Number Patterns and Mathematical Knowledge

Transcription

1 Number Patterns and Mathematical Knowledge Keywords Numbers, Sums, Square Pyramid, Exponents Abstract Sum/sum-of-squares identities. Using a simple algebraic fact. Consecutiveinteger relationships. Patterns and symmetry. Computer exploration and math learning. 1 Introduction This article concerns examples that are possible starting points for a school lesson. The idea is to present them in connection with related mathematics topics. In discovering the general truths implied by the examples, lessons can be drawn to highlight fundamental notions - proof, identities, and induction - or to develop skill with algebra. Start with the Pythagorean theorem and Fermat s conjecture, both below in equation form: x 2 + y 2 = z 2 (1) x n + y n = z n (2) There is a well-known number-relation that anchors this paper. It is only one of many numerical cases that satisfy equation (1), but it plays a dual role later, so we present it here: x = 3, y = 4, z = 5 (3) Many other interesting number-relations exist; two are: = (4) = 5 3 (5) Someone curious about (1) and (2) might ask a question like these. How many terms should appear in an expression involving squares summed? What do we know about relationships of numbers raised to the same or different powers and then added? 1

2 2 Discovery With rope, a straight stick, and knowledge [basically (3), (1) and its geometric meaning] it is possible to build a pyramid with 90 corners: see [18]. The numerical patterns in the many triples satisfying (1) are fascinating. Internet exposition about them [5] is complete and accessible. That means that teachers can use both examples and on-line text to offer new modes, interesting ways for people to go beyond names like Pythagorean triple and explore on their own in learning math. Much different series of numerical examples follow from the following two statements. The first is from a magazine column puzzle [2], while the second is from ancient knowledge transmitted via a textbook s preliminary statement in an exercise [12]: We are interested in finding a sequence of 2n + 1 consecutive positive integers, such that the sum of the squares of the first n + 1 integers equals the sum of the squares of the last n integers. The simplest such sequence is = 5 2 (6) The second goes beyond squares, raising the issue of higher powers: The cube of every positive integer is a sum of consecutive odd integers. [13] The sources for [2] are [11], an unavailable letter [16] mentioned there, and [17]. The last presents the first four equations involving summing sequences of adjacent integers-squared, while [11] presents the first five such by making use of an observation from [16]. All these appear in Table 1. Table 2 shows how four specific cubed positive integers, 5, 6, 7, and 8 can be written as sums of consecutive odd integers. (To make the mystery begin to vanish view [9].) Pattern thinking removes the magic from the expressions in the above two tables. The way to understand what is happening in these two situations precedes using words like sequences of adjacent integers-squared. We will see below that physical instances initially led to these and other observations. If these examples are put forward as discussion topics, students observations, their awareness of evident patterns in these tables, can lead back to topics used in the mathematics curriculum, specifically, mathematical induction and algebraic reasoning. 2

3 = 5 2 (7) = = 365 (8) = (9) = (10) = (11) Table 1: Consecutive Squared-Integer Sums 3

4 = 125 = 5 3 (12) = 216 = 6 3 (13) = 343 = 7 3 (14) = 512 = 8 3 (15) 3 Observations Table 2: Cubes as Sums of Consecutive Odd Numbers The first three Table 1 equations [(7) - (9)] motivate the following question. What value begins the series of consecutive numbers that are squared? (Gardner wrote that Linton [16] gave a formula for the lowest value in such an equation. But Gardner s book doesn t contain it and Linton s letter to him isn t available.) Instead of the lowest value, view this through pattern thinking. First look at the these equations in terms of symmetry. In each, there is one more on the left side than there are terms right of the equals sign. The symmetry is about the largest of the numbers (before squaring) on the left hand side. In essence the symmetry comes from counting down from that number on the left, and up from it on the right. Suppose one divides the numbers that are squared into three groups: those lower than the value closest to, and left of, the equals sign; that number itself; and those on the right side of the equality. In (7) - (9), but actually any consecutive squares equation, all terms are derived by counting from the particular value we just isolated (one of three groups). That value is always immediately left of the equals sign when there is one more term at its left than on the right. If we call the isolated value (with location just left of the equals sign when the larger number of consecutive squares is on the left side) the pivot, numbers to be squared on the left are obtained by counting down from 4

