Pokročilá odborná angličtina - zkouška
COURSE MATERIALS FOR STUDENTS
Hello, dear students,
here are some materials for those of you who cannot attend my seminars.
Fields of mathematics
(materials adapted from Mathematics, encyclopedia articles on http//en.citizendium.org)
1. What are the fields of Mathematics?
2. Match these concepts and the fields of Maths.
a) change, structure, quantity, space
b) algebra, geometry, analysis, arithmetic
3. Read the paragraph and find other subdivisions.
The major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.
Revision: What kinds of numbers do you know? Give examples. (Have a look at the Clock).
1. ............................................................. 2..............................................................
3.................................................................. 4.............................................................
QUANTITY
The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, whence such popular results as Fermat's last theorem. Two famous unsolved problems in number theory are the twin prime conjecture and Goldbach's conjecture.
As the number system is further developed, the integers are recognised as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalised to complex numbers. These are the first steps of a hierarchy of number systems that include the quaternions and octonions. Consideration of numbers larger than all finite natural numbers leads to the concept of transfinite numbers. In this formalism, infinite cardinal numbers, the aleph numbers, allow meaningful comparison of the size of infinitely large sets.
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4. What do you know about Fermat´s last theorem and Goldbach´s conjecture, and the twin prime conjecture?
5. What are quaternions and octonions?
6. Some problems concerning natural numbers.
a) is zero a natural number?
b) explain what a prime number is.
c) what is the difference between an odd and even number?
d) what is a square number?
e) what are the basic mathematical operations on numbers?
f) what are cardinal and ordinal numbers?
g) what are decimal numerals?
h) what is a perfect number?
i) what is a transfinite number?
j) what are aleph numbers?
7) The next short paragraph is about Peano axioms concerning natural numbers.
Have you ever heard about it? Try to give any information.
8) Now read the text and try to fill in the missing words.
recursion successor proceeds
element equivalent arbitrary topic
Peano axioms
During the 19th century the foundations of mathematics which, of course, include the concept of number became a major .............of discussion and research. In 1889 Giuseppe Peano published a system of axioms that characterizes the natural numbers. The axioms, essentially, state that eventually every natural number will be reached if one starts to count at 0 (or 1, if that is preferred) and ............... from that by stepping from one number to the next. The axioms are usually given as follows:
(1) 0 is a natural number.
(2) Every natural number has a unique .................
(3) 0 is not the successor of a natural number.
(4) Different natural numbers have different successors.
(5) If a property of natural numbers is such that:
0 has the property, and
if a natural number has the property then its successor has it as well.
Then every natural number has this property
The last axiom is ....................... to the following property:
(5a) Any non-empty set of natural numbers has a least ...............
In these axioms, the first (least) element is taken to be 0, but this is ............... It can be replaced by 1 (or any other number).
Axiom (5) (or (5a)) is the basis for proofs by induction, and for definition by .....................
9) Word study
There are some verbs from the text. Try to fill in the missing prepositions.
lead........st replace st..........st arise ....... .........st
count.......a number step.........st ..........st relate st...........st
dedicate ..........st start ..........st contain st...........st
Here is the listening part, you can find the video on the given web address.
Transcript - Lecture 34
OK. Good.
The final class in linear algebra at MIT this Fall is to review the whole course. And, you know the best way I know how to review is to take old exams and just think through the problems. So it will be a three-hour exam next Thursday. Nobody will be able to take an exam before Thursday, anybody who needs to take it in some different way after Thursday should see me next Monday. I'll be in my office Monday.
OK. May I just read out some problems and, let me bring the board down, and let's start. OK. Here's a question. This is about a 3-by-n matrix.
And we're given -- so we're given -- given -- A x equals 1 0 0 has no solution. And we're also given A x equals 0 1 0 has exactly one solution. OK.
So you can probably anticipate my first question, what can you tell me about m? It's an m-by-n matrix of rank r, as always, what can you tell me about those three numbers? So what can you tell me about m, the number of rows, n, the number of columns, and r, the rank? OK.
