Печатается по решению заседания кафедры английского языка

In this chapter you will deal with rational numbers. Let us begin like this.

John has read twice as many books as Bill. John has read 7 books. How many books has Bill read?

This problem is easily translated into the equation 2n = 7, where n represents the number of books that Bill has read. If we are allowed to use only integers, the equation 2n=7 has no solution. This is an indication that the set of integers does not meet all of our needs.

If we attempt to solve the equation 2n = 7, our work might appear as follows.

2n=7, 2n 7 2 7 7 7

2 = 2 , 2 x n = 2, 1x n= 2, n= 2 .

The symbol, or fraction, 7/2 means 7 divided by 2. This is not the name of an integer but involves a pair of integers. It is the name for a rational number. A rational number is the quotient of two in­tegers (divisor and zero). The rational numbers can be named by fractions. The following fractions name rational numbers:

2, 3, 5, 1, 4. a

We might define a rational number as any number named by n

where a and n name integers and n ≠ 0.

Let us dwell on fractions in some greater detail.

Every fraction has a numerator and a denominator. The deno­minator tells you the number of parts of equal size into which some quantity is to be divided. The numerator tells you how many of these parts are to be taken.

Fractions representing values less than 1, like 3 (two thirds) for example, are called proper fractions. Fractions which name a number

equal to or greater than 1, like 2 or 2 , are called improper fractions.

There are numerals like 1 2 (one and one second), which name a whole number and a fractional number. Such numerals are called mixed fractions.

Fractions which represent the same fractional number like

1 2 3 4

2, 4, 6, 8, and so on, are called equivalent fractions.

We have already seen that if we multiply a whole number by I we shall leave the number unchanged. The same is true of fractions since when we multiply both integers named in a fraction by the same number we simply produce another name for the fractional number.

1 1

For example, l x 2 = 2 We can also use the idea that I can be as

expressed a fraction in various ways: 2, 3, 4, and so on.

Now see what happens when you multiply 2 by 2. You will have

2 = 1 2 = 2 x 2 = 2x2 = 4 . As a matter of fact in the above

operation you have changed the fraction to its higher terms.

6 6 2 6 : 2 3

Now look at this: 8 : 1 = 8 : 2 = 8 : 2 = 4.

In both of the above operations the number you have chosen for 1 is

2 6

2 In the second example you have used division to change 8 to lower

terms, that is to 4 .The numerator and the denominator in this fraction are

relatively prime and accordingly we call such a fraction the simplest fraction for the given rational number.

You may conclude that dividing both of the numbers named by the numerator and the denominator by the same number, not 0 or 1 leaves the fractional number unchanged. The process of bringing a fractional number to lower terms is called reducing a fraction.

To reduce a fraction to lowest terms, you are to determine the greatest common factor. The greatest common factor is the largest possible integer by which both numbers named in the fraction are divisible.

From the above you can draw the following conclusion⁶: mathe­matical concepts and principles are just as valid in the case of rational numbers (fractions) as in the case of integers (whole num­bers).

6. 
   DECIMAL NUMERALS
   

In our numeration system we use ten numerals called digits. These digits are used over and over again in various combinations. .Suppose, you have been given numerals 1, 2, 3 and have been asked to write all possible combinations of these digits. You may write 123, 132, 213 and so on. The position in which each digit is written af­fects its value. How many digits are in the numeral 7086? How many place value positions does it have? The diagram below may prove helpful. A comma separates each group or period. To read 529, 248, 650, 396, you must say: five hundred twenty-nine billion, two hundred forty-eight million, six hundred fifty thousand, three hundred ninety-six.

But suppose you have been given a numeral 587.9 where 9 has been separated from 587 by a point, but not by a comma. The nume­ral 587 names a whole number. The sign (.) is called a decimal point.

All digits to the left of the decimal point represent whole numbers. All digits to the right of the decimal point represent fractional parts of 1.

The place-value position at the right of the ones place is called tenths. You obtain a tenth by dividing 1 by 10. Such numerals like 687.9 are called decimals.

You read .2 as two tenths. To read .0054 you skip two zeroes and say fifty four ten thousandths.

Decimals like .666..., or .242424..., are called repeating decimals. In a repeating decimal the same numeral or the same set of nume­rals is repeated over and over again indefinitely.

We can express rational numbers as decimal numerals. See how it may be done.

