By O. A. Ivanov (auth.)

The current booklet is unusual, even exact of its variety, a minimum of between arithmetic texts released in Russian. you have got prior to you neither a textbook nor a monograph, even though those chosen chapters from trouble-free arithmetic definitely represent a great academic instrument. it truly is my opinion that this can be greater than simply one other publication approximately arithmetic and the artwork of educating that topic. with out contemplating the particular issues handled (the writer himself has defined those in enough aspect in of the booklet as a complete, the Introduction), I shall try and show a basic inspiration and describe the impressions it makes at the reader. virtually each bankruptcy starts off by way of contemplating famous difficulties of effortless arithmetic. Now, each worthy hassle-free challenge has hidden in the back of its diverting formula what can be known as "higher mathematics," or, extra easily, arithmetic, and it's this that the writer demonstrates to the reader during this booklet. it really is hence to be anticipated that each bankruptcy should still include subject material that's faraway from user-friendly. the result of interpreting the booklet is that the fabric handled has develop into for the reader "three-dimensional" because it have been, as in a hologram, in a position to being seen from all sides.

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**Extra info for Easy as π?: An Introduction to Higher Mathematics**

**Example text**

Exercise. Establish directly an isomorphism between Problem 16 on the binomial coefficients and Problems 12-14 (generalized). ), we obtain the following particular identities: 2n = ~ (:} 3n = ~2k(:} 0= ~(-Il(:). The last of these may be rewritten as L (n)_ L (n) keven k - kodd k . There are a great many other relations among the binomial coefficients, many of which can be obtained by exploiting properties of the function Pn(x) := L~=o (~)xk. • Lemma 1. The identity Pn+l(x) = (x + I)Pix) is equivalent to (ntl) = (k~I)' This is immediate from (x + I) L~=o mxk Corollary.

Exercise. Establish directly an isomorphism between Problem 16 on the binomial coefficients and Problems 12-14 (generalized). ), we obtain the following particular identities: 2n = ~ (:} 3n = ~2k(:} 0= ~(-Il(:). The last of these may be rewritten as L (n)_ L (n) keven k - kodd k . There are a great many other relations among the binomial coefficients, many of which can be obtained by exploiting properties of the function Pn(x) := L~=o (~)xk. • Lemma 1. The identity Pn+l(x) = (x + I)Pix) is equivalent to (ntl) = (k~I)' This is immediate from (x + I) L~=o mxk Corollary.

Lemma 2. Ifa motion ofthe planefixes each oftwo distinct points. then it is either the identity transformation or the reflection in the straight line through those two points. " ····--1. ·. . \ \ ' ......... I :Ci I I Denote by A and B the two fixed points. Let C be any point on the line segment A B and denote by CI the image of C under the motion in question. Since ACI + CIB = AC+CB = AB, the point CI must also lie on the segment AB, whence it follows that in fact CI = C. A similar argument shows that every point of the line through A and B is fixed.