Poltergeists?

Years ago, my wife and I both worked in a building that had had a reputation for being haunted, even before the previous owner of the company we worked for had died. My wife was working late, sitting at the desk that used to belong to the now-deceased owner of the company, and her wedding and engagement rings started to feel a bit tight (which they do, occasionally). So she took them off and set them on the desk. (This was a Friday evening.) When she finally got done with what she was working on, she forgot to put her rings back on, and left them behind. When she got to my apartment (at this point in time, she was still living with her dad while we looked for a place to live together) I noticed that the rings were missing, and asked her about it. She realized that she'd left them behind, and said that she'd stop by the office and grab them on her way back to her dad's place. When I called her on Saturday, she said she hadn't been able to find them, so I went to the office to look for them. I turned that desk (figuratively) upside-down looking, and even searched the floor under it. THEY WEREN'T THERE. I would take oath on that. Both she and I looked and looked and looked, (she on Friday night, I on Saturday.) On Monday, I got into the office, and she was at the computer at that desk again, and I came in and sat across from her to chat before I clocked in. And there, right out in the open, not under anything or in any way hard to spot, were the rings,sitting there on the desk. I do not say that I believe in ghosts. I say that I have seen things that I cannot explain any other way. There is a difference.

Ask The Math Guy 3

The Math Guy is Jim Yanni, a 53 year old senior at Webster University looking to start a new career, who is majoring in Math and is a lifelong Math whiz. He is happy to field questions from readers, and is available for personal tutoring sessions for $20 an hour with a money-back guarantee: if he can’t answer your question, there is no charge for the session. (e-mail him at glenohhara@yahoo.com or via the Women’s Journals at info@womensjournals.com) I am most comfortable with questions relating to algebra or pre-algebra, and although I’m willing to consider questions from other branches of Mathematics I would be reluctant to offer tutoring in anything higher level than that. Our question for today is, “How do we know that an irrational number is irrational?” Two issues ago, I explained where the term “irrational” comes from, and why in non-Mathematical usage, it has come to mean “crazy”. To summarize, “rational” means “can be expressed as a ratio” and “irrational” means “can’t be expressed as a ratio.” So if a number can be expressed as a ratio, it can be expressed as a fraction. In mathematical terms, then, we say that if a number is rational, it can be expressed as a/b, with b ≠ zero. So let’s examine a specific irrational number; we can use \/¯2, but this argument would work for any root larger than one of any prime number. We are told that \/¯2 is irrational, and therefore the decimal representation of it never ends or forms a repeating pattern. The question is, “how can we be sure of that, given that it’s infinite? Maybe it starts a repeating pattern after the 2 billionth decimal place; how would we know?” We know because of the definition of a rational number. Suppose, for the sake of argument, that \/¯2 was a rational number. Then it could be expressed as a/b (b ≠ 0). That would give us the equation \/¯2= a/b. Squaring both sides of the equation, we would have a^2/b^2=2. Multiplying both sides of the equation by b^2, we get a^2=2b^2. But if a^2 has a factor of 2 in it, it has an even number of them, because it has twice as many factors of 2 as “a” does. The same is true for b^2, since it has twice as many factors of 2 as “b” does. But if b^2 has an even number of factors of 2, then 2b^2 has an odd number of factors of 2. Therefore, a^2 cannot equal 2b^2, since they have a different prime factorization. But for \/¯2 to be rational, it has to. Therefore, \/¯2 must be irrational.

