EXERCISES. Trigonometric, system, prism, cylinder, polygon, coordi­nate, co-functions, cosine, cotangent, cosecant.

I. Read the following words paying attention to the pronun­ciation:

trigonometric, system, prism, cylinder, polygon, coordi­nate, co-functions, cosine, cotangent, cosecant.

II. Add suffixes and translate the newly formed words into Russian:

-ly: particular, possible, frequent;

-cal: geometry, trigonometry;

-tion: define, coordinate, apply;

-ing: place, relate, permit.

III. Make up sentences of your own using the words and ex­pressions given below:

standard position, rectangular, consideration, frequently, interpolation, coordinate system, permit us to' find, have to be found, worth mentioning, worth studying, worth explain­ing, worth relating.

IV. Answer the following questions:

1. What is a trigonometric function of an angle? 2. What is one of the most important applications of the trigonometric functions? 3. How do we find the functions of any angle?

V. Translate into Russian:

On the basis of the definitions of the trigonometric func­tions it is easy to determine whether a given function of an angle is positive or negative.

For example, if θ is any angle in the second quadrant,

cos θ = x/r is a negative quantity because r is always posi­tive and x, in this case the abscissa of a point in the second quadrant, is negative.

VI. Translate into English:

Таблицы, в которых даются значения тригонометри­ческих функций, называются натуральными (natural) три­гонометрическими таблицами. Для вычисления значений синуса и косинуса могут служить одни и те же таблицы. Так например, sin 26° и cos 64° имеют одно и то же значение. В таблицах В. М. Брадиса значения аргумента синуса рас­положены в порядке возрастания сверху вниз, а значения аргумента косинуса снизу вверх.

SUPPLEMENTARY READING

TEXT

EUCLID

Euclid is known to us almost exclusively from those of his works which have survived.

Euclid lived in Egypt approximately 300 B.C. He taught in Alexandria and was the founder of its illustrious mathe­matical school. His chief extent work is the Elements in 13 books. Books treat of plane geometry, of proportion in gener­al, of the properties of numbers, of incommensurable magni­tudes, of solid geometry. Besides the Elements there are the Data — a collection of geometrical theorems.

Euclid's Elements has been translated into many languages, and is probably better known than any other mathematical book, with many of its blemishes removed and its deficiencies supplied, it is still widely used in Britain as a text-book of geometry, though attempts have been made for the last 150 years to supersede it.

The first printed edition of Euclid was a translation from Arabic into Latin, which appeared at Venice in 1482. The first printed Greek text was published at Rasel in 1533. The most recent edition is that of Heiberg in 5 volumes (1883-88).

TEXT

PYTHAGORAS

Pythagoras is for us at once the glorified and the actual founder of the philosophical school (which exercised a great influence on the course of ancient science). He was also a great mathematician. Pythagoras investigated harmonies and properties of numbers. His attention was turned to the odd and even, to prime numbers, square numbers and so.

The great mathematical discovery of Pythagoras is of course a hypotenuse theorem, where the square is equal to the sum of two squares. "Pythagorian numbers" are such numbers as are related in the way the theorem indicates. Various other theorems are closely connected with this cardinal one; these concern chiefly the squares of the various perpendiculars which may be let fall from different angles of the right-angled triangle upon the hypotenuse and sides.

TEXT

NEWTON

The greatest of natural philosophers, was born on 25th December 1642 — year remarkable in the history of science by the birth of Newton and the death of Galileo. He was born at the hamlet of Woolsthorpe in Lincolnshire in the family of a farmer.

He received his early education at the grammar-school at the same hamlet. On the 5th June 1661 he left home for Cambridge, where he was admitted the same year. He applied himself there to the mathematical studies. After a few years he began to make some progress in the methods for extending the science.

In 1666, the fall of an apple, as he walked in the garden at Woolsthorpe, suggested the most magnificent of his sub­sequent discoveries — the law of universal gravitation.

He accordingly abandoned the hypotheses for other stud­ies. He investigated the nature of light and was led to the conclusion that rays of light which differ in colour differ also in refractivity.

Newton became a professor of mathematics in 1669. In 1671 he resumed his calculations about gravitation.

