Thinking Toward Computer Basics

Sources and Credits From elementary concepts ... ... to computer-bafflers.

1. Reasoning About Solids - Adjoining Two Kinds of Pyramids

The following problem appeared in the 1980 PSAT. The answer believed to be correct by the Educational Testing Service experts was in fact incorrect.

Construct two pyramids with every edge of equal length, L. The first pyramid has a square base and four equilateral triangle faces. The second pyramid has an equilateral triangle base. It has three more equilateral triangles as faces. This object is a Tetrahedron. The task is to glue any triangular face of the first pyramid to one of the second, the tetrahedron. In doing this, completely match the two glued triangles together. This creates a single solid object. How many faces does the resulting solid have? Helpful Diagrams. Reasoning from visual images can be tricky. For an algebraic proof, please click: determinants.

2. Regular Solids - Objects for Games

The Tetrahedron is one of several regular solids. While it has four equal-area equilateral triangle faces. Indeed regular simply is another way to say all faces of the solid are the same in shape and area. Since the concept is about planar objects, like triangles (Tetrahedron) and squares (Cube), we can also use the word that describes this kind of match: faces in one regular solid are congruent. Such objects are known as the Platonic solids.

A Cube has six equal faces. Dice for games are in this shape. There dots indicate numbers on each face, one on up to six. Numbering the different faces of other equal-sided objects, creates other solids for game use: Platonic Dice.

3. Two Simple Classes of Numbers - Odd Viewed from Solids

In the first century A.D. the question How can the cubes be represented in terms of the natural numbers? was answered by the statement: Cubical numbers are always equal to the sum of successive odd numbers and can be represented this way.

A few examples show what this means:

13 = 1 = 1
23 = 8 = 3 + 5
33 = 7 + 9 + 11
43 = 13 + 15 + 17 + 19

The issue is to determine why this is so. To check your understanding, find the odd decomposition of 73. For visual clues this may help: Two-Cubed.

4. Two Simple Classes of Numbers - Odd Properties

How can you place nine apples in four baskets so there is an odd number of apples in each? Here is a visual Explanation.

5. Practical Versus School Mathematics - Fractions

In adding fractional quantity it isn't always necessary to take the same route. Think of two or three different and valid ways to add fractions in real life situations. Here are visuals about this: Unusual Views, Sport Statistics.

6. Man Versus Machine - Exponential Power

A tank attached to a chain attempts to tow a person holding the other end of it. The chain wraps three times completely around a permanently-anchored metal pole. The person pulls with a force of ten pounds. The tank pulls with a force of ten thousand pounds. Does the tank pull the chain out of the hands of the person? Show your reasons. Clues Tank Visual and Explanation.

7. Repeated Multiplication - Connecting Number and Symbol

Holding back a tank is allied to placing one grain on the first square of a a chess-board, two on the second, double that or four on the third, and so forth. The method used to represent Repeated Multiplication is the notion of power. Powers of an integer are just the number of times it is multiplied by itself. Exponents are small numerals placed slightly above and to the right of a number or other quantity. The signal indicates the number of times something is multiplied by itself.

Connections exist between raising numbers to powers and the inner working of digital computers. Exponents in notation began as a simple means to record large quantities. For instance, 106 is a product of six tens. It also is the quantity one million: a one with six zeroes after it. We write 2 times itself repeated again both as 2 * 2 * 2 and with a count of how many times 2 appears (three). The product is also written: 23.

Computers use 2n. The integer values n = 5, 8, 10, 16, 32, and 64 occur often enough in this field that many people know the following facts:

25= 32
and since 264=[(25)2]6 * 24

Some connections between computers and mathematical ideas begin by considering odd and even, inside and outside, connected and disconnected. This pointer locates four Visual Puzzles that bear on these ideas.

8. Repeat A Question - Power of the Internet

A number growing in size (placing one grain on the first square of a chessboard, doubling the grains on each sucessive square) is a concept that over centuries formed some key mathematical ideas (infinity, infinite series). Today the Internet fosters rapid communication. People can jointly discuss an issue using the world's computers. This has been going on since a visit to UCLA by UCSD Prof. R. Graham May 21, 1999. Here is a general view of that discussion.

An open problem (at least up to when this was written, June 24, 1999) is to find n, a second integer greater than one, with the property (8.1) that when 2 is raised to it and 3 is taken away from the result, the answer is evenly divisible by the n value. What is known is that the value 4700063497 does this. In other words 24700063497-3 is an integer times 4700063497.

Electronic mail (email) communication put this problem to this page's author. Inquiry by email led a correspondent suggesting listing the problem. Posting it to an email list on number theory led to at least four people sending informative messages in hours. This note continues the fact-seeking process through the Internet. (Informative email continues to arrive.) A recent message included: "D.H. Lehmer found such an n. Look at Mathematics of Computation about 10 years ago. ..."

Computer-communications stimulate learning about individuals involved in such work. [Related items are in mathematical books.]

Any n satisfying the property (8.1) must be odd. (Since the minus three causes the expression to be odd. No even number can exactly divide an odd.) That a solution can't be a multiple of three (substitute n=3k, find that for the property to be satisfied 3 has to evenly divide a power of 2 ... which is impossible), nor in several other families of integers is also known. UCLA Prof. E. Koutsoupias pointed out July 1, 1999 that Fermat's Little Theorem eliminates the possibility that such an n would be a prime.

This property written concisely in mathematical notation is:

n|[(2n)-3] (8.1)

Experts: Peter Montgomery wrote This is problem F10 in the 1981 Edition of Richard K. Guy's Unsolved Problems in Number Theory. He gives the solution 4700063497 = 19 * 47 * 5263229.
To refer to it later we write:
4700063497 = 19 * 47 * 5263229 (8.2)
June 11, 1999 Joe Crump pointed out that Number Sequences is an available resource and (8.2) appears in List [list begins a(1) = 1].
On June 24, 1999 Dr. Montgomery conveyed this number to me by electronic mail:

63130707451134435989380140059866138830623361447484274774099906755 (8.3)

On June 29, 1999 he stated that this solves the problem (see detail).

Prof. Noam Elkies mentioned a computer program June 5, 1999; using sources detailed in Computations, items sent me by email, I factored the above number to:

5 * 97 * 130166407115741105132742556824466265630151260716462422214638983 (8.4)

Factoris, by Xiao Gang (factors integers, polynomials) concluded 130166407115741105132742556824466265630151260716462422214638983 is prime; (8.4) confirms Montgomery message items. It seems striking that both the numbers found factor into three primes with two of them small (except see below). Further it is also striking that the ratio of the two is close to the ratio of their corresponding large prime factors. (8.3) is about 6.3 * 1064; (8.2) is about 4.7 * 109. The ratio (8.3)/(8.2) is about 1.34 * 1055. The ratio of their largest factors is about 2.47 * 1055.

In Chapter 3 of G. H. Hardy's Ramanujan, Twelve Lectures on Subjects Suggested By His Life and Work, Cambridge UK: The University Press, 1940, a round number is said to be one that "is the product of a considerable number of comparatively small factors." On the second page of that chapter appears "We find, if we try numbers at random from near the end of the factor tables, that f(n) [the number of distinct prime factors of a number n] is usually not 7 or 8 but 3 or 4; ...".

Now the questions about (8.3) begin.

9. Things Computers Can't Find - Power has Limits

In ancient times people looked at the properties of complex geometric images. One such, shown in the Kasner/Newman book, nests circles and inscribed regular polygons. The nested circles grow smaller. They are within inscribed regular polygons whose number of sides increases, beginning with three (equilateral triangle). Here is the image. An analytical solution for the inside-circle radius limit can be constructed by trigonometric reasoning. It is the infinite product:

Like past knowledge of π, mathematics has improved on what was known. For π, [2143/22]1/4 by Ramanujan Kanigel reference, is better than 3, 22/7, 3.14 and other approximations. This value is easy to find on a hand calculator. Still it is not the best that is known (for more on this see Product Paper). It happens that very slow convergence of the infinite product (9.1) makes it hard to compute, like the exponentials in 8.

10. Computers Use Numbers

Figure shows number representation: what is meant by decimal, binary, octal, and hexadecimal. Counting to other bases is part of daily life with time - sixty seconds; and shopping - dozens of eggs. A poem unites words, a great painting, and quantity recording tally and number systems.

For more, go to Association
9/27/13 Version ©2008 Allen Klinger