Ask The Math Guy 2

The Math Guy is Jim Yanni, a 53 year old senior at Webster University looking to start a new career, who is majoring in Math and is a lifelong Math whiz. He is happy to field questions from readers, and is available for personal tutoring sessions for $20 an hour with a money-back guarantee: if he can’t answer your question, there is no charge for the session. (e-mail him at glenohhara@yahoo.com or via the Women’s Journals at info@womensjournals.com) I am most comfortable with questions relating to algebra or pre-algebra, and although I’m willing to consider questions from other branches of Mathematics I would be reluctant to offer tutoring in anything higher level than that. There are many vocabulary words in Math that students are usually just told to memorize, words that either have no equivalent in day-to-day language, or, worse, mean something entirely different in day-to-day language than they do in Math. The Math Guy believes that if you can explain how the word came to have the meaning or meanings that it has, it will be easier to remember. Last time, I explained the origin of the terms rational and irrational as they apply to numbers. This issue’s question is, where does the term “Commutative Property” come from? It sounds like it should have something to do with getting to and from work. For those of you who have forgotten, the Commutative Property is the property that says that 3+5 is the same as 5+3, and 3x5 is the same as 5x3. In other words, it is the property that is in effect if the order in which you perform an operation doesn’t matter. As we saw above, addition and multiplication are Commutative. Subtraction is not (5-3 is not the same as 3-5) and neither is division (6/3 is not the same as 3/6.) The question is, why do we use the term Commutative for this property? The word “commute” in modern usage has the meaning of going to and from work. But the basic meaning of the word is “to change”. If you commute a prison sentence, you change it. Of course, you can only shorten it because legally, there is not the option of lengthening it. But the original meaning of the word is to change. Back in the late 1800s, a great many people started going to and from work by streetcar and subway; they paid for their fares using what we would call weekly or monthly passes. At the time, those were referred to as “commutation passes”, because they were changing a bunch of smaller fares for one larger fare that was a better buy. That’s how the word came to be applied to going to and from work. So in math, “commutative” means “we can change the order”.

Ask The Math Guy 1

The Math Guy is Jim Yanni, a 53 year old senior at Webster University looking to start a new career, who is majoring in Math and is a lifelong Math whiz. He is happy to field questions from readers, and is available for personal tutoring sessions for $20 an hour with a money-back guarantee: if he can’t answer your question, there is no charge for the session. (e-mail him at glenohhara@yahoo.com or via the Women’s Journals at info@womensjournals.com) I am most comfortable with questions relating to algebra or pre-algebra, and although I’m willing to consider questions from other branches of Mathematics I would be reluctant to offer tutoring in anything higher level than that. There are many vocabulary words in Math that usually students are just told to memorize, words that either have no equivalent in day-to-day language, or, worse, mean something entirely different in day-to-day language than they do in Math. The Math Guy believes that if you can explain how the word came to have the meaning or meanings that it has, it will be easier to remember. So today’s question is: what is the deal with “rational” and “irrational” numbers? Numbers can’t be sane or crazy; is it just that irrational numbers are kind of weird and hard to wrap your head around? The answer is: kind of. A “rational” number is simply a number that can be expressed as a ratio of two other numbers: One X for every three Y, or two P for every five Q, for example. Any number that can be expressed as a ratio can also be expressed as a fraction; fractions are simply a variation on the theme of ratios. Irrational numbers cannot be expressed as a ratio. There is no fraction that truly equals the square root of two, for instance. Therefore, it can never be fully expressed in decimal form; it can only be approximated. Granted, it can be approximated well enough to suit any practical need; 1.4142135623 is closer than you’re likely to ever need to know it, but it isn’t really the answer. If you multiply 1.4142135623 by itself, you get 1.999999979325598129, not 2. The ancient Greeks didn’t believe that there was such a thing as an irrational number; they were absolutely convinced that any number in nature could be expressed as a ratio. In fact, they considered it a crazy idea that there could be such a number. So that’s how the words “rational” and “irrational” came to mean “sensible” and “nonsensical”; the concept of an irrational number was considered a nonsensical idea, and eventually the concept spread to any nonsensical idea, and by extension, a rational idea was a sensible idea. So yes, after a fashion, it is true that “irrational” means what it does in Mathematics because an irrational number is a weird concept and hard to wrap your head around. But the Mathematical definition came first; the definition of “nonsensical” evolved from that, not the other way around.