5 it (those at right, up). There are other expressions not involving squares that are very similar to the above equations, for example (16) - (18). In all these cases there is a single number from which all others in that equation are found by counting down or counting up. Equation (19) states algebraically what takes place in the equations (16) - (18), using a to represent the largest offset from the pivot term t = 3 (16) = (17) = (18) (t a) + (t a + 1) t = (t + 1) (t + a) (19) Every term on the RHS of (19) has (after the letter t) some value between one and a; hence there are a-many entries there. Similar inspection of the terms on the LHS of (19) shows that there are a + 1 terms left of the equals sign. Expanding the elements in(19) lets us collect all terms with t s on the LHS, and all without t s on the RHS. That leads to these two equalities [equation (20). (a + 1)t a(t) = t = 2(sum of numbers from 1 to a) (20) Rewriting sum-of-numbers-from-1-to-a using summation notation, and division through by 2 yields this restatement of equation (20) a t/2 = i = a(a + 1)/2 (21) i=1 To check, substitute a = 3 in equation (21) and find the equation (18) pivot value. Equation (21) is often demonstrated by writing two rows of numbers 1 through a, one below the other but in reverse order. Vertically adding similarly-located values gets just a entries that all are (a + 1) in value, hence [since that addition yields sum-of-the-numbers-from-1-to-a twice] : a 2 i = a(a + 1) (22) i=1 To summarize, in an expression like (16)-(18) a number just left of the equals sign (side with larger number of terms) has value t. The value t begins a counting 5

6 down and up process that forms the equality. Finally, t the number of right (fewer entries) side terms, a, by: can be calculated from t = a(a + 1) (23) We count down a-many times to get the base which we will call b. Hence for linear [exponent one, general form (19)] cases, b is reached from subtracting a from t: 4 Squares b = t a = a(a + 1) a = a 2 (24) The linear/exponent-one case explained above makes the [2] question about consecutive squares like (7) - (9) straightforward. While we will get to the lowest value b formula, the best approach begins by finding the pivot value. The reasoning begins with an expression analogous to (19) stating the consecutive square property: (t a) 2 + (t a + 1) t 2 = (t + 1) (t + a) 2 (25) Again three groups of numbers appear. One group is just the pivot, t. The other two, are numbers below it, and those that are higher. The values to be squared in these last two groups come from differences or sums to t (they count down or up from it). That fact leads to using: (c d) 2 = c 2 2cd + d 2 = (c 2 + 2cd + d 2 ) 4cd = (c + d) 2 4cd (26) Applying (26) to (25) leads to: which simplifies: a t 2 = 4 ti (27) i=1 a t = 4 i (28) i=1 From the expression for sums of integers from 1 to n, (21), the result in (29) can be restated. In the linear case equation (23) results. Here applying (22) to (28) gives twice that: a t = 4 i = 2a(a + 1) (29) i=1 6

7 Just as in the linear case, we can use the observation that we count down from the pivot a, the number of terms appearing on the right side, to find the base b. That implies Linton s formula was: b = t a = 2a(a + 1) a = 2a 2 + a = a(2a + 1) (30) Direct substitution of a = 1, 2, 3 in equation (30) yields b = 3, 10, 21. This and the previous section used a form of symmetry analysis and algebraic properties (expanding the expressions from squaring a) the sum, and b) the difference, of two terms). 5 Values This section illustrates number-relationships involving equations of form (25). The general notion is that people see things about sequences of adjacent numbers squared and summed. In a teaching environment these results and interconnections can be used to stimulate students by drawing on their observations and pattern awareness skill. The key patterns involve symmetry, order - a slight generalization of what is above described by a, number of terms on right side; and total number of entries, i.e., (2a + 1). Instead of a let order signify smallest number of terms in a relationship between two sets of summed squares on consecutive values (this eliminates the right of the equals sign restriction). A reasonable word for the symbol b is either base or bottom: this refers to each consecutive-square equation s lowest-value-beingsquared. The observed values for order and base of equations (7) - (9) follow in Table 2. Notice that the product of order and number of entries is the base (entries refers to the total number of terms or elements in an equation). Stated algebraically, this is Linton s formula, (30), namely a(2a + 1) = b. order base number of entries Table 3: Product Pattern Table 4 shows symmetry about the pivot. It also displays pattern relationships between characteristics of different order equations. Notice the result from multiplying two items and dividing by a third. Those to be multiplied are number of left side terms and largest value on right. Division 7

