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Alfred S. Posamentier et al. - The Joy of Mathematics: Marvels, Novelties, and Neglected Gems That Are Rarely Taught in Math Class

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Alfred S. Posamentier et al. The Joy of Mathematics: Marvels, Novelties, and Neglected Gems That Are Rarely Taught in Math Class
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The Joy of Mathematics: Marvels, Novelties, and Neglected Gems That Are Rarely Taught in Math Class: summary, description and annotation

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Wouldnt it be great if all school teachers (from kindergarten through high school) would share the joy of mathematics with their students, rather than focus only on the prescribed curriculum that will subsequently be tested? This book reveals some of the wonders of mathematics that are often missing from classrooms. Heres your chance to catch up with the math gems you may have missed.
Using jargon-free language and many illustrations, the authors--all veteran math educators--explore five areas--arithmetic, algebra, geometry, probability, and the ways in which mathematics can reinforce common sense. Among other things, youll learn the rule of 72, which enables you to quickly determine how long it will take your bank account to double its value at a specific interest rate. Other handy techniques include an automatic algorithm for multiplying numbers mentally and a clever application that will allow you to convert from miles to kilometers (or the reverse) mentally. A delightful presentation of geometric novelties reveals relationships that could have made your study of geometry more fun and enlightening. In the area of probability there is a host of interesting examples: from the famous Monty-Hall problem to the counterintuitive probability of two people having the same birthday in a crowded room.
Finally, the authors demonstrate how math will make you a better thinker by improving your organizing abilities and providing useful and surprising solutions to common mathematics problems. Youll come away with an appreciation for math you never thought possible and a true appreciation for this queen of the sciences.

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The authors wish to acknowledge superb support services received from the - photo 1

The authors wish to acknowledge superb support services received from the publisher Prometheus Books, led by their editor in chief, Steven L. Mitchell, and his truly dedicated production coordinator, Catherine Roberts-Abel. We wish to also thank Senior Editor Jade Zora Scibilia for her highly meticulous editing and clever suggestions to make the presentation as intelligible as possible. Thanks is also due to Editorial Assistant Hanna Etu, and the typesetter, Bruce Carle. The cover design exhibits the talents of Nicole Sommer-Lecht. We're also very pleased with the indexing by Laura Shelley.

Each of the authors has many people to thank for their patience and support throughout this book-development process. In particular, Dr. Christian Spreitzer wants to thank Katharina Brazda for inspiring discussions, which resulted in some very creative contributions.

As we mentioned earlier Cevas theorem might well have been introduced to a - photo 2

As we mentioned earlier, Ceva's theorem might well have been introduced to a high school class, since it merely applies similarity relationships that are an integral part of the geometry curriculum. We offer one of many proofs available to justify Ceva's theorem. It is perhaps easier to follow the proof by looking at the left-side diagram in and then verifying the validity of each of the statements in the right-side diagram. In any case, the statements made in the proof hold for both diagrams.

Figure App1 Consider for which we have on the left triangle ABC with a - photo 3

Figure App.1.

Consider , for which we have on the left triangle ABC with a line (SR) containing A and parallel to BC, intersecting CP extended at S and BP extended at R.

The parallel lines enable us to establish the following pairs of similar triangles:

Now by multiplying I II and V we obtain our desired result This - photo 4

Now by multiplying I II and V we obtain our desired result This - photo 5

Now by multiplying (I), (II), and (V), we obtain our desired result:

This can also be written as AM BN CL MC NA BL A nice way to read this - photo 6

This can also be written as AMBNCL = MCNABL. A nice way to read this theorem is that the product of the alternate segments along the sides of the triangle made by the concurrent line segments (called cevians) emanating from the triangle's vertices and ending at the opposite side are equal.

Yet, it is the converse of this proof that is of particular value to use here. That is, if the products of the alternate segments along the sides of the triangle are equal, then the cevians determining these points must be concurrent.

We shall now prove that if the lines containing the vertices of triangle ABC intersect the opposite sides in points L, M, and N, respectively, so that The Joy of Mathematics Marvels Novelties and Neglected Gems That Are Rarely Taught in Math Class - image 7, then these lines, AL, BM, and CN, are concurrent.

Suppose BM and AL intersect at P. Draw PC and call its intersection with AB point N'. Now that AL, BM, and CN' are concurrent, we can use the part of Ceva's theorem proved earlier to state the following:

The Joy of Mathematics Marvels Novelties and Neglected Gems That Are Rarely Taught in Math Class - image 8

But our hypothesis stated that The Joy of Mathematics Marvels Novelties and Neglected Gems That Are Rarely Taught in Math Class - image 9.

Therefore, Picture 10, so that N and N must coincide, which thereby proves the concurrency.

For convenience, we can restate this relationship as follows: If AMBNCL = MCNABL, then the three lines are concurrent.

The ambitious reader may want to see other proofs of Ceva's theorem, which can be found in Advanced Euclidean Geometry by Alfred S. Posamentier (New York: John Wiley and Sons, 2002, pp. 2731.

When you think of arithmetic you typically consider the four basic arithmetic - photo 11

When you think of arithmetic, you typically consider the four basic arithmetic operations. With a little more thought, you tend to tag on the square root operation as well. Unfortunately, most of our school curriculum focuses on ensuring that we have a good mechanical command of the arithmetic operations and know the number facts as best we can to service us efficiently in our everyday life. As a result, most adults are not aware of the many amazing relationships that can be exhibited arithmetically with numbers. Some of these can be extremely useful in our everyday life as well. For example, just by looking at a number and determining if it is divisible by 3, 9, or 11 can be very useful, especially if it can be done at a glance. When it involves determining divisibility by 2, we do this without much thought, by simply inspecting the last digit. We shall extend this discussion to considering divisibility by a prime number, something that clearly is not presented in the school curriculum, with which we hope to motivate the reader to investigate further primes beyond those shown here. We truly expect that the wonders that our number system holds, many of which we will present in this book, will motivate you to search for more of these curiosities along with their justifications. Some of the units in this chapter will also provide you with a deeper understanding for our number system beyond merely arithmetic manipulations. Our introduction to a variety of special numbers will generate a greater appreciation of arithmetic than the typical school courses provide. Let us begin our journey through numbers and their operations.

WHEN IS A NUMBER DIVISIBLE BY 3 OR 9?

Teachers at various grade levels often neglect to mention to students that in order to determine whether a number is divisible by 3 or 9, you just have to apply a simple rule: If the sum of the digits of a number is divisible by 3 (or 9), then the original number is divisible by 3 (or 9).

An example will best firm up your understanding of this rule. Consider the number 296,357. Let's test it for divisibility by 3 (or 9). The sum of the digits is 2 + 9 + 6 + 3 + 5 + 7 = 32, which is not divisible by 3 or 9. Therefore, the original number, 296, 357, is not divisible by 3 or 9.

Now suppose the number we consider is 457,875. Is it divisible by 3 or 9? The sum of the digits is 4 + 5 + 7 + 8 + 7 + 5 = 36, which is divisible by 9 (and then, of course, divisible by 3 as well), so the number 457,875 is divisible by 3 and by 9. If by some remote chance it is not immediately clear to you whether the sum of the digits is divisible by 3 or 9, then continue with this process; take the sum of the digits of your original sum and continue adding the digits until you can visually make an immediate determination of divisibility by 3 or 9.

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