In 1696 he was appointed warden of the Mint and was afterwards promoted to the office of Master of the Mint in 1699, an office which he held till the end of his life. He took a seat in parliament in the year 1701 as the representative of his university.

A mathematical feat is recorded of him so late as 1716 in solving a problem prpposed by Leibnitz for the purpose, as he expressed it, of feeling that pulse of the English ana­lysts.

In 1699 Newton was elected a foreign associate of the Academy of Sciences. He died at Kensington on 20th March 1727 and was buried at Westminster Abbey, where a monu­ment was erected to his memory in 1731.

TEXT

LEIBNITZ

Distinguished for almost universal scholarship especially in philosophy and mathematics Leibnitz was born on 1st July 1646 at Leipzig, where his father was professor of Moral Philosophy. He attended the school in Leipzig, but learned much more from independent study. He spent some time also in Jena working at mathematics. He graduated at Altdorf, the university town of Nurnberg.

Some years later he invented a calculating machine and devised what was in many respects a noble method of calcu­lations. This gave rise to a controversy with Newton as to which of them first invented this valuable mathematical method.

In 1676 Leibnitz quitted the service of Mainz and was appointed a custodian of the library of Hanover. In 1687 he visited various cities in Germany, Austria and Italy.

Leibnitz was also a pioneer in the science of comparative philology. He died on 14th November 1716 at Hanover.

TEXT

GAUSS

Karl Friedrich Gauss, German mathematician was born at Brunswick, on 30th April 1777.

In 1801 he published an important work on the theory of numbers and other analytical subjects: Disquisitions Arith­metical. He was appointed as professor of Mathematics and director of the observatory at Gottingen. He also worked with equal­ly brilliant success in the science of geodesy and astronomy.

Later in life (in 1843-46) he published a collection of valuable memoirs on surface geometry. He also studied the problems arising out of the earth's magnetic properties. In 1833 he wrote his first work on the theory of magnetism.

In applied mathematics he investigated the problems con­nected with the passage of light through a system of lenses in 1846. Besides the researches already mentioned he wrote papers or works on probability, the method of least squares, the theory of biquadratic residues, constructed tables for the conversion of fractions into decimals and of the number of classes of binary quadratic forms, and discussed hyper- geometric series, interpolation, curved surfaces, all of which he printed in the seven volumes of his collected works.

Gauss died at Gottingen, on 23rd February 1855.

TEXT

SOPHIA KOVALEVSKAYA

The outstanding Russian mathematician Sophia Kovalevskaya was born in Moscow on February 15,1850, in a well- off family of an artillery general, Korvin-Krukovsky.

Sophia's childhood was spent in Polibino, where the fam­ily used to live the greatest part of the year. When Sophia was eight an experienced teacher was invited to Polibino to instruct her in arithmetic, grammar, literature, geography and history. Though she liked literature so much that it seemed that literature would become her ultimate object in life, the girl showed an unusual gift in mathematics and at the age of twelve puzzled her teacher by suggesting a new solu­tion for the determination of the ratio of the diameter of the circle to its circumference.

In 1867 Sophia and her elder sister were taken to St. Petersburg. There Sophia was allowed to go on with her studies privately. To attend lectures at the University a wom­an had to obtain a special permission, and even then by no means would she be allowed to take examinations, to say nothing of taking a degree.

This state of things remained unaltered despite the ef­forts of many scientists who voiced an urgent demand that women should be granted the right to education. The only way out for her was to go abroad, as some other Russian women did. But in this case there was a condition that the woman should be married. This made her marry Vladimir Kovalevsky, with whom she soon left for Vienna. There the Kovalevskys were given permission to attend lectures on physics at the Vienna University, but this did not satisfy Sophia. She made up her mind to go to the Heidelberg University to study under such scholars as Helmholz and Bunzen, as her intention was to take examinations for a Doctor's Degree in mathematics and mechanics. While in Heidelberg, she would attend eleven lectures a week, includ­ing eight lectures on mathematics and do a lot of practical work as well. In 1871 Sophia went to Berlin, where she read privately with professor Weierstrass, as the public lectures were not then open to women. During the four years spent in Berlin, Sophia succeeded not only in covering the university course of mathematics but also in writing three dissertations. In 1874 the University of Gottingen granted her a degree of Doctor of Philosophy in absentias excusing her from the oral examinations in consideration of the three dissertations sent in, one of which, on the theory of partial differential equati­ons, was one of her most remarkable works.

