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趣味數學
1)Acute-angled Triangles:
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新數學
ELEMENTARY MODERN MATHEMATICS
LESSON SIX
新數學
Hodam Kathonatics (4)
In 1654 the Chevalier de Mere', gambler and amateur mathematician, prom posed to Blaise Pascal a problem concern- ing the division of stakes in a game of dice. Pascal communicated the problem to Format, and from the correspondence between these two men arose what has subsequently become the modern theory of probability, Thug did a simple gam- bler's problem give birth to a powerful technique which constitutes the very foundation of mathematical statistics, and, through statistics, of much of the mathematics of economics and industry.
Consider the type of problem ezig- inally discussed, by Fascal and Fermate Suppose that two players, A and B, con- tribute equally to a stake of $12. They agree that the first player who makes 3 peints shall win the entire stake. After ▲ has won 2 points, and B has won 1, they agres to stop. How should the stake be divided?
* Offhand, this problem appears to be very simple. We may wall argue that since à has twice as many points as B. A's share should be twice B's, That is to say, A should take $8, and B. $4. But now suppons they were to play the nezt point--the oze they have agreed not to play. If A were to win this point, the whole atake of £12 would be his, If he were to lose, the score would then be 2 to 2, and they would split. the $12 evenly. Thus A is sure of getting 16- anyway And assuring that he has an even chance of winning the point, his share of the remaining $6 should be half that amount. In other words, ▲ should take $9 and B $3.
It is not difficult to see that the second solution is correct if A and B: are to stick to their original agree- ment concerning the winning of the stake, Had they agreed to divide the stake in proportion to their scores at any stage: of the game, the correct solution would “of course be the first one,
But we must not jump too quickly into the midst of difficulties.
Wa shall spend a few moments discussing Some of the basic principles of prebabi- litro
Laplace, an outstanding French math- ematician, once described the theory, of probability as nothing but "common sense reduced to calculation," Let us nes to what extent the following example juste ifies this description.
Two students are trying to decide! how to pass an afternoon. They finally agree to let their decision rest on the tess of a coin. Heade, they go to play football. Teils, they go to swim, and if the coin stands on edge,
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This story is not as meaningless as it may seem, for we can learn much from it. Common sense, basing its Judg- ment on Past experience, tells us that. the boys will be spared the necessity. of studying. In other worda, we know that the coin will not stand on edge, but that it will come to rest with either heads or tails showing. Moreover, if the coin is fair-if it does not have heads, Bay, on both sides-we are sure that the probability of heads and the probability of tails are equally likely.
New the theory of probability is based on the assumptions we make concern- ing such questions as these: What is the probability that the coin will stand on edge? What is the probability that it
will show either head or tails? What In the probability that it will show heads? What is the probability that It will show taila?
In order to discuss these questions in mathematical terme, it is necessary to assign numerical values to the vari- ous probabilities involved. Suppose, we denote by x the numerical value of the probability that the coin will show heads, Since it is equally likely that the coin will show tails, the probability of taile But we are sure that the must also be z coin will show either heads or taile. Hence 2x must have the Value of certainty of the probability that an event which in bound to occur will occur. We can choose for certainty any value we like. It is convenient to choose the value 1. That is to say, 2x - 1. Then the proba➡- bility that the coin will show heads is
that it will show taila ; and that it will show either heads or tails, 1+ 3, or 1..
We can generalize our definition of the measure of probability in the follow ing, ways Suppose that the number of ways in which a certain event can happen is h, and that the number of ways in which it can fail to happen is f. Suppose further that the ways in which the event can happen or fail are all equally likely. The probability that the event will happen is h/(h+f), the probability that it will Tail is f/(h+1), and the probabi lity that it will happen or fail is h/(h+1) + 1/(h+f)
Fiddles in Mathematics (3)
Kost of the surfaces met in every day life are "bilateral" or two-sided, A sheet of paper, for example, has two. sides. If a fly were placed on one side, he could get to a point on the other side by cutting through the paper-bow a fly could do this is besides the point
A sphere For by going over the edge. is a closed bilateral surface.
The fly could crawl all over the outside, and could get to the inside only by going, through the surface.
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Let us consider a simple example. of a unilateral surface--one which is First: somewhat easier to construct. take a long narrow, rectangular strip of paper and paste the ends together as shown in figure 1. The result is a oylindrical surface which has two sides and two edges, We shall raterj to this strip as 91.
Fig 1 An ordinary strip (51)
Fig 2 The Nobrus strip (Sa)
The
sting the ends Now before Pasting the ends of a similar strip together, give one of them a half-twist--a twist through 180
ine strip", resulting surface, called a "Mobize is a one-side surface with but one edge. An attempt has been made, in figure 2 to show what this surface looks like, but you had better actually construct one if you want to study its properties in detail. To convinoe yourself that it haa but one side, start at any point, and draw a line down the middle. Keep on drawing, without lifting the pencil from the paper, until you return to the] point from which you started. You will find that the single line has completely traversed what constituted, before the ends were pasted together, the two sides of the original rectangle piece of paper. We shall, for convenience, call the Mobius strip 32.
