I accidentally divided by zero and my paper burst into flames. Isaac Newton's birthday. I could only get arbitrarily close to my textbook. I couldn't actually reach it. I have the proof, but there isn't room to write it in this margin. I was watching the World Series and got tied up trying to prove that it converged. I have a solar powered calculator and it was cloudy. I locked the paper in my trunk but a four-dimensional dog got in and ate it. I couldn't figure out whether i am the square of negative one or i is the square root of negative one. I took time out to snack on a doughnut and a cup of coffee. I spent the rest of the night trying to figure which one to dunk. I could have sworn I put the homework inside a Klein bottle, but this morning I couldn't find it. Warning! It is against the rule to use these excuses in my classes! A. Ch.
"Do you love your math more than me?" "Of course not, dear - I love you much more." "Then prove it!" "OK... Let R be the set of all lovable objects..."
A group of Polish tourists is flying on a small airplane through the Grand Canyon on a sightseeing tour. The tour guide announces: "On the right of the airplane, you can see the famous Bright Angle Falls." The tourists leap out of their seats and crowd to the windows on the right side. This causes a dynamic imbalance, and the plane violently rolls to the side and crashes into the canyon wall. All aboard are lost. The moral to this episode is: always keep the poles off the right side of the plane.
CLEARLY: I don't want to write down all the "in- between" steps. TRIVIAL: If I have to show you how to do this, you're in the wrong class. OBVIOUSLY: I hope you weren't sleeping when we discussed this earlier, because I refuse to repeat it. RECALL: I shouldn't have to tell you this, but for those of you who erase your memory tapes after every test... WLOG (Without Loss Of Generality): I'm not about to do all the possible cases, so I'll do one and let you figure out the rest. IT CAN EASILY BE SHOWN: Even you, in your finite wisdom, should be able to prove this without me holding your hand. CHECK or CHECK FOR YOURSELF: This is the boring part of the proof, so you can do it on your own time. SKETCH OF A PROOF: I couldn't verify all the details, so I'll break it down into the parts I couldn't prove. HINT: The hardest of several possible ways to do a proof. BRUTE FORCE (AND IGNORANCE): Four special cases, three counting arguments, two long inductions, "and a partridge in a pair tree." SOFT PROOF: One third less filling (of the page) than your regular proof, but it requires two extra years of course work just to understand the terms.
ELEGANT PROOF: Requires no previous knowledge of the subject matter and is less than ten lines long. SIMILARLY: At least one line of the proof of this case is the same as before. CANONICAL FORM: 4 out of 5 mathematicians surveyed recommended this as the final form for their students who choose to finish. TFAE (The Following Are Equivalent): If I say this it means that, and if I say that it means the other thing, and if I say the other thing... BY A PREVIOUS THEOREM: I don't remember how it goes (come to think of it I'm not really sure we did this at all), but if I stated it right (or at all), then the rest of this follows. TWO LINE PROOF: I'll leave out everything but the conclusion, you can't question 'em if you can't see 'em. BRIEFLY: I'm running out of time, so I'll just write and talk faster. LET'S TALK THROUGH IT: I don't want to write it on the board lest I make a mistake. PROCEED FORMALLY: Manipulate symbols by the rules without any hint of their true meaning (popular in pure math courses). QUANTIFY: I can't find anything wrong with your proof except that it won't work if x is a moon of Jupiter (Popular in applied math courses). PROOF OMITTED: Trust me, It's true.
Is it a ship that saves the star? decrease ・・・・・・・ Do it sleep and do it get it hot for a moment? Now ..coming to the heart with Zun... I am a man. It is likely to do, do it sleep and do it get it?
Teachers went to classes too. In these required classes, they were trained in Nazi principles. Teachers who did not believe in the Nazi principles faced difficult decisions every day about what to teach. They wanted to teach their children the truth, but if they were caught being disloyal, they would be fired. The Nazis didn't need proof to fire someone. If a student accused a teacher of being disloyal, that teacher could be fired. Still, some teachers kept trying to find ways to teach their children the important things that they needed to learn.