"It isn't that they can't see the solution. It is that the can't see the problem." -- (G. K. Chesterton)
Mathematical Drought in Western World (500 - 1200):
The Islamic world represented fertile ground for both mathematical development and the preservation of ancient mathematical knowledge during the Middle Ages. The city of Baghdad (now the capital of Iraq) was established in 766 and became a flourishing intellectual center where, among other things, important works of the ancient Greeks where translated into Arabic for study. By the end of the ninth century, many important works by such Greek mathematicians as Euclid, Apollonius, and Diophantus had been translated into Arabic for study. The research center at Baghdad, called the Bayt al-Hikma , or House of Wisdom, drew scholars from all over the Arabian peninsula, who not only contributed to the translation of ancient documents, but also conducted much original research.
What did Herkimer call the headache he got when he accidentally spilled a load of wheat on the highway?
Answer: My grain headache.
Herky 's words of wisdom:
Birthdays are good for you - the more you have, the longer you live.
In just two days, tomorrow will be yesterday.
Reading: Section 2.8, pages 122-124.
Written: Pages 125-126/12-14, 19-33(odds), 42, 43.
Mathematical word analysis:
ABSOLUTE VALUE: From the Latin word absolvere (to free from). In a sense, absolute frees a number from its sign.
If you realize that the absolute value of a number is either the number or its opposite, then the concept of absolute value is easy to use. If you "read" mathematics, the following should make sense:
|x| = x if x is positive or zero.
|x| = -x if x is negative.
It is wise to read -x as "the opposite of x." For instance, if x = 7, then -x = -7. If x = -34, then -x = 34. Those who think
that -x must always represent a negative number are simply not reading mathematics correctly.
Let f be a function defined by y = f(x) = |x - 3| + 11.
OK, f is a rule that says "take a number, subtract 3, take the absolute value of what you have, then add 11. When you are asked to calculate f(43), for instance, you are simply being asked for the number produced by f when it operates on 43. In this case, f(43) = |43-3| + 11 = 40 + 11 = 51. As another example, f(-100) = |-100 - 3| + 11 = 103 + 11 = 114.
You should be able to determine that the smallest value y = f(x) can assume is 11. This is simply because |x - 3| cannot be negative, but it can be 0 when x = 3. the following should also make sense:
If x is equal to or greater then 3, then f(x) = x - 3 + 11 = x + 8.
If x is less than 3, then f(x) = -(x-3) + 11 = -x + 3 + 11 = -x + 14.
In other words, f can be considered a piecewise function that reads: If a number is 3 or greater, then the rule f says "take the number, and add 8." If a number is less than 3, then the rule f says "take a number, get its opposite, then add 14."
MATH IS POWER. AND, IT'S EASY TO READ AND WRITE IF YOU WILL LEARN THE LANGUAGE.
Problem: Express f(x) = | x - 9 | + 5 as a piecewise function without use of the absolute value bars.
Solution (with communication):
If x is 9 or greater, then | x - 9 | + 5 = x - 9 + 5 = x - 4.
If x is less than 9, then | x - 9 | + 5 = -(x - 9) + 5 = -x + 14.
Hence, we could write
f(x) = x - 4 if x is 9 or greater.
f(x) = -x + 14 if x is less than 9.
The function f is basically a two-part rule saying "Take a number. If the number is 9 or more subtract 4 from it. If the number is less than 9, take its opposite and then add 14."
Problem: Solve (a) | 2x + 5 | + 13 = 6; (b) | 5 - 2x | + 3 = 18.
Solutions (with communication):
(a) | 2x + 5 | + 13 = 6
==> | 2x + 5 | = -7.
The equation has no solution since the absolute value of a number can't be negative.
(b) | 5 - 2x | + 3 = 18
==> | 5 - 2x | = 15
==> 5 - 2x = 15 or 5 - 2x = -15
==> 2x = -10 or 2x = 20.
==> x = -5 or x = 10.