When the Kovalevskys returned to Russia they planned to live and work in St. Petersburg, but despite the efforts of Mendeleyev, Butlerov and Chebyshev, Sophia Kovalevskaya, a great scientist could not find a position there and was obliged to turn to journalism.

In 1878 Sophia gave birth to a daughter and as her hus­band was promised a lectureship at the Moscow University, she decided to take her Magister's Degree there. Great was her disappointment when she learned that her application had not been accepted, though her personal experience should have suggested her that there was no use in trying to get a degree in Russia. Again she went to Berlin to complete her work on the refraction of light in crystals, but the news of her husband's bankruptcy and suicide caused her to return home.

In 1883 she was given an opportunity to report on the results of her research at a session held in Odessa, but no post followed. Therefore, when she was offered lectureship at Stockholm University she willingly accepted the offer and went there with her little daughter.

In 1888 she achieved the greatest of her successes winning the highest prize offered by the Paris Academy. The problem set was: "to perfect in one important point the theory of a movement of a solid body about an immovable point." The solution obtained by her made a valuable addition to the results submitted by Euler and Lagrange.

In 1889 she was awarded another prize by the Swedish Academy of Science. Soon, in spite of her being the only woman-lecturer in Sweden, she was elected professor of me­chanics and held the post until her death.

Unfortunately Sophia Kovalevskaya did not live to reap the full reward of her labour, for she died on February 10, 1891, at the age of 41, just as she had attained the height of her fame and had won recognition even in her own country by election to membership of the St. Petersburg Academy of Sciences.

TEXT

NIKOLAI LOBACHEVSKY

Lobachevsky (1792-1856) will always be regarded as one of the greatest thinkers. Like Archimedos, Galileo, Coperni­cus and Newton, he is one of those who laid the foundations of science.

Lobachevsky became seriously interested in mathematics while still a schoolboy. He remained true to this science all his life.

During the 2000 years before Lobachevsky, geometry was based on The Elements of Euclid. Euclid gave the axiom on parallel lines asserting that there can be only one parallel to the given line through the point outside that line. It was this theory that Lobachevsky attacked.

Lobachevsky proved that there could be several parallels to the given line through a point outside that line.

The revolution in geometry achieved by Lobachevsky is the most significant example of the radical transformation which our conceptions of space have undergone. He demon­strated the need to study the properties of real space experi­mentally. Lobachevsky's ideas exercised a profound influence in the development not only of geometry but also of other mathematical sciences, as well as on mechanics, physics and astronomy.

TEXT

MATHEMATICIAN No. 1

At an international mathematical symposium a French scientist came up to academician Andrei Kolmogorov and said: "I thought that I would never be able to see you, I could not believe that all the books written by Kolmogorov actually belong to one person. I thought that it was a pen-name for a whole group of authors. I'm very glad indeed to meet you."

Kolmogorov's colleagues, many of whom are scientists of worldwide fame, call him "Mathematician No. 1." Kolmogorov has written more than 220 works. He is the creator of the modern theory of probability, and has applied it in di­verse fields of human activities, ranging from the theory of directed firing (during the war artillerymen used Kolmogorov's tables) to the theory of information.

Kolmogorov has lately begun to work on the application of mathematical statistics in the analysis of verse. His re­ports on this subject have stimulated great interest among linguists and poets.

One rarely meets Kolmogorov alone. He is always sur­rounded by his colleagues and pupils at the University, where he heads the department of the theory of probability, and in his summer cottage in the small Komarovka village near Moscow.

His cottage in Komarovka is like a hiking centre, a place for the start and finish of long hikes and skiing and boating excursions which he often organises with his pupils. New maths theories had often been outlined in talks by the camp fires. Kolmogorov's pupils return from these "maths hikes" not only with fresh ideas and high spirits. They also have rich impressions of landscapes and people in the Moscow countryside and of the architectural ancient relics there.