The Death of Balder

Loki's downfall was that he could not resist a challenge. When he cut off Sif's golden hair, and was forced to bargain with the dwarves for magical, dwarf-forged golden hair to replace it, it was not because he begrudged Sif her beauty and wished to mar it; it was because Thor, deep in his cups, had rashly boasted to Loki of the impregnability of the defenses of Sif's quarters.
Thus, when Balder was troubled by nightmares and Odin foresaw that Balder's death was fated to be soon and Freya obtained the promise of all things in the Nine Worlds not to harm him, Loki found the challenge irresistable. He would stand and watch the gods reassuring themselves of Balder's invulnerability by attacking him with every manner of weapon, and ponder the problem.
"Surely," he thought," Freya could not have gained the promise of EVERY substance. There are too many, and most of them are blatantly harmless. The trick, then, is to find some way to hurt him with something apparently harmless."
He thought of making a whip or a rope out of grass, but it occurred to him that Freya might have been clever enough to think of that. He didn't want to try something that might work and have it fail; it would blunt the effect of his prank if what finally drew blood was his fourth or fifth attempt. So he disguised himself as an old woman and went and spoke with Freya. Through his guile he tricked her into revealing that she had not bothered to obtain the promise of the mistletoe bush. The details of that conversation are recorded elsewhere, but that's a different tale entirely. As he hurried away, Loki thought to himself, "This is even better than I'd hoped. It will be far easier to make an arrow from that bush than to weave a whip from grass."
So he made his arrow, and brought it back to where the gods were sporting with Balder. It wasn't a particularly fine arrow; the mistletoe bush was truly not a particularly likely source of weaponry, but if it had been, Freya would not have ignored it. The arrow was on the small side, and rather flimsy, almost a dart, but it was strait and sharp. Loki found a short bow that he could fit the arrow to, and made ready to fire at Balder's shoulder. Then he hesitated. He had no intention of actually killing or even seriously injuring Balder; he planned to wing him, just to draw blood. Still, it occurred to him that as fearful as the gods were for Balder's safety, they might respond badly to seeing him blooded; indeed, they might do violence upon the one who had bloodied him, especially if that one were Loki. Most of the gods had reason to be short-tempered with Loki, for he had played almost all of them for fools at one time or another, and gods even more than men have little tolerance for that.
Then he spotted blind Hod.
He doubled up in silent laughter. As usual, Hod was taking no part in the game. Loki sauntered up to him and poked him in the ribs.
"That can only be Loki," said Hod.
"None other," replied Loki.
"Well?" said Hod.
"Why aren't you joining in? Why don't you throw things at your brother?" asked Loki.
"Because I can't see where he is," replied Hod, reasonably enough. "Besides, I have no weapon."
"Here," said Loki, "take the arrow that I was about to use. I'll stand behind you and guide your hand."
So Loki gave Hod the bow and arrow, and helped him aim at Balder's right shoulder.
"There now," said Loki, satisfied. "Release it just so and you'll hit him safely in the shoulder."
"That's silly," said Hod, "Why the shoulder? If I could hurt him at all, I wouldn't be doing this. Aim me centered, so I'll be sure not to miss."
Loki, realizing his mistake instantly and thinking quickly on his feet, hardly paused a moment. "Certainly, Hod," he said glibly as he adjusted Hod's aim for Balder's left shoulder. "There," he lied smoothly, "Now you're pointed right at his heart."
Hod, however, had begun to wonder. Loki's reputation as a trickster preceded him, and he might be up to almost anything. Then again, he might be doing just what he claimed--helping Hod to take part in a group activity for a change, when all the other gods had been ignoring him again. And what harm could he be up to? Balder WAS invulnerable. Hod didn't want to pass up his one chance at participating on a mere unfounded suspicion, nor did he wish to repay Loki's kindness with accusations. It did occur to him, though, that had Loki's intentions been less than honorable, Loki had adroitly manuvered him into being the one who had chosen to aim for the heart, taking Loki smoothly off the hook. Therefore, just in case his suspicions were justified when he let loose the arrow Hod actually aimed halfway between the two points Loki had given him. "This way," he thought, "I should still have enough margin for error not to miss and look foolish, and even if I can hurt him, it should just be a flesh wound-painful, but not dangerous."
The shot was a beauty, strait and powerful. It struck midway between Balder's two shoulders, at the base of his throat, and passed right through him. He gurgled once and was dead.
In roaring silence all the gods in the hall turned to stare at Hod and Loki. None could speak. Loki, as stunned as any of them, read in their eyes that they placed full blame for what had happened on his shoulders. None would have believed that he had meant no harm. He sensed that were he to try to speak it would only break the spell that held the hall in stasis, and he would be torn limb from limb. Carefully, slowly, he edged to the door, then turned and fled. He was gone before any could think to follow.

