In February 2017, a number theoretic problem was posed in Mathematics Magazine by Souvik Dey, a master’s student in India. The problem asked whether it was possible to represent a real number by a finite sum of elements in an open subset of the real numbers that contained one positive and one negative number. This paper not only provides a solutionto the original problem, but proves an analogous statement for elements of R2

Bouchat, Rachelle R.

Dominguez, Lewis T.

English

University of Dayton

University of Dayton eCommons Undergraduate Mathematics Day: Proceedings and Other Materials Department of Mathematics 1-1-2018 Finite Sum Representations of Elements in R and R2 Lewis T. Dominguez University of Dayton Rachelle R. Bouchat University of Dayton Follow this and additional works at: https://ecommons.udayton.edu/mth_epumd  Part of the Mathematics Commons eCommons Citation Dominguez, Lewis T. and Bouchat, Rachelle R., "Finite Sum Representations of Elements in R and R2" (2018). Undergraduate Mathematics Day: Proceedings and Other Materials. 31. https://ecommons.udayton.edu/mth_epumd/31 This Article is brought to you for free and open access by the Department of Mathematics at eCommons. It has been accepted for inclusion in Undergraduate Mathematics Day: Proceedings and Other Materials by an authorized administrator of eCommons. For more information, please contact frice1@udayton.edu, mschlangen1@udayton.edu. Elec. Proc. Undergrad. Math. Day5 (2018), 11–16.FINITE SUM REPRESENTATIONS OF ELEMENTS IN RAND R2LEWIS T. DOMINGUEZ AND RACHELLE R. BOUCHATCommunicated by Jonathan BrownAbstract. In February 2017, a number theoretic problem was posed in Mathematics Mag-azine by Souvik Dey, a master’s student in India. The problem asked whether it was possibleto represent a real number by a finite sum of elements in an open subset of the real numbersthat contained one positive and one negative number. This paper not only provides a solutionto the original problem, but proves an analogous statement for elements of R2.Keywords: Spanning setsMSC (2010): Primary 03F601. IntroductionThe work in this paper was inspired by a problem proposed by Souvik Dey in the February2017 Mathematics Magazine [1], namely:Let S be an open subset of the set R, such that S contains at least one positivenumber and one negative number. Then every real number can be written asa finite sum of (not necessarily distinct) elements of S.In this paper we will also consider an extension of this problem to R2. When consideringthe problem in R, we can think of the number line being broken into three distinct parts:{a : a < 0} ∪ {0} ∪ {a : a > 0}. Thinking of the problem posed by Souvik Dey, we can nowthink of it as saying that we must have an element from each of the sets other than {0}.Upon considering R2, we can then think of the coordinate plane as being broken into fivedistinct parts: {(a, b) : a, b > 0} ∪ {(a, b) : a < 0, b > 0} ∪ {(a, b) : a = 0 or b = 0} ∪ {(a, b) :a, b < 0} ∪ {(a, b) : a > 0, b < 0}. The four distinct sets, excluding {(a, b) : a = 0 or b = 0}correspond to the four quadrants in R2. Throughout this paper, we will use the notation:Q1 = {(a, b) : a, b ∈ R and a, b > 0} Q3 = {(a, b) : a, b ∈ R and a, b < 