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 12

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