When this value is evaluated, the semantics of Haskell no longer yield a meaningful value. In other words, further operations on the value cannot be defined in Haskell. Several ways exist to express bottoms in Haskell code.

There is some overlap among these topics, so I recommend reading the whole page. Ultimately, what are the sources of errors and of misunderstanding? What kinds of biases and erroneous preconceptions do we have? Two of my favorite historic discoveries are Einstein's discovery of relativity and Cantor's discoveries of some of the most basic rules of infinities.

These discoveries are remarkable in that neither involved long, involved, complicated computations. Both are fairly simple, in retrospect, to anyone who has studied them. But both involved "thinking outside the box" par excellence -- i. As philosopher John Culkin said, "We don't know who discovered water, but we are certain it wasn't a fish.

Errors in Communication Some teachers are hostile to questions. That is an error made by teachers. Teachers, you will be more comfortable in your job if you try to do it well, and don't think of your students as the enemy.

This means listening to your students and encouraging their questions.

A teacher who only lectures, and does not encourage questions, might as well be replaced by a book or a movie. To teach effectively, you have to know when your students have understood something and when they haven't; the most efficient way to discover that is to listen to them and to watch their faces.

Perhaps you identify with your brightest students, because they are most able to appreciate the beauty of the ideas you are teaching -- but the other students have greater need of your help, and they have a right to it. A variant of teacher hostility is teacher arrogance. Actually, most of the errors listed below can be made by teachers, not just by students.

However, most teachers are right far more often than their students, so students should exercise great caution when considering whether their teachers could be in error.

If you're a student with a hostile teacher, then I'm afraid I don't know what advice to give you; transfer to a different section or drop the course altogether if that is feasible. The remarks on communication in the next few paragraphs are for students whose teachers are receptive to questions.

For such students, a common error is that of not asking questions. When your teacher says something that you don't understand, don't be shy about asking; that's why you're in class!

If you've been listening but not understanding, then your question is not a "stupid question. Think of yourself as their spokesperson; you'll be doing them all a favor if you ask your question. You'll also be doing your teacher a favor -- your teacher doesn't always know which points have been explained clearly enough and which points have not; your questions provide the feedback that your teacher needs.

If you think your teacher may have made a mistake on the chalkboard, you'd be doing the whole class a favor by asking about it. To save face, just in case the error is your own, formulate it as a question rather than a statement.

For instance, instead of saying "that 5 should be a 7", you can ask "should that 5 be a 7?

Don't wait until the very end of the example, or until the end of class. As a teacher, I hate it when class has ended and students are leaving the room and some student comes up to me and says "shouldn't that 5 have been a 7? Now all your classmates have an error in the notes that they took in class, and they may have trouble deciphering their notes later.

In the early s, I managed a computer retail store. Several of my employees were college students. One bright your man was having difficulty with his Freshman college algebra class. I tutored him and he did very well, but invariably, he would say, "the professor worked through this problem on the board, and it was nothing like this.

I sure hope we got the correct answer. The professor was from the music department, and didn't normally teach college algebra he had been pressed into duty when over enrollment forced the class to be split.

During the class, he picked a problem from the assignment to work out on the board. Very early in the problem, he made an error. I don't recall the specifics, but I'm sure it was one of the many typical algebra errors you list.

Because of the error, he eventually reached a point from which he could no longer proceed.Write an equation in standard form for a line that has an undefined slope and passes through (-6,4). Thanks for your help, Ian Sol'n: if the slope is undefined, then the equation is y = answer.

Each one expresses the equation of a line, and each one has its own pros and cons. For instance, point slope form makes it easy to find the line's equation when you only know the slope and a single point on the line.

Standard form also has some distinct uses, but more on that later. DOSING. Use 1/2 teaspoon of oil, in each “portion” you make. 1/2 cup of oil = 24 teaspoons (48 doses) USING THE OIL IN YOUR RECIPE.

If you want to make 12 cookies, and your recipe calls for 1/2 cup of butter, use 6 teaspoons (1/8th cup) of canna oil, and make up . A) Write an equation in standard from and in slope intercept form for the line described through (3,10), parallel y= -4 B) Write an equation in sloper intercept form and in standard form for the line passing through (2,6) and perpendicular to x = SOLUTION: write the standard form of the equation of the line described: through (-1,3), perp.

to y=-1/5x + 5. This is from a worksheet that my stepson brought home from school. Algebra -> Coordinate-system -> SOLUTION: write the standard form of the equation of the line described: through . The standard form of the equation of a circle with center at (h, k) and radius r is.

Use completing the square to rewrite the given equation in standard form. So, h = 3, k = ±2, and r = 4.

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What I Wish I Knew When Learning Haskell ( Stephen Diehl )