TEXT

ABOUT COMMON FRACTIONS

"Fractio" is a Latin word meaning "to break". When a bone is fractured, it is broken. Fractions, in arithmetic, are used to express parts.

Several thousand years before the beginning of our era the Egyptians living in the valley of the Nile River, had a highly developed civilization. The fractions which they used were unit fractions. A unit fraction has 1 for its numerator. When the Egyptians wished to express the quantity which we call 3/4, they used 1/2+1/4.

The Babylonians, who lived in southwestern Asia, thought of the whole as being broken into sixty equal parts. Each of these sixty parts was thought of as broken into sixty equal parts. These fractions were called sexagesimal fractions, from "sexaginta" meaning 60. We still use the idea of these fractions in dividing our hour into 60 minutes and each min­ute into 60 seconds. The quantity which we call 3/4 would have been 45/60.

The Greeks used sexagesimal fractions too. They also used unit fractions which were represented by writing the denominator followed by an accent ('). 1/4 would have been written as 4 with an accent after it, thus 4'. When they used fractions which were not unit fractions they wrote the numer­ator once and the accented denominator twice. Using then symbols for 3 and 4 they would have written 3/4 as 34'4'. The Romans divided the whole into 12 parts. Englishmen still divide one foot into 12 parts or inches. In the seventh century of our era, a Hindu writer used the plan of writing the numerator over the denominator, 3/4 would then be 3/4.

The Arabs made one more change, inserting a bar between the numerator and denominator, giving us a present form

for writing fractions 3/4.

TEXT

THE TRUTH IN GODEL'S PROOF

Ever since the time of Euclid, 2,200 years ago, mathema­ticians have tried to begin with certain statements called "axioms" and then deduce from them all sorts of useful con­clusions.

In some ways it is almost like a game with two rules. First, the axioms must be as few as possible. If you can de­duce one axiom from the others, that„deduced axiom must be dropped. Second, the axioms must be self-consistent. It must never be possible to deduce two conclusions from the axioms with one the negative of the other.

Any high school geometry book begins with a set of axi­oms: that through any two points only one straight line can be drawn; that the whole is equal to the sum of the parts, and so on. For a long time, it was assumed that Euclid's axioms were the only ones that could build up a self-consistent ge­ometry to that they were "true".

In the 19th century, however, it was shown that Euclid's axioms could be changed in certain ways and that different "non-Euclidean geometries" could be build up as a result. Each geometry was different from the others, but each was self-consistent. After that it made no sense to ask which was "true". One asked instead which was useful.

In fact there are many sets of axioms out of which a self- consistent system of mathematics could be built; each one different, each one self-consistent.

In any such system of mathematics you must not be able to deduce from its axioms that something is both so and not- so, for then the mathematics would not be self-consistent and would have to be scrapped. But what if you make a state­ment that you can't prove to be either so or not-so?

Suppose, that I say: "The statement I am now making is false."

Is it false? If it is false, then it is false that I am saying something false and I must be saying something true. But if I am saying something true then it is true that I am saying something false, and I am indeed saying something false. I can go back and forth forever. It is impossible to show that what I have said is either so or not-so.

Suppose you adjust the axioms of logic to eliminate the possibility of my making statements like that. Can you find some other way of making such neither-so-nor-not-so state­ments?

In 1931, an Austrian mathematician, Kurt Godel present­ed a valid proof that showed that for any set of axioms you can make statements that cannot be shown to be so from those axioms and yet cannot be shown to be not-so either. In that sense, it is impossible to work out, ever, a set of axioms from which you can deduce a complete mathematical system.

Does that mean that we can never find "truth"? Not at all.

First: Just because a mathematical system isn't complete doesn't mean that what it does contain is "false". Such a system can still be extremely useful provided we do not try to use it beyond its limits.

Second: Godel's proof applies only to deductive systems of the types used in mathematics. But deduction is not the only way to discover "truth". No axioms can allow us to deduce the dimensions of the solar system. Those dimensions were obtained by observations and measurements — another route to "truth".

TEXT

MATHEMATICS—HANDYMAN FOR ALL SCIENCES

Karl Friedrich Gauss, the famous 18th century mathema­tician, once called mathematics the queen of sciences. In our view, its role is much loftier.