See, do you want to tell me first what m is? How many rows in this matrix? Must be three, right? We can't tell what n is, but we can certainly tell that m is three.
OK. And, what do these things tell us? Let's take them one at a time.
When I discover that some equation has no solution, that there's some right-hand side with no answer, what does that tell me about the rank of the matrix? It's smaller m. Is that right? If there is no solution, that tells me that some rows of the matrix are combinations of other rows.
Because if I had a pivot in every row, then I would certainly be able to solve the system.
I would have particular solutions and all the good stuff. So any time that there's a system with no solutions, that tells me that r must be below m. What about the fact that if, when there is a solution, there's only one? What does that tell me? Well, normally there would be one solution, and then we could add in anything in the null space. So this is telling me the null space only has the 0 vector in it.
There's just one solution, period, so what does that tell me? The null space has only the zero vector in it? What does that tell me about the relation of r to n? So this one solution only, that means the null space of the matrix must be just the zero vector, and what does that tell me about r and n? They're equal. The columns are independent.
So I've got, now, r equals n, and r less than m, and now I also know m is three.
So those are really the facts I know.
n=r and those numbers are smaller than three.
Sorry, yes, yes. r is smaller than m, and n, of course, is also.
So I guess this summarizes what we can tell.
LISTENING 1. Lecture 34 MIT
http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/detail/lecture34.htm
Listen for the first time to the first part of the lecture and answer the following Qs. Compare your answers with your neighbour.
- What is the topic of the lecture?
- When can students take an exam?
- What is the first question the lecturer asks?
- What does it mean that there is only one solution?
- Is m smaller than three?
- How many rows are there in the matrix?
- What does the letter r stand for?
- What does the letter n stand for?
Listen for the second time and fill in the missing words.
Transcript - Lecture 34
OK. Good.
The final class in linear algebra at MIT this Fall is to review the whole course. And, you know the best way I know how to review is to take old exams and just 1. __________ the problems. So it will be a three-hour exam next Thursday. Nobody will be able to take an exam before Thursday, anybody who needs to take it in some different way after Thursday should see me next Monday. I'll be in my office Monday.
OK. May I just read out some problems and, let me bring the board down, and let's start. OK. Here's a question. This is about a 2. ___________ matrix.
And we're given -- so we're given -- given -- A x equals 1 0 0 has no solution. And we're also given A x equals 0 1 0 has exactly one solution. OK.
So you can probably 3. ________________ my first question, what can you tell me about m? It's an m-by-n matrix of rank r, as always, what can you tell me about those three numbers? So what can you tell me about m, the number of rows, n, the number of 4. ___________, and r, the rank? OK.
See, do you want to tell me first what m is? How many rows in this matrix? Must be three, right? We can't tell what n is, but we can certainly tell that m is three.
OK. And, what do these things tell us? Let's take them one 5. ____________.
When I discover that some equation has no solution, that there's some 6. ____________ with no answer, what does that tell me about the rank of the matrix? It's smaller m. Is that right? If there is no solution, that tells me that some rows of the matrix are 7. ______________ of other rows.
Because if I had a 8. ______________ in every row, then I would certainly be able to solve the system.
I would have particular solutions and all the good stuff. So any time that there's a system with no solutions, that tells me that r must be below m. What about the fact that if, when there is a solution, there's only one? What does that tell me? Well, normally there would be one solution, and then we could add in anything in the 9. ____________. So this is telling me the null space only has the 0 vector in it.
There's just one solution, period, so what does that tell me? The null space has only the zero vector in it? What does that tell me about the relation of r to n? So this one solution only, that means the null space of the matrix must be just the 10. __________________, and what does that tell me about r and n? They're equal. The columns are independent.
So I've got, now, r equals n, and r less than m, and now I also know m is three.
So those are really the facts I know.n=r and those numbers are smaller than three.
Sorry, yes, yes. r is smaller than m, and n, of course, is also.
So I guess this summarizes what we can tell.
Now go through the text again and underline the signaling devices and rhetorical questions.