Point-and-Line Proof for the Sum of the CubesAuthor(s): Barbara TurnerSource: The Two-Year College Mathematics Journal, Vol. 12, No. 4 (Sep., 1981), pp. 270-271Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/3027076 .

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ff(t) dt = xf(x) - af(a) - fx)g(y) dy.

Therefore,

d j~xf(x) - af(a) -ffx) g(y)dy} d {f f(t) dt = fx),

and it follows that

ff(x)dx= xf(x) - G(f(x)) + C.

Point-and-Line Proof for the Sum of the Cubes Barbara Turner, California State University, Long Beach, CA

An interesting combinatorial proof of the well-known identity

n2 ( ,21

can be geometrically obtained by counting points of intersection of two sets of lines in the plane.

Let Sn = {(X1X2, . . . , Xn) and T, = {Y Y2' . . . ,yn) each be sets of n points in one plane such that no three of the 2n distinct points are collinear, and no line connecting points in Sn is parallel to a line connecting points in Tn. The total number of Sn-lines (i.e., lines connecting points of Sn), is the number of combina- tions of n objects taken 2 at a time, (n) = n(n - 1)/2. Geometrically, xl determines (n - 1) lines joining xl to the other (n - 1) points, the point x2 determines (n - 2) additional lines joining it to the points X3, X4, . . ., xn, etc. Thus, the total number of Sn-lines (and Tn-lines) equals

n- n(n -) (n -1) + (n -2) + ***+ 2 + I E i= (2)=n)

i=1 1

These (n) Sn-lines intersect the (n) Tn-lines in exactly

()2 2 (,=)

points. We now count these points of intersection in another way by considering the

points of intersection of Sk-lines with Tk-lines. We determine the increase in the number of points of intersection when one point is added to each set, with Sk+I = Sk U {Xk+1) and Tk+l = Tk U {Yk+1)} The addition of the point Xk+1 adds k new lines to the lines determined by Sk. These new lines intersect the (k) Tk-lines in k(2) points. Also, the addition of Yk+ 1 to Tk adds k new lines to the lines determined by Tk. These new lines intersect the (k+ 1) Sk+ -lines in k(k+ 1) points.

270

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Thus, the total increase in the number of points of intersection, by adding one point to Sk and one point to Tk, is k(k) + k(k, l) = k3.

Now, the sets Sn and Tn can be built up, starting with the sets S, = (xl} and T, = {y,}, adding one point at a time to each set. Since S, and T, determine no lines, there are no points of intersection at the first step. When S, and T, are enlarged to S2 = (xl, x2} and T2 = { Y I, Y2}, one point of intersection is introduced. When S2 and T2 are enlarged to S3 = (X1,X2,x3} and T3 = { Y 'Y2' Y3}, 23 points of intersection are added. Continuing in this manner, we see that the Sr-lines will intersect the T -lines in EI-?i3 points. Therefore,

n-i i3(fn)2

Other geometrical proofs of these and related formulas are presented by Martin Gardner [Mathematical Games, Sci. Amer. (September 1972) 114-117] and Warren Page [Math. Mag. (to appear)].

Vector Identities from Quaternions William C. Schulz, Northern Arizona University, Flagstaff, AZ

Most students are not too familiar with the four-dimensional linear algebra of quaternions given by Hamilton a century ago. Thus, most students cannot appreci- ate how much of three-dimensional vector analysis was presented in terms of quaternions during the period 1850-1900. (For further discussions of historical interest, see Birkhoff and Mac Lane, A Survey of Modern Algebra; Kelland and Tait, Introduction to Quaternions, London, 1873; and Kline, Mathematical Thought from Ancient to Modern Times.) Our objective, in addition to briefly introducing quaternions into a classroom discussion, is to illustrate how elementary vector identities, such as the formula for the vector triple product

a X (b X c) = (a . c)b - (a * b)c,

can be obtained from quaternions. A quaternion is an expression of the form A = ao + ai + aj + a3k, where the

coefficients {ao,al,a2, a3) are real, and multiplication (denoted o) of the units (i,j, k) is defined in the table below. For our purposes, a quaternion will be written A = ao + a, where a = ali + aj + a3k is a vector in three-dimensional space. It is easily verified that the quaternions constitute a four-dimensional vector space over the reals when equality, addition, and scalar multiplication are defined in terms of their scalar and vector components.

Cross Product Quaternion Product Dot Product x i j | k o | j k jk

X 0 k I-j i- 1 0 0

j -k 0 i j -k - | j 0 1 0

k | -i 0 k j -i I kO? 0l

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