by José Ferreira Alves on Young Towers and Yuri Lima on Markov Partitions. These mini-courses will be introductory and require only some familiarity with Uniformly Hyperbolic Dynamics and will be spread out over a period of 2 to 3 weeks, and be accompanied by some additional tutorial sessions, in order to give participants time to actually study the material and consolidate their knowledge. The mini courses will then be followed by a week of research level seminars describing recent results on these topics. Mini-Course Lecturers: José Ferreira Alves, U. of Porto, Portugal Yuri Lima, UFC, Brazil Speakers: Jerome Buzzi (Paris) Sylvain Crovisier (Paris) Peyman Eslami (Rome) Carlos Matheus (Paris) Ian Melbourne (Warwick) Snir Ben Ovadia (Weizmann) Yakov Pesin (Penn State) Omri Sarig (Weizmann) Agnieska Zelerowicz (F, Maryland) Hong-Kun Zhang (F, Amherst) Online - ICTP [email protected] 15 Nov 2021 - 10 Dec 2021

            » Markov Partitions and Young Towers in Hyperbolic Dynamics | (smr 3642)

            Watch the video: Class 10 Maths Chapter 2 Polynomials. Graph of Quadratic Polynomial Part 1 (December 2021).