8 ( count down) pivot = (count up ) 3 4 = = = Table 4: Symmetry Pattern is by number of right side terms. The result is exactly base of next-higher-order equation. Also, notice that right side counting up from a pivot makes the largest value be t + a. The result from multiplying two properties, dividing by a third, can be put in algebraic terms. We do that using subscripts that let us know which equation we are at, either a or a + 1 order: (t a + a a )(a a + 1)/a a = b a+1 (31) Since (30) holds in general, (31) can be rewritten using b a+1 = (a + 1)(2a + 3) : (t + a)(a + 1)/a = (a + 1)(2a + 3) (32) Algebraic manipulation here again leads to t = 2a(a + 1) as in (29). 6 Cubes as Sum of Adjacent Odds Up to now this paper has put forth connections between linear and squared terms. It early on mentioned a past issue: representing a cube, as in the Nicomachus statement [12]. Again this is a situation where numeric examples are easily brought forth, one where hand calculators may be employed, and a candidate for instruction about proof. You can establish that the statement holds for all positive integers using mathematical induction [3], [4]. But to do that you need to change by leaving the symbolic realm. This topic requires visual thinking. It is difficult to understand within the domain of mathematical symbols for exponents. But cube has real physical meaning. You can construct a model using eight or twenty-seven dice to explain 2 3 and 3 3. Another method is to visualize either of two things. A Cartesian system with three coordinates handles 2 3 by providing the eight octants. Likewise a three-dimensional tic-tac-toe game represents 3 3. The last paragraph, math-induction, and recognizing the difference between numbers being cubed that are odd and even could lead to the ability to write down expressions like (12)-(15) for any positive integer cubed. Your students can 8

9 learn to do that and then go out into the world armed with the puzzling (to others) question, how can you represent... as the sum of adjacent odd numbers? Whether it leads to it rapidly or not depends on the prior paragraph, and access to such material as [1], [15], [8] or reference to [10], [9]. 7 Square Sum of Summed Squares This issue involves piling spheres as done with marbles in the photo. The pile creates a pyramid with square cross -section: one at the top, a two-by-two array below that, and so forth to the n-th level (n 2 ). The goal is to choose n so that for some integer m the next equation holds: n 2 = m 2 (33) The word description of the preceding equation is that a series of consecutive squares summed begins with unity and the result is a square. As the number of elements summed increases, an interesting fact appears. The total is only square when the last number squared is twenty-four [7]. (For more detail on this question see [6]; proof is beyond the scope of this item.) In mathematical statements: 1) there is an integer-pair (n, m) satisfying (33) (existence); and, 2) there is only one integer-pair (n, m) satisfying (33) (uniqueness). These statements show two ways mathematicians look at situations. Another issue is behind this. The expression (21) or its alternate form (22) and possibly (26) are thought of as truths designated as important by being labeled identities. There is another identity, a summing squares relationship that equates the sum of squares beginning at one and on up through the integers to a value n, as the product of three factors in n: n 2 = n(n + 1)(2n + 1)/6 (34) The values 12, 18, 24, 30 and 36 all are evenly divided by 6. Of them, the only one that could be an n yielding equation (34) square is n = 24. The following table shows this. Although we haven t established the uniqueness of n = 24 as the only value solving 34 that has been done elsewhere; see [6]. Whether one seeks materials for communicating further using numerical illustrations, seeks another domain for mathematical induction, or wishes to address open computational questions for an honors class, there is much more that can be said. To establish equation (34) begin with the first few values of such a sum as n increases from unity. One can find these as 1, 5, 14, 30, and 55 by direct computation and then notice each satisfies the right hand side of equation (34) 9