Thought for the Day: What the Problem is with Modern American Education

I had a thought yesterday; in my Educational Psychology text, we were covering the four styles of parenting (Diana Baumrind): Authoritarian parenting (in which parents provide structure but little affection and too little opportunity for children to develop their independant ability to provide their own structure), authoritative parenting, (in which parents provide both a reasonable level of structure and plenty of affection, but still allow kids to learn to be self-motivated), indulgent parenting, (in which parents provide plenty of affection but insufficient structure) and neglectful parenting, (in which parents don't provide enough of either structure or affection.) Of course, authoritarian parenting is held up as the ideal to strive for, which is certainly reasonable. But one thing that is NOT addressed is, if one must err, should one err in the direction of authoritarian or indulgent? After all, these styles are not simply isolated points; they exist on a continuum, and (ignoring neglectful, which has nothing positive to be said about it) one could be in the authoritarian range, but tending more towards authoritative, or one could tend more toward indulgent. Which is the less harmful way to err?It also occurred to me that those styles could apply to teaching as well as parenting. Decades ago, American schools were unquestionably authoritarian; nowadays, we have not only recognized the value of the authoritative style, but we are so concerned with NOT being authoritarian that we err consistently on the side of indulgent. The problem with this is twofold: first, this is probably a large part of the reason why so many kids never learn the basic skills in math, reading, and writing that they need; we're afraid to hurt their feelings by telling them that they have failed, so instead we let them slide by with incomplete skills; second, it is always possible, if one starts out by erring on the side of authoritarianism, to ease up a bit, gradually, until one reaches a good balance. But if one starts out erring on the side of indulgent and finds that one's class is out of control, it is almost impossible to reassert control; certainly, it cannot be done gradually, but would need to be done by harshly and punitively overshooting the desired level of control, and then easing back up again. That process would be extremely difficult and painful for all concerned; certainly, it would be more painless (and therefore more appropriate) to choose the first method.Also, since we definitely err in the direction of indulgent with young children these days (after all, they can usually be dealt with fairly easily without too much harshness, and we are definitely afraid of dampening their spirits by too much structure) as an educational system, we are guilty of using the more painful second mentioned system: we start kids out by treating them indulgently, and then, when they get older and harder to manage, we try to tighten the reins on them. But when a teenager has been given a certain amount of freedom and autonomy as an elementary school student, he or she is going to resent being given less as an older student, and will resist and fight authority. This leads to middle and high schools that are out of control and downright dangerous, in large part because we didn't establish a solid basis of authority and regimentation when the kids were young and more easily taught control.This is not to say that we need to return to the days of totally authoritarian teaching; the authoritative style is certainly preferable. But we need more authoritarianism and less indulgence in our schools than we now have; what we have now is leading to kids who are, indeed, used to thinking for themselves and being autonomous, but who have no idea of how to structure their lives, no idea of how to accept outside structuring, no ability to stick to unpleasant tasks that need to be done, and no idea of why they SHOULD do so. This is doing no one any favors.

Lyrics to "The Ostrich", by Steppenwolf:

We'll call you when you're six years old
And drag you to the factory
To train your brain for eighteen years
With promise of security
But then you're free
And forty years you waste to chase the dollar sign
So you may die in Florida
At the pleasant age of sixty nine

The water's getting hard to drink
We've mangled up the country side
The air will choke you when you breathe
We're all committing suicide
But it's alright
It's progress folks keep pushin' till your body rots
Will strip the earth of all it's green
And then divide her into parking lots

But there's nothing you and I can do
You and I are only two
What's right and wrong is hard to say
Forget about it for today
We'll stick our heads into the sand
Just pretend that all is grand
Then hope that everything turns out ok

You're free to speak your mind my friend
As long as you agree with me
Don't criticize the father land
Or those who shape your destiny
'Cause if you do
You'll lose your job your mind and all the friends you knew
We'll send out all our boys in blue
They'll find a way to silence you

But there's nothing you and I can do
You and I are only two
What's right and wrong is hard to say
Forget about it for today
We'll stick our heads into the sand
Just pretend that all is grand
Then hope that everything turns out ok.