A lot of names. That is just one way to describe *number*. Names
for *how much*, *how many*, or in terms of a concept,
*quantity*. Or a bunch of signs. Names for signs, in a word
*symbols*. Things those words enable that are
True ... and
Strange.

The words
*dozen, hour, minute, and week* all are names. The last three are
*amounts, measures, units* of time. Some names are of quantities of
any kind: words like *five*, *six*, or *eight*. They
give ways to say how many are present
of *most objects*
whether it is
eggs, minutes, seconds or days. Number-*symbols* are shorter than words.
They came from picturing quantity, but in many cases the symbols are far
different than those early pictures. This page deals particularly with two
quantities that are connected to digital computers.
*Five*
reflects the fingers of a hand, while *sixteen* indicates some
computers' number of ones and zeroes.

An overview table at
Sixteen lists some symbols with math and
computing language about them. The table parts can be individually
reached from the last five of
six pointers in 16_Links. Some ways of writing
sixteen are at Tallied.

The numeric symbol *five* repeats in the "Figure
Five in Gold" painting by Demuth, found in the New York City Metropolitan Museum
of Art. Inspired by William Carlos Williams' poem *The Great Figure*, the
image of fives and fire engine headlights, assigns power to the vehicle.
That power belongs to humanity's invention of numbers.

Five and sixteen, six and eight, all four are called *whole* or *natural* (both words are in use). They contrast with things like five-sixths or five-eighths that
express *fractional* or *proportional* quantities, and to
which we give the name *ratios*. There are many ways to write fractions.
In one place where writing began, Egypt, a dot on a symbol showed that
it was that fractional part, not that whole number. There general
fractions like five-sixths and five-eighths could not have been expressed
in a simple way (Egyptians wrote them
as sums of *unit* fractions like - using * for the dot - *3 + *6 +
*6 + *6 to represent what today is commonly written 5/6).

## Poem-Inspired PaintingSimilarly Related Art |
## The Great FigureWilliam Carlos Williams
Among the rain and lights I saw the figure 5 in gold on a red fire truck moving tense unheeded to gong clangs siren howls and wheels rumbling through the dark city |
## Five Symbols |

This link between words and an image involving numbers is natural. Both are kinds of language, ways to communicate. Languages' kinships are detected by the stability of their words for

Numbers came about and still act today to increase people's power in commerce. (Getting good at business means keeping track of small differences.)

One way to view mathematics is as building on or extending numbers. There are many mathematical concerns. They all result from thinking. That thinking started when someone had a new idea. Nevertheless the importance, and the actual origin of those ideas, is in solving some problem. One way to see that notion is to consider finding the whole numbers, for instance one through ten, or one up to eighteen, or even twenty, from simple combining operations on exactly four fours (all must be used each time the result is obtained). Gardner [1, p. 51] puts "simple" this way: the arithmetical signs for addition, subtraction, multiplication, and division together with the square-root sign (repeated as many finite times as desired, parenthesis, decimal points and the factorial sign. (Factorial n is written n!. It means 1 x 2 x 3 x ... x n.) A decimal point may also be placed above .4, in which case it indicates the repeating decimal .4444 ..., or 4/9.

For whatever reason, I gave up on finding five from four fours. Anyone similarly inclined can see the solutions below (Fig. 7 The game of fours, from Gardner [1, p. 52]): .

The solving a new problem notion
is true of even simple ideas like *odd*: sometimes all we need to
do is show something true in the case of an odd number, and then also
true when dealing with evens, to prove its universality.

The
Apples & Baskets puzzle, the second example,
tests understanding of
*odd* and *even*, the mathematical form of the idea behind digital
computers, "*on* or *off*."

A few examples show what this means:

1

2

3

4

Can you see why this is so? Can you give the decomposition of 7

10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, __?

Little hint. Big hint. To see what this example is really about please click Sequence Answer or Short Item on Representation. For more questions, please click Some Problems. To see some other ways to represent numeric information, please click Help Sequence?

Pizza knowledge comes in two ways. First, hard earned: Solutions. A generalization appeared in

More on mathematical things can be found from links in Math Sources. Big numbers (and scientific notation) are at Size. An older version of this page is at Dozen; another is at In Progress.

Gardner, Martin, * The Numerology of Dr. Matrix*, NY:
Simon and Schuster, 1967.

It is difficult to get the news from poems yet men die miserably every day for lack of what is found there. -William Carlos Williams

11/02/06 Version |
http://www.cs.ucla.edu/~klinger/association.html | ||||||

©2006 Allen Klinger |