Unit II

In the world of mathematics

Grammar: Revision of Modal Verbs. Modal Compound Predicate. Reading. Text A. Welcome to Mathematics! Text B. Philosophy of Mathematics Summary Writing. Text C. Mathematical Beauty

Topical Vocabulary

I. Read and memorize

applied mathematics – прикладная математика

calculus – математический анализ; дифференциальное и интегральное исчисление

complex number – комплексное число

computability theory – теория вычислимости

computational complexity theory – теория сложности вычислений

concept of continuity – понятие непрерывности

continuous quantity – непрерывная величина

decimal number – десятичное число

denominator – знаменатель

derivative – производная функции

differential equation – дифференциальное уравнение

fiber bundle – пространство расслоения

group theory – теория групп

imaginary number – мнимое число

infinite-dimensional – бесконечно мерное

integer – целое число

natural number – натуральное число

numerator – числитель

numerical analysis – вычислительная математика

polynomial – многочлен

polynomial equation – полиномиальное уравнение

property – свойство

quotient – частное

rational number – рациональное число

real analysis – действительный анализ

real numbers – вещественное число

real-valued function – действительная функция

raising to a power – возведение в степень

rounding errors – ошибки (погрешности) округления

set theory – теория множеств

smoothness – гладкость

spatial relationship – пространственное расположение

variable (quantity) – переменная (величина)

taking a square root – извлечение квадратного корня

II. Which terms from the Topical Vocabulary do these definitions refer to?

1. An analytical form of the problem of investigating which values of variables in functions give equal results. 2. A whole number without a fractional or decimal component. 3. The study of the properties of sets, pre-eminently infinite, disregarding the properties of the elements in those sets. 4. А value that may change within the scope of a given problem or set of operations. 5. А number with square that is negative. 6. А measure of how a function changes as its input changes. Loosely speaking, it can be thought of as how much one quantity is changing in response to changes in some other quantity. 7. A branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.

Reading Section

Pre-reading discussion.

1. Are you good at mathematics? What is your favourite branch? Why?

2. Mathematics is considered to be beautiful. What do you think the beauty of mathematics lies in?

3. What do you know from the history of mathematics?

4. How did mathematics influence other sciences and our everyday life?

5. Carl Friedrich Gauss, a German mathematician, once said, “Mathematics is the queen of sciences”. How would you comment on this statement? Do you agree with him? What other famous mathematicians’ expressions on mathematics do you know?

Text A. Welcome to Mathematics!

Mathematics is pure. It does not rust or decay. It only needs your thought to make it work. Mathematics is “the science of structure, order and relation that has evolved from elemental practices of counting, measuring and describing the shapes of objects”. Simply put, mathematics is the study of quantity, structure, space and change.

The word “mathematics” comes from the Greek μάθημα (máthema) which means “science, knowledge, or learning μαθηματικός” (mathematikós) means “fond of learning”.

The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction and for astronomy. The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC. Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today.

The study of structure starts with numbers; familiar kinds of numbers include natural numbers (ℕ), integers (ℤ), rational numbers (ℚ), real numbers (ℝ), and complex numbers (ℂ). In mathematics, natural numbers are the ordinary counting numbers 1, 2, 3, … The integers are formed by the natural numbers including 0 (0, 1, 2, 3, …) together with the negatives of the non-zero natural numbers (–1, –2, –3, …). A rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number. The type of number we normally use, such as 1, 15.82, –0.1, ³/₄, etc…positive or negative, large or small, whole numbers or decimal numbers are all real numbers. They are called “Real Numbers” because they are not imaginary numbers. A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i ² = −1.

The study of space originates with geometry. It is dealing with spatial relationships. The modern fields of differential geometry and algebraic geometry generalize geometry in different directions: differential geometry emphasizes the concepts of functions, fiber bundles, derivatives, smoothness, and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations. Group theory investigates the concept of symmetry abstractly and provides a link between the studies of space and structure. Topology connects the study of space and the study of change by focusing on the concept of continuity.

Understanding and describing change in measurable quantities is the common theme of the natural sciences, and calculus was developed as a most useful tool for that. The central concept used to describe a changing variable is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods to solve these are studied in the field of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. For several reasons, it is convenient to generalize to the complex numbers which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions, laying the groundwork for quantum mechanics among many other things. Many phenomena in nature can be described by dynamical systems and chaos theory deals with the fact that many of these systems exhibit unpredictable yet deterministic behavior.

In order to clarify and investigate the foundations of mathematics, the fields of set theory, mathematical logic, and model theory were developed.

When computers were first conceived, several essential theoretical concepts were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, information theory, and algorithmic information theory. Many of these questions are now investigated in theoretical computer science. Discrete mathematics is the common name for those fields of mathematics useful in computer science.

An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis and prediction of phenomena and is used in all sciences. Numerical analysis investigates the methods of efficiently solving various mathematical problems numerically on computers and takes rounding errors into account.

Mathematics has wide applications in engineering, physics, chemistry and most of the other sciences. The major discoveries and inventions, the major theories and applications would never be possible without mathematics.

And it is widely used in both Information Technology and communication. These would be nonexistent if there had been no developments in mathematics over the centuries.

Accountants, economists and business people use it every day. The weather is predicted using powerful mathematics modeling. And even your favorite computer game has at its heart lots of mathematical equations that work out how everything moves and behaves.