"When you can measure what you are talking about and express it in numbers, you know something about it." -- (Lord Kelvin, 1824-1907)
Henry Wadsworth Longfellow (1807-1882): He was an accomplished mathematician who created problems in poetic settings. He resigned his position of modern languages at Harvard University to find time to write - and his writing included some mathematics, a subject for which he had a deep appreciation.
Can you find the number of water lilies in this problem created by Longfellow?
One-third of a collection of beautiful water lilies is offered to Mahadev, one-fifth to Huri, one-sixth to the Sun, one-fourth to Devi, and six which remain are presented to the spiritual teacher.
When Herkimer was a waiter, what did he say to a customer who ordered a lobster tail?
Answer: "Once upon a time there was a handsome lobster who ... "
Things Herky would like to know:
If 7-11 is open 24 hours a day, why are there locks on the doors?
Is it true that cannibals don't eat clowns because they taste funny?
Reading: Section 6.1, pages 323-325
Written: Problems provided on handout (sheet provided in class).
Mathematical word analysis:
AVERAGE: From the Arabic word awariyah (damaged goods). In the Arab world, money lost due to goods damaged by shipping was shared (averaged) by merchants.
We live in a mathematically-designed universe. The basic laws of algebra, which are definitely part of mathematics, are important in understanding the world in which we live. The Pythagoreans (500 B.C.) believed that all things are numbers, and that numbers rule the universe. Is there anything you can think of that does not somehow relate to number sense? Things to think about:
(a+b)2 is not equal to a2 + b2. Does your number sense suggest that (5+2)2 = 52 + 22?
(a-b)2 is not equal to a2 - b2. Does your number sense suggest that (5-2)2 = 52 - 22?
÷(x2) is not necessarily x. Does your number sense suggest that ÷[(-2)2] = -2?
If x2 = 25, then x = 5 is not a necessary conclusion. That is, x2 = 25 ==> x = 5 is a false statement. By saying the statement is false, you are not saying that 52 is not 25. You are saying that x = 5 is not a necessary consequence of the premise. Note that x = 5 ==> x2 = 25 is a true statement.
Mathematics is a language. The behavior of our universe (so well described by mathematics) certainly suggests that mathematics is a very logical language. Interestingly, English is not a totally logical language. As a simple example, the words know and no are pronounced exactly the same way. There is a k in know, but there is no k sound when the word is spoken.
OK, we have just been introduced to zero and negative exponents. These are new concepts defined in terms of known concepts, in the same way that when you learn a new word, it can make sense only if it is defined in terms of words you do know. What follows are just some statements involving the new types of exponents:
(x-7)/(y-6) = (1/x7)/(1/y6) = y6/x7.
0.00007 = 7x10-5.
(x-3)4 = (1/x3)4 = 1/x12 = x-12. (Note that we can use laws of exponents and multiply the exponents in the original expression.)
80 + (1/2)-2 = 1 + 4 = 5.
MATH IS POWER. LEARN THE LANGUAGE. USE IT EFFECTIVELY.
Problem: How many digits are contained in the simplified version of the number (0.1)-15?
Solution (with communication):
(0.1)-15 = (10-1)-15
The number contains 16 digits.
Problem: Is x-1+y-1 = (x+y)-1?
Solution (with communication):
If x = 1 and y = 1, then x-1 + y-1 = 1/x + 1/y = 1/1 + 1/1 = 2.
If x = 1 and y = 1, then (x+y)-1 = 1/(x+y) = 1/(1+1) = 1/2.
Conclusion: x-1+y-1 is not equal to (x+y)-1.
Note: From a purely algebraic approach,
x-1+y-1 = 1/x + 1/y = (x+y)/(xy), and
(x+y)-1 = 1/(x+y).