Mathematics is a handyman working for all sciences. Today economists, linguists and psychologists resort to its services. Mathematics does its job so well that other sciences depend to some extend on this handymen.

Mathematics follows the changes taking place in various fields of knowledge and in this connection sets itself definite targets. At the same time, scientists in other spheres must closely follow the progress made in mathematics since it is impossible to keep abreast of latest developments in, say, physics without making use of mathematics.

The recent sensational achievements in biology — genet­ics, in particular, are closely linked with progress in mathe­matics. It would be impossible to decipher the genetic code, the code of heredity had we not had such terms as coding, transmission of information and so on.,

Mathematics today is often occupied with "strange" things. One of the leading mathematicians in the world, Andrei Kolmogorov, is making a thorough study of matters pertaining to higher nervous activity and to poetry. Of course mathematicians do not at all intend to entrust machines to write poetry for us. But Andrei Kolmogorov applies mathematics to analysing the problems of writing verse.

Mathematics itself experiences a very strong influence of other sciences. When Kolmogorov tried to apply the mathematical methods of the theory of information to study works of literature he had to alter the very definition of in­formation. In doing so he arrived at several new conceptions in keeping with which the theory of information was wrested from the theory of probability and rested on conceptions stemming from mathematical logic. Later Andrei Kolmogorov completely changed his point of view on the content of the theory of probability. Now he tries to substantiate it proceed­ing from the theory of information, from the new approach to this theory to which he had been prompted by his study of literary works.

In the 30s of this century it seemed that mathematics only studied continuous functions, differentials, integrals, differential and integral equations.

Yet, during the war the first electronic computers were made. Few people know that once there were two points of view regarding mathematical machines. There were machines of discrete action and machines of continuous action, repro­ducing functions and processes. The upper hand was gained completely and unequivocally by the discrete machine, by cipher computers, because any discrete alphabet makes it possible to record the most diverse phenomena with suffi­cient precision.

Modern electronic machines are designed on this pattern. They use a language possessing a small number of letters, but by alternating these letters, they can describe highly intri­cate processes.

The changes in mathematics connected with progress in other sciences and the changes in the sciences embracing mathematics and connected with the progress of mathematics are reflected in the way mathematics is taught. We still pay much attention to differential and integral calculus, which is only beginning to enter our secondary schools, but we are not stopping at this. Both in secondary and higher schools more and more attention is being paid to discrete mathematics connected with the new views on the world around us — views combined in the term "cybernetics".

TEXT

ORDINARY VS. BINARY NUMBERS

What is the difference between ordinary numbers and bi­nary numbers and what are the advantages of each?

The ordinary numbers we use are "ten-based". That is, they are written as powers of ten. What we write as 7291 is really 7X10³ plus 2x 10² plus 9X10¹ plus 1x10⁰. Remem­ber that 10³=10x 10X10=1000; that 10²=10x 10=100; that 10¹=10 and 10⁰—1, so that 7291 is 7x1000 plus 2x100 plus 9x 10 plus 1. We say this when we read the number aloud. It is "seven thousand two hundred ninety (nine tens) one".

We have grown so accustomed to the use of powers of ten that we just write the digits by which they are multiplied, 7291 in this case, and ignore the rest.

But there is no magic about powers of ten. The power of any other number higher than one would do. Suppose, for instance, we wanted to write number 7291 in terms of powers of eight. Remember that 8⁰=1; 8¹=8; 8²=8x8=64; 8³= =8X8X8=512; and 8⁴=8x8x8x8=4096. The number 7291 can then be written as 1 X8⁴ plus 6x8³ plus 1 X8² plus 7X8¹ plus 3x8⁰. (Work it out and see for yourself). If we write only the digits we have 16173. We can say, then, that (8-based) =7291 (10-based).

The advantage of the 8-based system is that you only need to memorize seven digits besides 0. If you try to use the digit 8, you might have 8x8³ which is equal to 1 X8⁴, so you can always use a 1 instead of an 8. Thus 8 (10-based) =10 (8-based); 89 (10-based) = 131 (8-based) and so on. On the other hand, there are more total digits to the number in the 8-based system than in the 10-based system. The smaller the base, the fewer different digits but the more total digits.

