Computer Mathematics In-Progress

Computers use only two symbols, zero and one. They replace familiar decimal numbers by strings of zeroes and ones. The quantity two looks like a decimal ten (a one with a zero at its right). The quantity eight is a one symbol followed at its right by three zeroes). Other quantities like sixteen (one with four zeroes at the right) and thirty-two (one and five zeroes) are composed taking advantage of the notion that doubling is the same as adding another zero at the right.

The same zero/one symbols can be adjoined in ways that signal that a non-number is represented. Some specifics are alphabetic lower-case letters, capital letters, and punctuation marks. Even colors can be signalled by zero/one symbols.

Here are a few challenges. They are areas to think about, things that are more like entry points to fundamental ideas than processes to master (like long division).

### Today Computers Do Many Things

#### Support World-Wide Communication

*Source:
http://rsd.gsfc.nasa.gov/goese/autogvar/goes8/full_earth/color/latest.color.jpg

### Mathematical Concepts and Principles

Each of the following images links to a mathematical problem or concept -- click on an image to view its contents

Adding or multiplying numbers when there is no last item is a little like the idea of the rabbit trying to catch a snail who has a one mile head start. The puzzle goes that while the rabbit runs to the half-mile point the snail moves forward. That means that when the rabbit makes it to the three-quarter mile point, the seven-eighths marker, etc., there always will be distance remaining to catch the snail. The same kind of reasoning led to the view that the continually-smaller circles inside the increasing-number-of-side polygons in the following figure get to a smallest circle ... if we go far enough in the process. This idea is called limit in mathematics.

### Four and Five Sided

Mathematics

Writing
Clicking on the Gustavus Raven image at the left just yields a larger image. The image is here because this figure symbolizes power. Mathematics knowledge is also a source of power. Mankind has developed because of social forces. Exercise by talking things over with others. Speaking

Mathematical Ideas

Whether dealing with symbols or the underlying idea, mathematics deals with aspects of the world. Some things known from the past, e.g., the circle, go back to practical discoveries (the wheel). The symbol π the ratio of a circle's boundary length (circumference) to its diameter.

Many mathematical ideas are joined to special symbols used to concisely portray them. Sometimes the symbol blocks thinking, the notions that led to the concepts being understood in the first place. One such symbol is the ratio of the length of the boundary or circumference of a circle to its diameter, the line through the circle center which defines its widest crossing. That quantity is called π and pronounced by the sound pi.

Finding π to greater accuracy was a significant task in mathematics up to a few hundred years ago.

Adding or multiplying numbers when there is no last item ... a situation usually summarized in a shorthand way by the word infinite, leads to two special symbols: Σ represents addition while Π indicates multiplication.

 Computer Mathematics In-Progress http://www.cs.ucla.edu/~klinger/inprogress.html 11/19/2008 Version