2. Calculation

Most instances of original uses of digital computers involve making something
happen visually. Browsers, spreadsheets, and the checkbook graphic user
interface are all examples. The same is true of the mathematical and
computing innovations described in chapter one figures.
Using a visual representation as a symbolic shortcut for an idea is a
frequent device in mathematics:
examples include simple use of a letter from some alphabet as p standing for the ratio of a
circle's circumferenced to its diameter, x being
the unknown quantity in word problem algebraic representation, and q standing for an angle. This chapter concerns the role of
visual displays, in particular, symbols, signs, and abbreviations, in computing.

2.1

The approach of using a single item like p or
x to stand for a complex idea occurs in many computing instances. The easiest
cases to consider come from old or traditional signs from written media. For
instance, exponents. In that case relative position of two numerals has special meaning: 2^{3} is not 3^{2}. By contrast, a less commonly
understood instance is the Unix computer operating system's use of the
imperative sign "!" to indicate letting the user take over control.
The general situation is that computers extend reliance on symbols.

The rapid change that is a hallmark of computing involves so many abbreviations and special meanings attached to symbols
that a viable learning strategy becomes a professional necessity.
The power of digital technology like root computing
a. Egyptian and strange cancellation.
b. Odd Apples
c. Cubes
two-hexadecimal-digit codes for computer screen colors
requires particular left-right positioning for each tone.
Gillings, Richard J., * Mathematics in the Time of the Pharaohs*,
Cambridge, MA: The MIT Press, 1972; NY: Dover Publications, Inc., 1982,
ISBN 0-486-24315-X.

Golomb, Solomon W., Polyominoes, (with more than 190 diagrams by
Warren Lushbaugh), NY, Scribner, 1965.

Nelsen, Roger B., Proofs without words: exercises in visual thinking,
Washington, DC: The Mathematical Association of America, 1993.

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

In the first century A.D. the question

A few examples show what this means:

1

2

3

4

The issue is to

How can you

In adding fractional quantity it isn't always necessary to take the same route. Think of

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.

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

Connections exist between raising numbers to powers and the inner working of digital computers.

Computers use 2

2^{5}= 32 | 2^{10}=1024 |
2^{64}=[(2^{5})^{2}]^{6} * 2^{4} |
2^{64}>10^{6} |

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.