If you used a 20-based system, the number 7291 becomes 18 X 20² plus 4X20¹ plus 11x20⁰. If you wrote 18 as # and 11 as % you could say that # 4% (20-based)=7291 (10- based). You would have to have 19 different digits in a 20- based system but you would have fewer total digits per num­ber.

Ten is a convenient base. It gives us not-too-many dif­ferent digits to remember and not-too-many separate digits in a given number.

What about a number based on powers of two — a 2- based number? It is this which is a "binary number", from a Latin word meaning "two at a time".

The number 7291 equals 1X2¹² plus lx2^u plus lx2¹⁰ plus 0X2⁹ plus 0x2⁸ plus 0x2⁷ plus lX2⁶ plus lx2⁵ plus 1x2⁴ plus 1X2³ plus 0x2² plus 1X2¹ plus 1x2⁰. (Work it out and see, remembering that 2⁹, for instance, is nine two's multiplied together: 2x2x2x2x2x2x2x2x2=512). If we write only the digits we have 1110001111011 (2-based) =7291 (10-based).

Binary numbers contain only l's and 0's, so that addi­tion and multiplication are fantastically simple. However, there are so many digits altogether in even small numbers like 7291 that it is fantastically easy for the human mind to become confused. A computer, however, can use a two-way switch. In one direction, current-on, it can symbolize a 1; in the other di­rection, current-off, a 0. By manipulating the circuits so that the switches turn on and off in accordance with binary rules of addition and multiplication, the computer can perform arithmetical computations very quickly. It can do it much more quickly than if it had to work with gears marked from 0 to 9, as in ordinary desk calculators based on the decimal or 10-based system.

APPENDIX

SIGNS USED IN MATHEMATICS

+ plus

— minus

X times; multiplied by

: divided by; the ratio of... to... = sign of equality; equals, is equal to

Examples: a—b a equals b, a is equal to b

2x3=6 twice three is six

4x5=20 four multiplied by five are (make) twenty

15: 5=3 fifteen divided by five is (equals) three

3: 6=2: 4 three is to six as two to four

3/8 three eighths

0.3 three tenths; nought point three

0.024 nought point nought two four % per cent

25 % twenty five per cent

() parentheses

[ ] brackets

{ } braces

∞ infinity

< is less than

> is greater than

≤ is less than or equal to

≥ is greater than or equal to

x;1/x; x²; x³; x⁴; x^-1; xⁿ eks; one over eks; eks squarred; eks cubed; eks to the fourth power; eks to the minus one; eks to the en

y=f(x) wai is a function of eks

SHORT MATHEMATICS DICTIONARY

absolute value — The numerical value of a number, regard­less of the sign of the number.

acute angle — An angle less than 90° and more than 0°.

acute triangle — A triangle having all acute angles.

amount — The whole; the total; quantity.

angle — A figure formed when two straight lines inter­sect at a point.

approximate number — A number that is not exact but whose accuracy is sufficient for the purpose desired.

arc — Any part or a section of a circumference of a circle.

area — The number of square units contained in the sur­face of a plane figure.

bar graph — A graph made up of parallel bars whose lengths represent given quantities drawn to scale.

base (of a geometric configuration) — A line or surface upon which a plane or solid figure rests.

bisect — To cut into two equal parts, to divide in half.

circle — A closed plane curve all points of which are the same distance (called the radius) from a point within (called the centre).

circle graph — A graph in the form of a circle in which the angles (parts) indicate relations to each other and to the whole.

circumference — The curved line bounding a circle; the length or distance around a circle.

coefficient — A number written in front of an algebraic expression.

common denominator — A number into which all the given denominators divide evenly.

compasses or compass — An instrument for drawing cir­cles and arcs.

cone — A solid figure having a circular base and curved surface which comes to a point at the vertex.

congruent triangles — Triangles that have the same size and shape and can be made to coincide.

consecutive numbers — Numbers that follow one another, such as 1, 2, 3, 4, etc.

corresponding parts — Angles or sides of triangles which are placed in the figure in the same positions.

cube — A rectangular solid with 6 equal square faces or, the product obtained by multiplying a number by itself three times.

cylinder — A stolid figure with bases made of two equal circles and with curved sides.

