Assignment 9

"No employment can be managed without arithmetic, no mechanical invention without geometry." -- (Benjamin Franklin)

Math History Tidbit:
The effects of time have destroyed many great mathematical works of significance. One survivor is the Rhind Papyrus, from approximately 1800 B.C., which provides much information about ancient Egyptian mathematics. It is about 1 foot high and 18 feet long, and rests in a British museum. The Rhind Papyrus contains 85 problems written in hieratic form. Many of the problems involve the concepts of counting and measuring.

Egyptian arithmetic operations didn't include fractions that contained numerators other than 1. The Rhind Papyrus contains a table that allows the reader to represent fractions as a sum of fractions with a numerator of 1. For instance,

2/97 = 1/56 + 1/679 + 1/776.

Herkimer's Corner

When Herkimer was a clergyman, what did he say to the parishioner whose name he couldn't remember?

Answer: "I can't remember your name, but your faith is familiar."

Things Herky would like to know:

How did a fool and his money get together in the first place?

Why is the word abbreviation so long?

Reading: Review Section 2.3, as necessary,

Written: Pages 87-88/58-63 (Write these up neatly.)

Items for reflection:

Mathematical word analysis:
FRACTION: From the Latin word frangere (to break). In this sense, a fraction is a broken portion of some whole.

To determine a line, one needs two basic things: The slope and a point on the line. It is true that if you are given two points on the line, you can "determine" the line, but to describe it with an equation you need to calculate the slope.

Example: What is the equation of the line with slope 3 that contains the point (2,15).

Response: y - 15 = 3(x - 2) ==> y = 3x +9.

Example: What is the equation of the line containing (3,6) and (-1,14).

Response: The slope of the line is (6-14)/[3-(-1)] = -8/4 = -2. The equation of the line is

y - 6 = -2(x - 3) ==> y = -2x +12.

On another note, thinking ahead....

Preparing for a test can be a relatively easy and equational experience if you

(1) have been a learner rather than a memorizer.

(2) organized your materials (notes, homework, handouts, etc.) in a neat and orderly manner.

(3) understand that mathematics is a language.

Problem: Identify the slope, y-intercept, and x-intercept of the line 8x + 2y = 13.

Solution (with communication):

8x + 2y = 13 ==> 2y = -8x + 13

==> y = -4x + 6.5.

The slope of the line is -4.

x = 0 ==> y = 6.5 (the y-intercept).

y = 0 ==> 8x = 13 ==> x = 13/8 (the x-intercept).

Problem: Two positive single-digit integers have a sum of 15. Find the integers.

Solution (with communication):

If the desired integers are represented by x and y, then x + y = 15 ==> y = -x + 15. This line contains all pairs of numbers that add up to 15. However, we only want pairs (x,y) that are positive single-digit integers. The only possibilities are (6,9), (7,8), (8,7), and (9,6). These are points on the line y = -x + 15.

Responding to the question as asked, there are two integer sets that meet the specified conditions. They are {6,9} and {7,8}.

Note: Watch your notation here. Don't confuse number sets (use of brackets) with points represented as ordered pairs (use of parentheses).