Vocabulary notes

by no means – semmiképpen, sehogy

purpose of measuring – a mérés célja

in other words – másszóval, máshogy

sufficient – elegendő

interval – időköz, intervallum

precisely one – csakis egy

nested intervals – beágyazott intervallumok

to preserve – megőriz

in terms of – vmiben kifejezve/ megadva

quotient – hányados

in such a way – íly módon

preceding – előbbi, megelőző

commensurable – arányos, összemérhető

to assume – feltételez, feltesz

entirely – maradéktalanul, teljesen

segment – szegmens, darabja / része vminek

endpoint – végpont

to increase – növekszik, növekedés

by definition –a meghatározás szerint

respectively – illetve

similarly – hasonlóan

axis – tengely

to hold – vonatkozik, érvényben van

Exercises

I. Answer the following questions on the text.

1. What numbers are irrational?

2. What sequence is called sequence of "nested intervals"?

3. What is a real number?

4. What is the basic postulate of geometry?

5. What does the sequence of nested intervals define?

II. Change the sentences to negative and to question form.

1. The number is an irrational number.

2. This conclusion is rather unexpected.

3. The rational numbers are entirely sufficient.

4. The addition of two irrational numbers and can be defined in terms of the two sequences of nested intervals defining and respectively.

5. The new sequence of nested intervals defines + .

6. There exist numbers which cannot be rational.

III. Form derivatives from the following words and translate them:

precise, similar, numeral, respect, integer, geometry, determine, densely, change, axis, capable, entire, square, commensurable

IV. Find in the text the following words and word combinations. Guess their meanings. Make up your own sentences with them.

purpose of measuring, there exist, in other words, rational points, theoretical viewpoint, to consider, to be contained in, nested intervals, by definition, a real number, greater than, to be defined in

Part II Lesson 1 geometry

Geometry allows us to explore the properties of space in terms of plain (two-dimensional) figures and solid (three-dimensional) figures.

We can use geometric techniques to draw a line of an exact length, bisect a line, bisect an angle, construct a triangle and calculate the area of a sphere.

Map-making, surveying, designing architecture and computer circuitry all depend on geometry in their precise use of angles, figures and volumes.

The principles of geometry were laid down by the Greek mathematician Euclid (330 BC – 275 BC). He compiled out of the disorganized geometry of his day a set of rules concerning space and shapes that seemed so basic and true that no one changed it for two thousand years.

Under his guidance geometry became an organized body of knowledge concerning the relations, properties and measurements of lines, angles, surfaces and solids.

Even today most textbooks on geometry follow the plan of Euclid’s writings, often using his own diagrams, methods of proof and ways of stating geometrical truths.