0}Q2 = {(a, b) : a, b ∈ R and a < 0, b > 0} Q4 = {(a, b) : a, b ∈ R and a > 0, b < 0}Using this notation, we now propose an analogous statement for R2:Let S be an open subset of the set R2, such that S contains at least one pointin each of Q1, Q2, Q3, and Q4. Then every element of R2 can be written asa finite sum of (not necessarily distinct) elements of S.In the next section of the paper, we will consider some of the background material thatwill be necessary for the proof of this main result, seen later as Theorem 3.1.12 L. T. Dominguez and R. R. Bouchat2. Background MaterialTo better understand the problem presented by Souvik Dey, as well as the extension toR2, we will need to understand what an open set looks like in Rn. Specifically, we will needto consider what an open set looks like in R2 using the standard Euclidean distance. Let usfirst consider what an open set looks like in Rn, as in the book of Walter Rudin (Definition2.8 in [2]).Definition 2.1.(1) Given ε ∈ R+ and (a, b) ∈ R2, the setN(a,b),ε ={(x, y) :√(x− a)2 + (y − b)2 < ε}is called an open neighborhood around (a, b).(2) A set E ⊂ Rn is called an open subset of Rn provided that for each x ∈ E, thereexists ε ∈ R+ such that Nx,ε ⊂ E.If we consider what this definition means in R2, this implies that every open set has theform ∪iN(ai,bi),εi , where each of the N(ai,bi),εi are open disks. Hence, every open set in R2 canbe thought of as an arbitrary union of open disks.Another property that will be integral to the paper is the density of the rationals in thereal numbers. This can seen in the following theorem (Theorem 1.20 in [2]).Theorem 2.1. If x, y ∈ R and x < y, then there exists q ∈ Q such that x < q < y.Essentially, the fact that the rationals are dense in the real numbers implies that between anytwo distinct real numbers, you can find a rational number. Moreover, this can be extendedto the following theorem about the density of the Q2 in R2.Corollary 2.1. Fix ε ∈ R+, and let (a, b), (c, d) ∈ R2 such that a 6= c and b 6= d. Then thereexists (p, q) ∈ Q2 such that (p, q) ∈ {(x, y) : a < x < c and b < y < d}.Proof. The result follows immediately by projecting the points (a, b) and (c, d) first ontothe x-axis and secondly onto the y-axis, and then applying Theorem 2.1.This result says that for any rectangular region in R2, we are guaranteed to be able to finda point in the interior of this region whose x- and y-coordinates are both rational.3. The Main ResultWe begin by considering an open set, S, in R2 that contains an element of each of the fourquadrants. By considering Definition 2.1, it immediately follows that any open set S ⊂ R2that contains a point from each of the four quadrants must also contain a set of the form:N(a1,b1),ε1 ∪N(a2,b2),ε2 ∪N(a3,b3),ε3 ∪N(a4,b4),ε4where ε1, ε2, ε3, ε4 ∈ R+ and (a1, b1) ∈ Q1, (a2, b2) ∈ Q2, (a3, b3) ∈ Q3, and (a4, b4) ∈ Q4.Furthermore, by setting ε = min{ε1, ε2, ε3, ε4}, we get thatN(a1,b1),ε ∪N(a2,b2),ε ∪N(a3,b3),ε ∪N(a4,b4),ε ⊂ S.In order to prove the main result, we will first show that (0, 0) can be written as a linearcombination of elements from the open set S.Finite Sum Representations of Elements in R and R2 13Lemma 3.1. Let S be an open set in R2 containing the points (a1, b1) ∈ Q1, (a2, b2) ∈ Q2,(a3, b3) ∈ Q3, and (a4, b4) ∈ Q4. Then there exist n,m, s, t ∈ N such thatn(p1, q1) +m(p2, q2) + s(p3, q3) + t(p4, q4) = (0, 0)where (pi, qi) ∈ N(ai,bi),ε with (pi, qi) ∈ Q2 for i ∈ {1, 2, 3, 4} and some ε ∈ R+.Proof. Let S be an open set in R2 containing the points (a1, b1) ∈ Q1, (a2, b2) ∈ Q2,(a3, b3) ∈ Q3, and (a4, b4) ∈ Q4. Then, by Definition 2.1, there exists ε ∈ R+ such thatN(a1,b1),ε ∪N(a2,b2),ε ∪N(a3,b3),ε ∪N(a4,b4),ε ⊂ Swhere N(ai,bi),ε ⊂ Qi for i ∈ {1, 2, 3, 4}. Furthermore, by Corollary 2.1 we may find (pi, qi) ∈Q2 such that (pi, qi) ∈ N(ai,bi),ε for i ∈ {1, 2, 3, 4}. Furthermore:(p1, q1) =(n11n12,m11m12)(p3, q3) =(−n31n32,−m31m32)(p2, q2) =(−n21n22,m21m22)(p4, q4) =(n41n42,−m41m42)where nij,mij ∈ N for i ∈ {1, 2, 3, 4} and j ∈ {1, 2}. Set n0 = n12n21 and m0 = n22n11.Then it follows that the linear combinationn0(p1, q1) +m0(p2, q2) =(0,n12n21m11m22 + n11n22m12m21m12m22)Letting c1 = n12n21m11m22 + n11n22m12m21, we get(3.1) n0(p1, q1) +m0(p2, q2) =(0,c1m12m22).Similarly, set s0 = n32n41 and t0 = n31n42. Then it follows that the linear combinations0(p3, q3) + t0(p4, q4) =(0,−(n32n41m31m42 + n31n42m32m41)m32m42).Letting c2 = n32n41m31m42 + n31n42m32m41 we get:(3.2) s0(p3, q3) + t0(p4, q4) =(0,−c2m32m42).Now consider u0 = m12m22c2 and v0 = m32m42c1. Then it follows that:(3.3) u0(0,c1m12m22)+ v0(0,−c2m32m42)= (0, 0).Combining together Equations 3.1, 3.2, and 3.3 provides(3.4) u0n0(p1, q1) + u0m0(p2, q2) + v0s0(p3, q3) + v0t0(p4, q4) = (0, 0).14 L. T. Dominguez and R. R. BouchatSince nij,mij > 0 for all i ∈ {1, 2, 3, 4} and j ∈ {1, 2} it follows that u0n0, u0m0, v0s0, v0t0 ∈N. Setting n = u0n0, m = u0m0, s = v0s0, and t = v0t0 the result follows immediately fromEquation 3.4.It will also be essential that we can scale a point in R2 to be arbitrarily close to (0, 0). Todo this, we will need the following two functions.Definition 3.1. Let x ∈ R.(1) The floor function, denoted bxc, assigns to the input x the greatest integer that isless than or equal to x.(2) The ceiling function, denoted dxe, assigns to the input x the least integer that isgreater than or equal to x.These two functions can now be used to perform the necessary scaling.Lemma 3.2. Let (z1, z2) ∈ R2, and let ε ∈ R+. Then there exists β ∈ N such that1βb|zx|+ 1cb|zy|+ 1c(z1, z2) ∈ N(0,0),ε.Proof. Let (z1, z2) ∈ R2, and let ε ∈ R+. Set ∆ =|zx|+ |zy|εb|zx|+ 1cb|zy|+ 1c. Then√(zx∆b|zx|+ 1cb|zy|+ 1c)2+(zy∆b|zx|+ 1cb|zy|+ 1c)2≤√(zx∆b|zx|+ 1cb|zy|+ 1c)2+√(zy∆b|zx|+ 1cb|zy|+ 1c)2=|zx|+ |zy|∆b|zx|+ 1cb|zy|+ 1c= εChoosing β = d∆e produces the desired result.Now, we are ready to prove the main result of the paper, which is an extension of theoriginal problem posed in the February 2017 edition of Mathematics Magazine (see [1]).Theorem 3.1. Let S be an open set in R2 containing the points (a1, b1) ∈ Q1, (a2, b2) ∈ Q2,(a3, b3) ∈ Q3, and (a4, b4) ∈ Q4. Furthermore, let (z1, z2) ∈ R2. Then there exist n,m, s, t ∈N such thatnx1 +mx2 + sx3 + tx4 = (z1, z2)where xi ∈ N(ai,bi),ε ⊂ S for i ∈ {1, 2, 3, 4} and some ε ∈ R+.Proof. Let S be an open set in R2 containing the points (a1, b1) ∈ Q1, (a2, b2) ∈ Q2,(a3, b3) ∈ Q3, and (a4, b4) ∈ Q4. Then from Lemma 3.1, there exists n0,m0, s0, t0 ∈ N andε ∈ R+ such