decagon — A polygon having ten sides and ten angles.

decimal fractions — A part of a whole expressed by using a decimal point.

degree — A unit used in measuring angles. 360°=one complete rotation.

diameter — A straight line drawn through the centre of a circle and dividing the circle into two equal parts.

digit — Any one of the ten numbers from 0 to 9.

dimension — A linear measurement such as the length, width, height of a figure.

equation — A statement showing the equality of two quantities.

equivalent fractions — Fractions having different forms but equal values.

evaluate — To determine the value of an unknown letter in a formula; to find the value of an algebraic expression by substituting in the arithmetic values of the literal quanti ­ ties.

exponent — The; small number or letter written slightly above and to the right of a number or letter to indicate how many times the number is to be multiplied by itself.

factor — One of two or more numbers which when multi­plied together give a certain product.

formula — A statement of a general rule expressed by means of letters and numbers.

graph — A representation of relationships by means of lines, bars, circles or symbols.

height — The distance from the top to the base of an object.

hemisphere — One half of a sphere.

hexagon — A plane figure having six sides and six angles.

hypotenuse — The side opposite the right angle in a right triangle.

isosceles triangle — A triangle having two equal sides.

like terms — The terms of an algebraic expression contain­ing the same letter;.

lowest terms — When both the numerator and denomi­nator of a fraction are reduced as far as possible.

maximum — The greatest value of a quantity.

metric system — A system of weights and measures based on the decimal system.

minimum — The smallest value of a quantity.

monomial — An algebraic expression consisting of a sin­gle term.

negative number — A number whose value is less than zero and which is preceded by minus sign.

obtuse angle — An angle containing more than 90° but less than 180°.

obtuse triangle — A triangle containing an obtuse angle.

octagon — A plane figure containing 8 sides and 8 angles.

parallel lines — Lines that extend in the same direction and are the same distance apart no matter how far extended.

parallelogram — A four-sided figure (quadrilateral) whose opposite sides are parallel.

pentagon — A plane figure with 5 sides and 5 angles.

per cent — A value expressed in hundredths using the per cent sign (%) or the words "per cent".

perimeter — The sum of the lengths around a plane fig­ure.

perpendicular lines — Lines which intersect so as to form right angle.

pi (π) — The ratio of the circumference of a circle to its diameter; π=3.14159 or 22/7.

polygon — A plane figure having any number of sides and angles.

positive number — A number whose value is greater than zero and which is sometimes preceded by plus sign.

protractor — An instrument marked off in degrees used for measuring or making off angles of a given size.

pyramid — A solid figure having triangles for faces.

quadrilateral — Any four sided plane figure.

radius — The distance or straight line from the centre to the circumference of a circle.

ratio — The comparison by means of a division of two like quantities.

rectangle — A quadrilateral whose opposite sides are equal and which has four right angles.

rectangular solid — A solid figure whose six faces are rectangles.

regular figure — A plane figure whose angles are equal and whose sides are equal in length.

right angle — An angle which contains 90°; one fourth of a rotation.

round number — A number that is approximate to a certain extent, not accurate.

scale drawing — A drawing that is the exact shape of an object but which is reduced or enlarged in size in a definite ratio.

scalene triangle — A triangle in which no two sides are equal.

secant — A line drawn through a circle and extending beyond it.

sector — The portion of a circle between two radii and an arc.

semicircle — Half of a circle.

signed number — Positive and negative numbers, directed numbers.

solid figure — A figure having three dimensions: length, width and height.

sphere — A circular solid such that all points on a surface are the same distance from the centre.

square — A rectangle all of whose sides are equal, or the product obtained by multiplying a number by itself two times.

straight angle — An angle containing 180°.

symbol — A representation by means of a sign or a letter.

symmetry — The correspondence of parts such as lines or points.

term — A member of an expression.

trapezoid — A quadrilateral having two parallel sides.

triangle — A closed plane figure with three sides and three angles.

triangular prism — A solid figure having 3 rectangular facesand 2 parallel triangles for bases.

unlike terms — The terms of an algebraic expression con­taining different letters.

vertex — The point of intersection of the sides of an angle.

volume — The number of cubic units in a solid figure.