10 n n + 1 2n + 1 RHS of (34) Table 5: Product for Squares Summed for the corresponding n values, 1, 2,..., 5. So since (34) is known to be true at some value, it can be proven using mathematical induction [3], [4], beginning with the established pattern that it is true at these five n values. That is, the five numeric patterns suggest seeking proof by establishing the inductive step via this relationship: n(n + 1)(2n + 1) + (n + 1) 2 (n + 1)(n + 2)(2n + 3) = (35) 6 6 Evaluating the expressions on each of the two sides of (35) by multiplying out terms gives the same result. Hence (34) is true in general. 8 Conclusion Mathematics comes from observations about patterns in numbers and nature. This paper described such relationships, beginning with one that extends the Pythagorean relationship common in school-mathematics. The paper presented the general form of relationships among sums of squares of consecutive integers. It explained the starting values in such an ascending series by displaying a new notion in pattern thinking. That notion was the pivot as a starting point for counting down and up. Theme pivot develops pattern thinking with respect to symmetry within a number series. That approach was used to set up similar starting value relationships among sums of consecutive integers. Having resolved issues about squares and first-power sums involving consecutive integers we presented material about representing cubed positive integers as sums of adjacent odd numbers. These three topics relate to a conjecture mathematics has considered to be very important [14], because they ask questions about sums of numbers raised to different exponents. The final section concerned a change of focus from consecutive or adjacent values to summing squares beginning at unity. Material there covered three themes. The first was in regard to the sum being square in sample numerical instances. The 10

11 second was the difference between instances/examples and existence/uniqueness. Finally, this topic was used to address the central role of proof in mathematics with particular reference to mathematical induction. Throughout the material suggestions appear for class dialog and self-study. 9 Acknowledgement Taylor B. Kang and Michelle M. Amagrande of California State Polytechnic University, Pomona, observed the relationships described in Values. Richard Korf, University of California Los Angeles (UCLA), told me of the problem Watson solved. David G. Cantor, UCLA and Center for Communications Research, San Diego, suggested documenting Squares. The referee suggested that this material had, through the range of exponents, a relationship to Fermat s conjecture. 10 Captions Table 1. Consecutive Squared-Integer Sum Relationships Table 2. Cubes as Sums of Consecutive Odd Numbers Table 3. Product Pattern Table 4. Symmetry Pattern Table 5. Product for Squares Summed References [1] Lynn Arthur Steen. The Science of Patterns. Science, pages , April [2] Brain ticklers. The Bent of Tau Beta Pi, Spring 2004, p. 44; tbp.org/pages/publications/bts/sp04.pdf, [3] Anonymous. Mathematical Induction from Wikipedia. wikipedia.org/wiki/mathematical\_induction, [4] Alexander Bogomolny. Mathematical Induction. cut-the-knot.org/induction.shtml, [5] E. W. Weisstein. Pythagorean Triple. MathWorld; wolfram.compythagoreantriple.html,

12 [6] E. W. Weisstein. Square Pyramidal Number. MathWorld; mathworld.wolfram.com/squarepyramidalnumber.html, [7] G. N. Watson. The Problem of the Square Pyramid. Messenger. Math. 48, pages 1 22, [8] Keith J. Devlin. Mathematics: The Science of Patterns. W. H. Freeman and Co., NY, [9] Allen Klinger. Cube s decomposition. cubes.jpg, [10] Allen Klinger. A Cubed Positive Integer is a Sum of Consecutive Odd Numbers [11] Martin Gardner. The Numerology of Dr. Matrix. Simon and Schuster, NY, [12] Thomas L. Naps. Introduction to Data Structures and Algorithm Analysis, 2nd Edition, p. 50, prob. 4. West Publishing Co., St. Paul, Minnesota, [13] Nicomachus. Arithmetica. not known. [14] J. J. O Connor and E. F. Robertson. History Topic Fermat. google.com/search, [15] Roger B. Nelson. Proofs Without Words Exercises in Visual Thinking. The Mathematical Association of America, [16] Russell Linton. Unpublished letter to Martin Gardner, [17] T. H. Beldon. Runs of Squares. The Mathematical Gazette, pages , December [18] William Dunham. Journey Through Genius: The Great Theorems of Mathematics. John Wiley & Sons, Inc., NY,