that(3.5) n0(p1, q1) +m0(p2, q2) + s0(p3, q3) + t0(p4, q4) = (0, 0)where (pi, qi) ∈ N(ai,bi),ε ⊂ S with (pi, qi) ∈ Q2 for i ∈ {1, 2, 3, 4}. Moreover, Lemma 3.2provides a β ∈ N, for any ε ∈ R+, such that1βb|zx|+ 1cb|zy|+ 1c(z1, z2) ∈ N(0,0),ε.Finite Sum Representations of Elements in R and R2 15Thus, we can find a β ∈ N such that1∆(z1, z2) + (pi, qi) ∈ N(pi,qi),ε′ ⊂ N(ai,bi),ε ⊂ Swhere ε′ ∈ R+ and ∆ = βb|zx|+ 1cb|zy|+ 1c. It follows that ∆ ∈ N andx1 =14n0∆(z1, z2) + (p1, q1) ∈ S x3 =14s0∆(z1, z2) + (p3, q3) ∈ Sx2 =14m0∆(z1, z2) + (p2, q2) ∈ S x4 =14t0∆(z1, z2) + (p4, q4) ∈ S.Furthermore,n0∆x1+m0∆x2+s0∆x3+t0∆x4 = (z1, z2)+∆ [n0(p1, q1) +m0(p2, q2) + s0(p3, q3) + t0(p4, q4)]Then, by Equation 3.5n0∆x1 +m0∆x2 + s0∆x3 + t0∆x4 = (z1, z2) + ∆(0, 0) = (z1, z2).The claim follows by setting n = n0∆, m = m0∆, s = s0∆, and t = t0∆.We can further use this main result to return to the original problem posed by SouvikDey.Corollary 3.1. Let S be an open subset of the set R, such that S contains at least onepositive number and one negative number. Then every real number can be written as a finitesum of (not necessarily distinct) elements of S.Proof. Let S be an open subset of R such that there exists a, b ∈ S with a < 0 andb < 0. Moreover, let r ∈ R. Then (r, 0) ∈ R2. Since S is an open subset of R, it follows thatS×S = {(x, y) : x, y ∈ S} is an open subset of R2. Moreover, (a, a), (a, b), (b, a), (b, b) ∈ S×Swith (b, b) ∈ Q1, (a, b) ∈ Q2, (a, a) ∈ Q3, and (b, a) ∈ Q4. By Theorem 3.1, there existn,m, s, t ∈ N and x1,x2,x3,x4 ∈ S × S such that(3.6) nx1 +mx2 + sx3 + tx4 = (r, 0).Let (xi)1 denote the first coordinate of xi for i ∈ {1, 2, 3, 4}. Then from Equation 3.6, itfollows that:n (x1)1 +m (x2)1 + s (x3)1 + t (x4)1 = r.Since xi ∈ S × S for i ∈ {1, 2, 3, 4}, (xi)1 ∈ S for i ∈ {1, 2, 3, 4} and the result follows.4. ConclusionFrom the initial problem posed by Souvik Dey (see [1]), we were able to generalize theproblem to R2. There is still the possibility to extend this problem to Rn. We end with thefollowing conjecture regarding this general problem.Conjecture 1. Let S be an open set in Rn containing the points (x1i, x2i, . . . , xni) wherei ∈ {1, 2, . . . , 2n} such that each of these points lies in one of the 2n distinct regions ofRn determined by the (n − 1)-dimensional hyperplanes partitioning Rn. Furthermore, let(z1, z2, . . . , zn) ∈ Rn. Then there exist c1, c2, . . . , c2n ∈ N such that2n∑i=1ci(x1i, x2i, . . . , xni) = (z1, z2, . . . , zn)′16 L. T. Dominguez and R. R. BouchatREFERENCES[1] S. Dey, Problem 2011, Mathematics Magazine, 90 (2017), No. 1, p.75.[2] W. Rudin, Principles of Mathematical Analysis, International Series in Pure and Applied Math-ematics (1976).Lewis T. Dominguez,Department of Mathematics, Indiana University of Pennsyl-vania, Indiana, PA 15701, USAE-mail address: l.t.dominguez@iup.eduRachelle R. Bouchat, Department of Mathematics, Indiana University of Penn-sylvania, Indiana, PA 15701, USAE-mail address: rbouchat@iup.eduReceived December 29, 2017; revised January 29, 2018.

Finite Sum Representations of Elements in R and R2

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Finite Sum Representations of Elements in R and R2

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