General Requirement for any Course taught by me.

1. Absenteeism. Unless you call me beforehand or produce evidence that you are truly sick, I regard absenteeism as a personal insult, because you regard my lectures as not as important as other things in your life. If you are too sleepy, come to my class and sleep. I don't mind. You may absorb something during your dreams by osmosis.

2. Homework. Very important. Nobody passes my courses by missing more than 20 % of my homework assignments. I am sometimes lenient on deadlines. Missed homework problems must be redone, but only for 70% credit.

The purpose of my teaching is to develop your technical skills, as well as your presentation, and self-motivation to investigate interesting problems. Therefore be prepared to spot and correct my mistakes and even ignorance on some subject.

The way I grade: Getting the right answer at the end of the book may get you only 70 % or less, if your presentation is too brief or simply illogical. Even if your logic is right, you may not get 100 % because your logic is not the most elegant or most concise. Math is an art of pursuing elegance. I grade not by subtraction of points from your mistakes. I simply ask myself: is this problem done in a way that deserves an A-, B+, or F, etc. Then I give you a percentage accordingly. 90%+ is A, 85%+ is A-, 80 is B+, 75 is B, 70 is B-, 65 is C+, 60 is C, 55 is C-, 50 is D, < 50 is F. At end of the semester I compute your semester grade by a formula such as 40% for homework, 30 % for 2 mid-term tests, and 30% for a final, comprehensive, all-inclusive 2-hour examination (an exam is more serious than a test.) A make-up test is permitted only if you are very, very, sick that day.

3. Presentation. Computer programs must have 20 % text devoted to comments, and mathematical proofs must have 20 % to 50 % devoted to English phrases such as "thus", "since", "WLOG (without loss of generality)", "let us assume", "let x be an arbitrary integer", "... it behooves us to consider...", "..by a stroke of genius...", "necessary", "sufficient", "suppose...", "QED", etc. Diagrams must be drawn whenever meaningful, applicable, or possible. There must be some scale on the x- and y-axes on every diagram. Parabolas must not look like a capital U, ellipses must not look like footballs or eggs.

You must leave 1 inch margin on the top and left of the page. Proofs should be short and elegant, otherwise cut up into lemmas. They must be logical, and readable like an English article, with periods at the end of each line. Equations do not always have to start on a new line, if you write in paragraph form, but do not let any equation or formula to be broken into two lines. Equal signs in a series of equations on different lines must be aligned. For Geometry or word problems, you must restate the problem in your homework, so that when I grade your homework I do not have to look at the book. I want a diagram for every geometry problem, even if the book already has it. Diagrams must be on the right side of the paper.

If I give you hints with "..." signs for you to fill in the details, you must copy everything I wrote, explaining any details, as well as filling in the "..." parts.

Name and date should be on upper right corner, and you should fold your paper backward, folded along a vertical middle line.

Do not mutilate and tear the left upper corner. Use a staple. Do not do multiple columns on your paper unless you draw a vertical dividing line with a ruler. Leave a blank line between problems, and make the amount of indentation of each line meaningful.

Surround your final answer to a problem with a rectangle, underline, or highlight with flourescent yellow or pink.

Each page should have a chapter and section number in a rectangle, on the upper left. Problem numbers, with a,b,c,d if necessary, must occur on the left of the red line, encircled.

Do not use paper with frazzled edges. Paper must be thick enough so that my red ink will not seep through to the opposite side. Do no write on both sides of the paper.

For fractions, don't use slashes unless you know what it means. For example 1/x+1 is not the same as 1/(x+1). The former has to mean (1/x)+1, but you may be thinking of the latter. In other words, the denominator of the slash only includes products, but not plus or minus terms. Another example: a/b * c means a/(b*c) i.e., a divided by the product of b and c. If you want to multiply c with the quotient of a divided by b, don't write a/b*c, but write (a/b)*c.

4. On your font:

You must do the opposite of what my professor did: he had a way to make 9, g, P, p, q, r, all look alike, and r, v, n, g, all look alike.

You must make your t and + look different. Let your lower case a, c, e, m, n, o, etc. be as tall as 1/3 to 0.4 the line height, i.e., the gap between the blue lines. Superscripts and subscripts should be slightly smaller. In a statement such as "32 x dy + 24 y dx = 0", make sure there is a space before the dy, but no space between the d and y, no space between the 3 and 2, and space before and after the + and = signs. For fractions, You should vertical space half a line, thus leaving more room for the numerator and denominator. If each of them contain superscripts and subscripts, or more fractions, then you must do more vertical space, on top of the current line. When you write a fraction, draw the middle line about as high as a hyphen. The numerator and denominator must not use too small a font as to crunch the whole fraction into the gap between the blue lines.

Your I must have caps and shoes (called serifs), and your 1 should have a little hook. Your little L must have a tilt upwards at the bottom, and your r must look different from little L or p. Your zero should be narrow, your big O must be wide and fat. Your theta should be narrow. Too bad you may have to give up some American freedom in creating your own fonts.

You must not use a,b,c for variables (use x,y,z instead).

You must not use ink or ball point for math. Use only dark lead pencils, such as Ticonderoga #1 or Venus Velvet #2. Mechanical pencils must be 0.5 mm or thinner, HB lead or darker. Never use red ink or ball point.

5. The following 'proof' is not acceptable:

Prove that (1 - x³) / (1 - x) = 1 + x + x² if x is not = 1.
Bad Proof: 1 - x³ = (1 + x + x²) (1 - x)
1 - x³ = 1 + x + x² - (x + x² + x³)
1 - x³ = 1 + x + x² - x - x² - x³
1 - x³ = 1 - x³
0 = 0

The above proof is illogical because you are arguing backward. The presentation is wrong. There are 2 ways of presenting it correctly:

Proof 1: (1-x) (1+x+x²) = 1+x+x² -(x+x²+x³)
= 1 - x³.
If 1-x is not = 0, then we can divide both sides by 1-x, so that (1-x³)/(1-x) = 1+x+x² provided that x is not 1.

Proof 2: (1 - x³)/(1-x) = 1 + x + x² is implied by
1 - x³ = (1 + x + x²) (1 - x) if x is not 1,
which is implied by
1 - x³ = 1 + x + x² - (x + x² + x³) which is implied by
1 - x³ = 1 + x + x² - x - x² - x³ which is implied by
1 - x³ = 1 - x³ which is obviously true.

Notice: The bad proof above is made right only if you insert "which is implied by" at the end of every line.

An equation is a statement. An expression is something that does not have an equal or unequal sign. The equal or unequal sign, or <, >, £, ³, are verbs. We have to say "equals", or "is equal to", because they are verbs, but a simple "equal" is wrong, because it is not a verb. Strictly speaking you have to say "4 equals 2 squared", not "4 equals 2 square".

When you write:
a < b
= c.
I take it to mean "a is less than b, and b is equal to c, hence a < c." The equal sign in the 2nd line does not tell me "a = c", but rather "b = c", because the two lines make only one sentence: "a < b, which = c ". You may write " a < b = c" all in one line.