31 4 4 x 4 16

100 = 0.31. 25 = 4 x 25 = 100 = 0.16

The digits to the right of the decimal point name the numerator of the fraction, and the number of such digits indicates the power of 10 which is the denominator. For example, .217 denotes numerator 217 and a denominator of 10³ (ten cubed) or 1000.

In our development of rational numbers we have named them by fractional numerals. We know that rational numerals can just as well be named by decimal numerals. As you might expect, calcula­tions with decimal numerals give the same results as calculations with the corresponding fractional numerals.

Before performing addition with fractional numerals, the fractions must have a common denominator. This is also true of decimal nu­merals.

When multiplying with fractions, we find the product of the numerators and the product of denominators. The same procedure is used in multiplication with decimals.

Division of numbers in decimal form is more difficult to learn because there is no such simple pattern as has been observed for mul­tiplication.

Yet, we can introduce a procedure that reduces all decimal-divi­sion situations to one standard situation, namely the situation where the divisor is an integer. If we do so we shall see that there exists a simple algorithm that will take care of all possible division cases.

In operating with decimal numbers you will see that the arithme­tic of numbers in decimal form is in full agreement with the arith­metic of numbers in fractional form.

You only have to use your knowledge of fractional numbers.

Take addition, for example. Each step of addition in fractional form has a corresponding step in decimal form.

Suppose you are to find the sum of, say, .26 and 2.18. You can change the decimal numerals, if necessary, so that they denote a com­mon denominator. We may write .26 = .260 or 2.18 = 2.180. Then we add the numbers just as we have added integers and denote the common denominator in the sum by proper placement of the decimal point.

We only have to write the decimals so that all the decimal points lie on the same vertical line. This keeps each digit in its proper pla­ce-value position.

Since zero is the identity element of addition it is unnecessary to write .26 as .260, or 2.18 as 2.180 if you are careful to align the de­cimal points, as appropriate.

6. 
   THE DIFFERENTIAL CALCULUS
   

No elementary school child gets a chance of learning the differential calculus, and very few secondary school children do so. Yet I know from my own experience that children of twelve can learn it. As it is a mathematical tool used in most branches of science, this form a bar between the workers and many kinds of scientific knowledge. I have no intention of teaching the calcu­lus, but it is quite easy to explain what it is about, particularly to skilled work­ers. For a very large number of skilled workers use it in practice without knowing that they are doing so.

The differential calculus is concerned with rates of change. In practical life we constantly come across pairs of quantities which are related, so that af­ter both have been measured, when we know one, we know the other. Thus if we know the distance along the road from a fixed point we can find the height above sea level from a map with contour. If we know a time of day we can determine the air temperature on any particular day from a record of a ther­mometer made on that day. In such cases we often want to know the rate of change of one relative to the other.

If x and y are the two quantities then the rate of change of y relative to x is written dy/dx. For example if x is the distance of a point on a railway from London, measured in feet, and y the height above sea level, dy/dx is the gradi­ent of the railway. If the height increases by 1 foot while the distance x in­creases by 172 feet, the average value of dy/dx is 1/172. We say that the gradi­ent is 1 to 172. If x is the time measured in hours and fractions of an hour, and y the number of miles gone, then dy/dx is the speed in miles per hour. Of course, the rate of change may be zero, as on level road, and negative when the height is diminishing as the distance x increases. To take two more examples, if x the temperature, and y the length of a metal bar, dy/dx—:—y is the coeffi­cient of expansion, that is to say the proportionate increase in length per de­gree. And if x is the price of commodity, and y the amount bought per day, then -dyldx is called the elasticity of demand.

For example people must buy bread, but cut down on jam, so the de­mand for jam is more elastic than that for bread. This notion of elasticity is very important in the academic economics taught in our universities. Professors say that Marxism is out of date because Marx did not calculate such things. This would be a serious criticism if the economic "laws" of 1900 were eternal truths. Of course Marx saw that they were nothing of the kind and "elasticity of demand" is out of date in England today for the very good reason that most commodities are controlled or rationed.

The mathematical part of the calculus is the art of calculating dy/dx if y has some mathematical relations to x, for example is equal to its square or loga­rithm. The rules have to be learned like those for the area and volume of geo­metrical figures and have the same sort of value. No area is absolutely square, and no volume is absolutely cylindrical. But there are things in real life like enough to squares and cylinders to make the rules about them worth learning. So with the calculus. It is not exactly true that the speed of a falling body is proportional to the time it has been falling. But there is close enough to the truth for many purposes.

