 # Creative Mathematics

H. S. Wall
Edition: REV - Revised, 1
Pages: 218
https://www.jstor.org/stable/10.4169/j.ctt5hhb5f

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1. Front Matter
(pp. i-vi)
2. Foreword
(pp. vii-x)
Harry Lucas Jr.

The University of Texas Press in Austin published the first edition ofCreative Mathematicsin 1963. After H.S. Wall’s death in 1971, his son Hubert Richard Wall assigned the book’s copyright to us at the Educational Advancement Foundation (EAF), and we produced photographic reprints of it in 2006. At our request the Mathematical Association of America (MAA) agreed to update the book and produce this revised edition published in 2008.

Although elements of H.S. Wall’s teaching method are described by the author in the Preface, it may be helpful to situate his point of view in an historical context. This...

3. Preface
(pp. xi-xiv)
H.S.W.
(pp. xv-xvi)
5. Short Biography of H. S. Wall
(pp. xvii-xxii)
Albert C. Lewis
6. 1 Numbers
(pp. 1-10)

We assume the existence of certain things callednumbers, some of which are calledcounting numbers, and we take for granted certain statements concerning numbers, calledaxioms. The first few axioms are

If each ofxandyis a number, then$x + y$(readx plus y) is a number called thesumofxandy. The association withxandyof the sum$x + y$is calledaddition.

If each ofx, y,andzis a number, then$x + (y + z)$is$(x + y) + z$.

0 is a number such that ifxis a number, then 0+xisx.

Ifx...

7. 2 Ordered Number Pairs
(pp. 11-22)

The statement thatPis apointmeans thatPis an ordered number pair (x, y) having a first numberxcalled theabscissaofPand a second numberycalled theordinateofP. Apoint setis a collection each element of which is a point.

A plane writing surface, e.g., a blackboard, imagined of indefinite extent, may serve as a model for the set of all points. From the set of all horizontal lines on the surface select one and designate it$\underline 0$(read 0-horizontal) and from the set of all vertical lines on the...

8. 3 Slope
(pp. 23-30)

The statement thatfis the straight line ofslope mcontaining the point (a, b) means thatmis a number andfis the simple graph such that for every numberx

$f\left( x \right) = m \cdot (x - a) + b$.

(i) Show that if (x, y) and (u, v) are points of the straight line of slopemcontaining the point (a, b), then

$m = \frac{{y - v}}{{x - u}}$.

(ii) Sketch the straight line of slope 1 containing the point (0, 0) and the straight line of slope –1 containing the point (0, 0).

(iii) Supposecis a positive number. Determine a numbermsuch that the straight...

9. 4 Combinations of Simple Graphs
(pp. 31-36)

The statement that$f+g$is thesumof the simple graphfand the simple graphgmeans that there is a number common to theX-projection offand theX-projection ofgand$f+g$is the simple graph whoseX-projection is the common part of theX-projection offand theX-projection ofgsuch that, ifxis in this common part, the ordinate of that point of$f+g$whose abscissa isxis$f(x) + g(x)$, or

$(f + g)(x) = f(x) + g(x)$.

Example. Iffis the subset of the horizontal line$\underline 1$whoseX-projection is the set of all nonpositive numbers...

10. 5 Theorems about Simple Graphs
(pp. 37-42)

We now state a theorem about simple graphs followed by a number of other theorems which, since they may be proved using the main theorem, are labeledcorollaries. We suggest that the reader take the main theorem for granted temporarily and use it as an axiom to prove the corollaries. Then return to the main theorem from time to time and try to prove it.

Theorem.If the simple graph f has X-projection the interval[a, b]and has slope at each of its points, then there is a point P of f such that no point of f is...

11. 6 The Simple Graphs of Trigonometry
(pp. 43-54)

Following the pattern by which the simple graphsLandEwere developed fromH, simple graphsAandTmay be developed from$\Omega$, where (see Figure 6.1)

$\Omega = \frac{1}{{1 + {I^2}}}$.

There are enough differences and similarities between the two developments to make the work interesting. We shall give a broad outline, leaving the details to the reader.

If [a, b] is an interval, we denote by [$\Omega$;a, b] the point set to which (x, y) belongs only ifxbelongs to [a, b] andyis 0,yis$\frac{1}{{1 + {x^2}}}$, oryis a number between 0 and$\frac{1}{{1 + {x^2}}}$....

12. 7 The Integral
(pp. 55-66)

The statement that the simple graphf is bounded on the interval[a, b] means that theX-projection offincludes [a, b] and that there exist horizontal linesαandβsuch that every point offwhose abscissa is in [a, b] is betweenαandβ.

Suppose that the simple graphfis bounded on [a, b]. The statement that${}_is$is an inner sum for f on[a, b] means that there exists a finite collectionDof nonoverlapping intervals filling up [a, b] such that, if the length of each interval inDis multiplied...

13. 8 Computation Formulas Obtained by Means of the Integral
(pp. 67-78)

The approximation formulas forLandAwere obtained by experimentation and conjecture. The integral furnishes a method for obtaining such formulas for many simple graphs.

We need a few preliminaries. First, it is convenient to make the agreement that

$\int\limits_a^a {f = 0}$and$\int\limits_b^a {f = - \int\limits_a^b f }$.

With this agreement,$L\left( x \right) = \int\limits_1^x {\frac{1}{I}}$for every positive numberxand$A\left( x \right) = \int\limits_0^x \Omega$for every numberx. Second, the following lemma, which we leave for the reader to prove, will be useful.

Lemma.Suppose each of f and g is a simple graph with X-projection the interval [a, b] having property S at each of its points and$g\left( x \right) \ge 0$...

14. 9 Simple Graphs Made to Order
(pp. 79-82)

We consider here the problem of constructing simple graphs, mainly as combinations ofI,E,S, andC, that have certain prescribed properties.

If each ofgandhis a simple graph whoseX-projection is the set of all numbers having propertySat each of its points and if (a, b) is a point, does there exist a simple graphfcontaining (a, b) such that$f' = g \cdot f + h?$

The simple graphEhas the property that$E'=E$. Let us try to constructffromE:

$f = \upsilon E[u]$

Since$f = \upsilon E[u]u' + \upsilon 'E[u]$, we require that

$\upsilon E[u]u' + \upsilon 'E[u] = g \cdot \upsilon E[u] + h$

and

$\upsilon (a)E(u(a)) - b$

Now, Now, (i) is true...

(pp. 83-92)

The integral is closely tied to the derivative. For example, to express a simple graphfas an integral by the formula$f(x) = f(a) + \int_a^x {f'}$requires that$f'$be integrable on the interval over which the integral is extended. To find a more general kind of integral, we begin by generalizing the notion of length of an interval.

Definition. The statement that$\left. g \right|_a^b$is theg-lengthof the interval [a, b] means thatgis a simple graph whoseX-projection includes [a, b] and

$\left. g \right|_a^b = g(b) - g(a)$

In particular, theI-lengthof [a, b] is$b-a$, the ordinary length of [a, b].

Suppose that...

16. 11 Simple Surfaces
(pp. 93-118)

The statement thatfis asimple surfacemeans thatfis a collection, each element of which is an ordered pair (P, z), whose first memberPis a point and whose second memberzis a number such that no two ordered pairs infhave the same first member. The second member of that ordered pair infwhose first member isPis denoted by$f\left( P \right)$) (readf of P) or, if$P = (x,y)$, by$f(x,y)$) (readf of x and y).

Problem. Generalize to simple surfaces some of the ideas concerning simple graphs such as slope,...

17. 12 Successive Approximations
(pp. 119-134)

Supposeais a positive number and we wish to prove the existence of a positive numberxwhose square isa. If${x_0}$is any positive number, then$x_0^2 > a,x_0^2 = a$, or$x_0^2 < a$. In the first case,${x_0}$is too large and

${x_0} > \frac{a}{{{x_0}}}$.

The average$(1/2)\left\{ {{x_0} + (a/{x_0})} \right\}$is less than${x_0}$and greater than$a/{x_0}$and may be a better approximation to the hypothetical numberxwhose square isathan either${x_0}$or$a/{x_0}$. The same is possibly true in case$x_0^2 < a$. Thus, the number${x_1}$, defined by

${x_1} = \frac{1}{2}\left\{ {{x_0} + \frac{a}{{{x_0}}}} \right\}$,

may be a letter approximation to$x$than${x_0}$is. Then${x_2}$...

18. 13 Linear Spaces of Simple Graphs
(pp. 135-160)

In this chapter we outline some selected theorems about the space of points, i.e., the plane of ordered number pairs, and then try to lead the reader to extend them to a space in which the word “point” is interpreted to mean “continuous simple graph withX-projection the interval [a, b],” i.e., a space in which “point” means a certain kind of set of ordinary points.

To make the extension easier to accomplish, we make some changes in notation.

Iffis a point, i.e., an ordered number pair, we denote the point by$(f(1),f(2))$. (Apoint fis determined...

19. 14 More about Linear Spaces
(pp. 161-174)

The space in whichpointmeans number sequence, with addition of the number sequenceaand the number sequencebdefined by

${(a + b)_p} = {a_p} + {b_p}$$p = 1,2,3, \ldots$,

and multiplication ofaby the numberkdefined by

${(k\cdot a)_p} = k{a_p}$,$p = 1,2,3, \ldots$,

is a linear space. We denote this linear space ofallnumber sequences byW. Ifais a point ofWfor which there is a number$k_1$such that

$\sum\limits_{p = 1}^n {a_p^2} \le {k_1}$,$n = 1,2,3, \ldots$,

andbis a point ofWfor which there is a number$k_2$such that

$\sum\limits_{p = 1}^n {b_p^2} \le {k_2}$,$n = 1,2,3, \ldots$,

then

$\sum\limits_{p = 1}^n {{{({a_p} + {b_p})}^2}} = \sum\limits_{p = 1}^n {a_p^2} + \sum\limits_{p = 1}^n {b_p^2 + 2\sum\limits_{p = 1}^n {{a_p}{b_p}} }$

$\le {k_1} + {k_2} + 2{\left\{ {\sum\limits_{p = 1}^n {a_p^2} .\sum\limits_{p = 1}^n {b_p^2} } \right\}^{1/2}} \le {({k_1}^{1/2} + {k_2}^{1/2})^2}$so that$a + b$has the same property. Also, ifk...

20. 15 Mechanical Systems
(pp. 175-186)

We consider applications of simple graphs to the analysis of measurable physical things that may vary with time. Each numbertis regarded as the measure, in some convenient unit, of thetimefrom some specified instantτ,after τif$t > 0$,before τif$t < 0$. SupposeGis a number set each element of which is so regarded. For eachtinG, suppose$f(t)$is the measure (a number) of some physical thing at timet(i.e., at the time fromτdetermined byt). Thenfis a simple graph whoseX-projection isG.

Example. Suppose a...

21. Integral Tables
(pp. 187-188)
22. Index of Simple Graphs
(pp. 189-190)
23. Glossary of Definitions
(pp. 191-195)