Isaac Newton

1. fragile

2. calculus

3. impact

4. inertia

1 pacifist	A arms which can kill
	many people
2 weapons of	В something that prevents
mass destruction
3 notion	С somebody who doesn't
	believe in war
4 multi­	D involving time and space
dimensional
5 bend	E idea
6 obstacle	F curve
Niels Bohr
1 spectral line	A basically
2 in essence	В in disagreement
3 groundbreaking	С light or dark band of
	particular wavelength,
	used for identifying
	substances
4 contradictory	D revolutionary
5 complementarity	E a principle according to
	which things cannot be
	studied as having
	contradictory properties
	at the same time
В Use the words	in the box to replace
the words in blue.
■ patent ■ numerals ■ operations
■ theorem ■ symbols ■ integers
V	J

5 Gravity Albert Einstein

1. Mathematicians are interested in natural numbers and numbers which are zero itself or more or less than zero.

2. A legal document to protect an inventor's rights ensures nobody steals the invention.

3. In maths you use systems to combine numbers together in different ways.

4. We use these marks to indicate addition, subtraction, multiplication or division.

5. Statements or rules are used in mathematics.

6. We still use Roman numbers nowadays.

1. С Find ten words or phrases in this word search using the clues.

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Clues

2. principle

3. an angle of 90 degrees

4. a number larger than zero

5. a path round a planet for example

6. obviously separate from others

7. proof

8. to give off energy

9. when for example, the Moon covers the Sun

10. obvious

11. the quality of some things of having contradictory properties

Before you read

Discuss these questions with your partner.

Can you think of any ways that we use it in our everyday lives?

H A Vocabulary

a. Match these words and phrases with their definitions.

A money gained or lost

В rewriting

What is algebra?

1. linear algebra

2. reunion

3. matrices

4. profit and loss

5. transposing

6. operators

С linear equations

D coming back together

E arrangement of mathematical elements

F what goes into something and what comes out

7. input and output G signs used in maths

b. Match the words to make phrases.

1. arithmetical A quantities

2. unknown В spaces

3. abstract С system

4. vector D algebra

И Reading 1

Algebra

Algebra originated in the Middle East. Earlier than 1000 BC, the Babylonians developed an arithmetical system for solving problems that could be written algebraically. This was in advance of other systems, notably that of the Ancient Egyptians, who were able to solve the same problems, but did so by using geometry. The word algebra comes from Arabic and translates into English as reunion. It describes a system of mathematics which performs calculations by firstly rewriting, that is, transposing them, and then reducing them to their simplest form.

Algebra is the branch of mathematics which studies the structure of things, the relationship between things and quantity. It looks different from arithmetic when it is written. Arithmetic uses numbers and the four operators (plus, minus, multiply and divide). Algebra uses symbols, usually letters, and the operators. Actually, it is not very different from arithmetic;

what can be done in algebra can be done in arithmetic. There are good mathematical reasons, however, why algebra is used instead of arithmetic.

Firstly, by not using numbers, mathematicians are able to set out arithmetical laws. In this way they are able to understand the system of numbers more clearly. Secondly, by using algebra, mathematicians are able to perform calculations where unknown quantities are involved. This unknown is usually represented by x. Solutions can then be applied not just to the immediate problem, but to all problems of the same nature by the use of a formula. A common algebraic problem to solve in school exams would be, for example: find x where 3x + 8 = 14. A third reason for the use of algebra rather than arithmetic is that it allows calculations which involve change in the relationship between what goes into the problem and what comes out of it, that is, between input and output. It is an algebraic formula which allows a business to calculate its potential profit (or loss) over any period of time.

It is possible to classify algebra by dividing it into four areas. Firstly, there is elementary algebra in which symbols (such as x and y, or a and b) are used to denote numbers. In this area, the rules that control the mathematical expressions and equations using these symbols are studied. Then, there is abstract or modern algebra in which mathematical systems consisting of a set of elements and several rules ( axioms) for the interaction of the elements and the operations are defined and researched. Thirdly, there is linear algebra (linear equations) in which linear transformations and vector spaces, including matrices, are studied. Finally, there is universal algebra in which the ideas common to all algebraic structures are studied.

Like all branches of mathematics, algebra has developed because we need it to solve our problems. By avoiding the use of numbers we are able to generalise both the problem and the solution.

S s

Pronunciation guide

equation /ikweijsn/

linear /'lima/

matrices /'meitrisiiz/

И В Comprehension

Read the text and decide if the following statements are true or false.