Finally, if one of the ende is turged through a full twist (through. 360) before Pasting, the resulting, surface, like Sl has two sides and two edgea. We shall refer to this strip illustrated in fig. 3 as 53.
Fig. 3 The strip S...
And now get out your scissors, for have more to do. Suppose we out the bilateral surface 51 along a line mid-}
between the eigas. It is not dif Waya ficlut to see that we obtain two separ- ate strips, identical with the original one except they are half as wide. But what of we out the Möbius atrip $2 in the same manner? Anyone who can prediot the result before actually carrying out the experiment must have batter intuit- ive powera than average. For the result in a single strip—not two-trice : long and half as wide as the original one. Furthermore, it is no longer unilateral, but is a bilateral strip of the type S3. And what if this strip 53 is cut down the middle? Here the result consists of two interlocked sur- faces of the type 33, each of them equal in length to the strip from which they were cut and half as wide.
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For a final surprise, take a new strip 52 the unilateral Kobus strip- and cut it along a line which runs par- allel to the edge and a third of the width of the strip from the edge. Keep outting until you are back at the start- ing point, The result? Two interlooked surfaces, one equal in length to the orginal strip, the other twice as long. The shorter one is again of type 82, the lengar one of type $3.
Try the experiments described and invent variations of your own. Prediot- ing the results will give your intuition goed practice.
perfolding Geometry
Now luet us go over to the? study of properties of figures. In Geometry, the simplest figu re on the plane is made up of
three sides and three angles, (An area bounded by 3 straight lines. It is called a "Triangle." In this lesson ▼ we are going to learn some sim- ple properties of the triangle.
There are three kinds of triangles:-
現代數學
Riddles in mathematics (2)
»lerical curiosities.
Let us think for a moment of the shape of a quiet, er anchor-ring, Is that part which constitutes the hols inside or sutside the ring? We gener ally avoid the wordy phrase used hers and speak of "the hele in a ring", implying, however unconsciously, that it is inside. But is the inside really inside or outside? If we go on to debate the question without first set tling upon some sort of definition of "inside" and "outside", sur argument is likely to be quite fruitless.
The problem of what constitutes the inside and the outside of a ring is the concern of the student of "topology!! er "analysisoitus" (literally, the analysis of situation, er position). Ordinary plane and solid geometry are essentially quantitative, dealing an they do with the sizes of things-the lengths of lines, the areas of surfaces, and the volumes of selida. Topology; en the ether hand, is a kind of geometry which igneren siges and concentrates en auch qualitative questions ze whether
osrtain point is inside, en, er outside a certain closed curve er zurface.
oircle
The same circle distorted
To be more specific, consider tha circle shewn in the above figure, The
student of plane geometry is interested in such things as the number of inches in the circunference of the circle, er in the number of inches the distance from the centre 0 to the paint P, OT in the number of square inches in the) area of the circle. The topalegiat, en the other hand, is interested in this Bart er question: The paint P is invide the circle, the point Q on the cirols. and the paint R outside the circle. Now suppass the ofrole is drawn on a sheet of rabbar and then stretched and diste erted in any way, shape or manners-me provided it is not tern, yes Patills lie inside the curve? Loss Q still lis en the curva? Doas R still lie entside the curve? The answer to all three of these questions is abriously you, but.... this problem is an elementary, one,
The science of topology is relat ivaly young. The first systematio werk in the subject appeared about the middle
A triangle with all angles
less than a right angle, 2) Right-angled Triangles:
A triangle with one of its
angles equal to a right angle. 3) Obtuse-angled Triangles:
A triangle with one of its angles greater than a right angle.
These triangles can be obtained by paperfolding inthe following way:-
a) Aright-angled triangle can be obtained by folding a piece of rectangular paper. Mark 2 points B,C, on the sides ofthe rectangular paper as shown in the following figure, Folding along BC will resut in getting a right-angled triangle on une folding (see figure below)
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of the 19th century. But in 1736, over- a hundred years earlier, Fuler published the first single result of any tapslage 1oal consequence, Let us look at his problen.
› In the German tem af Königsberg ran the river Pregel. In the river verä two islands, connected with the mainland and with each other by seven bridges, as shown in the felleying diagram.
(D)
1 frequent topic of conversation in the town was whether or not it was possible for a person to Bet out for a walk from any point in the town, orosa each bridge once and only once, and return to his starting point, No one had ever found a way to do this, but en the other hand, no one had ever been. able to prove that a way did not exist. Fuler heard of the problem and want about its solution in a systematio manner. He noted--and here the topaloge ical method oreeps in--that the problem is unchanged if the somewhat complicated figure above is replaced by the simpla diagram in the following.
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Then the original problem is equivas lent to thiaï is it possible to start u at any point and trace this diagram with a pencil without lifting the pencil from the paper and without retracing any. portien of the diagram? Buler proved not only that this is impossible, but went on to establish additional résulta for diagrams of a more general nature. Incidentally, the diagram above can be traced in the manner indicated if the bridge BD is replaced by one from A ̈to