2. Calculation

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: 23 is not 32. 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 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. (Nicomachus theorem)

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= 32 210=1024 264=[(25)2]6 * 24 264>106

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.