1. Algebra is a mathematical system which rewrites a problem making it as simple as possible.

2. Written down, algebra differs to arithmetic in the operators it uses.

3. Algebra has some advantages to offer T Q the mathematician. F Q

4. Algebraic formulae are primarily

of use in businesses. F Q

5. Universal algebra combines all the structures from the other three areas. F

Before you listen

Discuss these questions with your partner.

Do you do arithmetic, algebra and geometry at school?

If so, which do you like best, and why?

И С Listening 4)))

Listen to a teacher talking to a class.

Then listen again and complete

the sentences.

1. Algebra is a branch of mathematics that

uses mathematical to

describe variables.

2. In a mathematical statement, letters are often

used to represent a(n) which

is not fixed.

3. A(n) is a mathematical

statement containing letters or symbols to represent numbers.

4. A term is a number or a(n) of a

number and one or more variables.

5. An expression is a collection of numbers,

variables and positive or

negative, of operations that make mathematical and logical sense.

Before you read

Discuss these questions with your partner.

What is the job of an engineer?

-» Why does an engineer need to know maths?

H D Vocabulary

Complete the sentences below with

words and phrases from the box.

accumulation of quantities

methodology ■ infinitesimal

differential calculus ■ integral calculus

vast ■ vital

tangent latter

coordinate ■ chord

sake ■ distinction

1. A line segment joining two points on a curve is a

2. A is a line or surface that

touches another.

3. The area of maths used to determine areas, volumes and lengths is called

4. The area of maths relating to changes in variable is called

5. If something is close to zero it is

6. You need to eat well for the

of your health.

7. There is a amount of

knowledge to learn in sciences.

8. There are two theories - one from ancient

times and a modern one. The

the modern one, is widely accepted now.

9. She claimed the of having

solved the equation.

10. A is a number that identifies

a position relative to a straight line.

11. is the system of methods

followed in an area of study.

1. 2 measures areas under a

curve, distance travelled, or volume displaced.

13. If something is it is of the

utmost importance.

H Reading 2

Gottfried Leibniz*

Gottfried Leibniz was born and lived most of his life in Germany. He made visits to both Paris and London, for the sake of learning and study, but spent the vast majority of his working life as an employee of German royalty, as a philosopher, engineer and mathematician. It is for the latter that he is best remembered. Ilis greatest achievement was as an inventor of calculus, the system of notation which is still in use today. Leibniz is remembered as an inventor, not the inventor of calculus. In England, Isaac Newton claimed the distinction, and was later to accuse Leibniz of plagiarism, that is, stealing somebody else’s ideas but stating that they are original. Modern-day historians however, regard Leibniz as having arrived at his conclusions independently of Newton. They point out that there are important differences in the writings of both men. Newton, it must be said, was very protective of his achievements and jealous of others' success. It is important to mention that Leibniz published

his writings 011 calculus three years before Newton published his most important work.

Leibniz was the first to use function to represent geometric concepts. Among other terms, Leibniz used what is now everyday language in mathematics to describe these concepts. Words such as tangent and chord, were first used by Leibniz. He also saw that linear equations in algebra could be arranged into matrices. It was in this significant piece of work on calculus that he introduced mathematics and the world to the word coordinate. lie also made important advances in algebra and logic in ways that still today, three hundred years later, have an impact on mathematics.

Leibniz' importance for modern mathematics can be understood through his work. He was especially interested in infinitesimal calculus. This is an area of calculus developed from geometry and algebra. It is divided into two parts. There is differential calculus, which is concerned with measuring rates of change of quantities. And there is integral calculus, which studies the accumulation of quantities. That is, Leibniz was looking at ways of measuring the speed and the distance travelled, for example. Today, calculations of this type are used not only in mathematics but in every branch of science and in many fields which apply a scientific methodology, such as economics and statistics.

Despite the disagreements between Leibniz and Newton, modern mathematicians recognise each of them as being vital to the development of modern mathematics. Newton was certainly the first to apply calculus to the problems of physics. In mathematics itself, it is to Leibniz that we look for our system of writing equations and for the language we use to refer to the concepts. While both reached their understanding without the benefit of reading each other's work, it remains a fact that Leibniz was first to publish.

Pronunciation guide

Gottfried Leibniz /'gotfriid 'laibnits/ infinitesimal /infinitesimal/ plagiarism /'pleid33iiz3m/

И E Comprehension

Read the text and answer the questions