The differential calculus goes a lot further. Think of a bus going up a hill which gradually gets steeper. If x is the horizontal distance, and y the height, this means that the slope dy/dx is increasing. The rate of change of dy/dx with regard to y is written d^2y/dx². In this case it gives a measure of the curvature of the road surface. In the same way if x is time and distance, d^2y /dx² is the rate of change of speed with time, or acceleration. This is a quantity which good drivers can estimate pretty well, though they do not know they are using the basic ideas of the differential calculus.

If one quantity depends on several others, the differential calculus shows us how to measure this dependence. Thus the pressure of a gas varies with the temperature and the volume. Both temperature and volume vary dur­ing the stroke of a cylinder of a steam or petrol engine, and the calculus is needed for accurate theory of their action.

Finally, the calculus is a fascinating study for its own sake. In February 1917 I was one of a row wounded officers lying on stretchers on a steamer going down the river Tigris in Mesopotamia. I was reading a mathematical book on vectors, the man next to me was reading one on the calculus. As anti­dotes to pain we preferred them to novels. Some parts of mathematics are beautiful, like good verse or painting. The calculus is beautiful, but not because it is a product of "pure thought". It was invented as a tool to help men to cal­culate the movement of stars and cannon balls. It has the beauty of really efficient machine.
VIII. RAYS, ANGLES, SIMPLE CLOSED FIGURES
1. You certainly remember that by extending a line segment in
one direction we obtain a ray. 2. Below is a picture of such an ex-­
tension.


3. The arrow indicated that you start at point M, go through point N, and on without end. 4. This results in what is called ray MN_, which is denoted by the symbol MN. 5. Point M is the endpoint in this case. 6. Notice that the letter naming the endpoint of a ray is given when first naming the ray.

7. From what you already know you may deduce that drawing two rays originating from the same endpoint forms an angle. 8. The com­mon point of the two rays is the vertex of the angle.

9. Angles, though open figures, separate the plane into three distinct sets of points: the interior, the exterior, and the angle. 10. The fol­lowing symbol  is frequently used in place of the word angle.

11. 
    The angle pictured above could be named in either of the follo­wing ways: a) angle LMN (or LMN); b) angle NML (or ∟NML).
    
12. 
    The letter naming the vertex of an angle occurs as the middle
    letter in naming each angle. 13. Look at the drawing below.
    

11. 
    Ray PA (PA) and ray PB (PB) form a right angle, which means
    

that the angle has a measure of 90° (degrees). 15. Since PC

→

measure less than 90°. 16. Since PD (except for point P) lies in the exterior of APB, we say that APD is greater than a right angle and call it an obtuse angle with a degree measure greater than 90°.

17 A simple closed figure is any figure drawn in a plane in such a way that its boundary never crosses or intersects itself and encloses part of the plane. 18. The following are examples of simple closed figures. 19. Every simple closed figure separates the plane into three distinct sets of points. 20. The interior of the figure is the set of all points in the part of the plane enclosed by the figure. 21. The exterior of the figure is the set of points in the plane which are outside the figure. 22. And finally, the simple closed figure itself is still another set of points.

angles or the measure of the sides. 26. This is true of triangles — geometric figures having three sides — as-well as of quadrilaterals, having four sides.

27. 
    In the picture above you can see three triangles. ∆ABC is referred to as an equilateral triangle. 29. The sides of such a triangle all have the same linear measure. 30. ∆DEF is called an isosceles triangle which means that its two sides have the same measure. 31. You can see it in the drawing above. 32. ∆ALMK being referred to as a right triangle means that it contains one right angle. 33. In ∆MKL, M is the right angle, sides MK and ML are called the legs, and side KL is called the hypotenuse. 34. The hypote­nuse refers only to the side opposite to the right angle of a right triangle. Below you can see quadrilaterals.
    

 

circles. This ratio is expressed d 17. Since d = 2r (the length of a diameter is equal to twice the length of a radius of the same circle), the follo­wing denote the same ratio.

18. The number d or 2r which is the same for all circles, is designated by π 19. This allows us to state the following:

20. By using the multiplication property of equation, we obtain the following: