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:
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
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:
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
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:
|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:
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:
On June 29, 1999 he stated that this solves the problem
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)
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
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
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
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
©2008 Allen Klinger