"Let no man ignorant of geometry enter." -- (Sign over Plato's Academy in Athens.)
Euclid (c. 300 B.C.) is the "father of geometry." His major work, the Elements, formulated laws of space and properties of space that we teach in geometry today. Euclid attempted to develop an organized form of thinking through deductive reasoning. In a deductive system, a few basic statements (postulates) are accepted as true. These are then used to establish the truth of additional statements (theorems) that reflect the logical consequence of either a postulate or some previously established theorem.
What did Herkimer tell Cinderella when her photographs didn't arrive on time?
Answer: "Don't worry, Cindy. Some day your prints will come."
IVY STICKER ...a nurse.
WELDON BERGER ... a cook at McDonalds.
Reading: Section 1.4, pages 26-29.
Written: Pages 29-31/24-36.
Mathematical word analysis:
NEGATIVE: The Latin root is negare (to deny). In this sense, denying or invalidating a positive number.
Here are some examples of communication in mathematics.
On pages 29, problem #19 presents the equation -xy + 3x = 30, and asks for the value of y when x = 15. OK, there are various ways this problem can be done, but here is a presentation that could be understood by someone reading the solution process.
-xy + 3x = 30
==> xy = 3x - 30
==> y = 3 - 30/x. (Note the period. This is the end of a sentence.)
x = 15 ==> y = 3 - 30/15 = 3 - 2 = 1.
The original equation is a conditional statement containing two variables. We have concluded that if x has a value of 15, then the conditional statement will be true if y = 1.
Suppose now that I want to solve the equation a/b = c + d/w for w. There are different ways to do this, but if you can read and understand the language of mathematics, you should be able to comprehend the following process:
a/b = c + d/w
==> d/w = a/b - c = (a-bc)/b
==> w/d = b/(a-bc)
==> w = bd/(a-bc).
To understand what is above, it is important to know the basic order of binary number operations. A binary operation combines two numbers to produce a third number. Binary operations are addition, subtraction, multiplication, division, and exponentiation. The order of operations is as follows:
- Exponentiation has highest priority.
- Multiplication and division are then performed in the order in which they appear, left to right.
- Addition and subtraction are then performed in the order in which they appear, left to right.
- Parentheses () can be used to alter the order of operations.
Calculators are programmed to follow this order of operations. Here are some examples. Make sure you understand them.
5+30/10 = 5+3 = 8.
80/4x2 = 20x2 = 40.
80/(4x2) = 80/8 = 10.
4-6x2+8-30/15 = 4-12+8-2 = 0.
4+3x52 = 4+3x25 = 4+75 = 79.
4+(3x5)2 = 4+152 = 4+225 = 229.
(5-2)2 = 32 = 9. [Note that the original expression is not 52-22 = 21.]
Mathematics is a language. Learn it well. Math is power. Pythagoras (572 - 497 B.C.) and the Pythagoreans were the first to realize that the amazing mathematical design of the universe (as they knew it).
Problem: Solve a2x - b = cx + d for x.
Solution (with communication):
a2x - b = cx + d
==> a2x - cx = b + d
==> x(a2 - c) = b + d
==> x = (b + d)/(a2 - c), if a2 - c is not zero.
Problem: The formula for the area of a trapezoid is A = (1/2)(b1 + b2)h. Solve the formula for b1.
Solution (with communication):
A = (1/2)(b1 + b2)h
==> 2A = (b1 + b2)h
==> 2A/h = b1 + b2
==> b1 = 2A/h - b2.