Iamma mathemetican

[Iamme]

No I'm not. I'm lying. I just wanted you to read this.:D

Here is a neat trick I just learned from a caller who called this in to the syndicated radio show called "On Your House", by the Carey Brothers (a home repair-type show) :

To change a decimal to *any* desired fraction, this is all you have to do; multiply the decimal by the bottom number in your fraction. Suppose I want to change .289 to 64ths. You multiply .289 X 64. Then you take that number and put *it* over 64. That's it.

If you want to convert to 32nds, 16ths 8ths, 5ths, 3rds...whatever...you do the same math as I said.

Let's do a fraction you can readily tell the answer to: Suppose the decimal is .5 and you want to know how many 1/2's that is? Easy eh? The answer is 1/2. Let's do the caller's math, to test him out. You take .5 (the decimal) and multiply it by the lower part of the fraction (2). The answer = 1. You then put the 1 over the 2. There you go!

[Donks]

Uhmmm... wow.

[Bruce]

..................... I think you need to remove one "m" from your name.

[kalen]

.289 x
------ = ---
1 64





Very mysterious.

[Kaylee]

Iamme, there's an excellant chance you are joking but you did just remind me why I hated school (a long, long time ago). It was all about learning the right answers but never about why the answers were right. :mad: What. A. Waste. Of. Time.

So just in case someone here has rusty math, I'm going to treat your post as if you had posted seriously.

* The reason the trick works is because you are multiplying the first number (the one expressed in decimals) by 1.

*The number 1 can be expressed in many different ways. Using your examples it can be expressed as both 64/64 and 2/2. 64/64 is just another way of saying, how many groups of 64 items each are within the total quantity of 64? And the answer is, just 1. (If the question had been 128/64 or how many groups of 64 items each are within the total quantity of 128, the answer would have been 2.)

* Any number multiplied by 1 equals itself. Because multiplying is just another way of saying how many groups of the first number (quantity) you want. Since we want don't want to actually change the original amount we multiply by 1. But since we do want to change how the number will look, we use a version of 1 that looks like the new way we want our number to appear.

In your last example you said you wanted to take the decimal .5 and change it to a fraction showing how many 1/2s it was equivalent to. So we do this:

.5 * 2/2 = 1/2

BTW, we know that decimals are just a faster way to write fractions of 10. We use fractions of 10 so often, that’s why we created the short cut method of writing it -- the decimal system.

So we could have multiplied the numbers this way also, it would have just been another way of multiplying the same thing:

5/10 * 2/2 = 10/20

10/20 can be reduced to 1/2, because its really the same value.

In your first example you wanted to take the decimal .289 and change it to a fraction of 64.


.289 * 64/64 = 18.496/64 or
2.89/10 *64/64 = 184.96/640

184.96/640 can be reduced to 18.496 / 64


PS to the rest of the forum -- I haven't read all of Iamme' posts, but I'm willing to bet he's quite young. Don't forget, some schools have some truly atrocious teachers, especially for the areas of arithmetic, science and grammar {Sigh} I still remember one of my teachers (2nd grade?) saying how scary fractions were … it never did get much better…

[Iamme]

Bruce----I have been called "lame" on this board more than I care to know.:D

Shera---I was one of the stupid ones in school that just so happened to luck out in math with A's all the way through pre-calc. Good explanation on your part. But I have been out of school for over 30 years and frankly I , like the Carey brothers, who are no idiots, can ever remember how you can so easily convert a decimal to not just a fraction...but to *any* desired fraction.

And now that I have you on the line...can you recall how that crazy money transaction goes that is called "The case of the missing dollar"? That's always a hoot in trying to bring high I.Q. wannabes back down the ladder a notch, when they hear that story and try to figure out where the dollar went when the 3 guys show up at a ski resort (or whatever) and when the rooming fee is paid, by the 3, there finally is this dollar that 'disappears' in the transaction.

[LostAngeles]

OMG! Holy crap! It's the actual method used to convert a decimal into a fraction that I learned in elementary school! Good God man! You are a genius!

Now prove that a nilpotent matrix does not have an inverse, because I think I just blew that on my Linear Algebra midterm an hour ago.

[Cabbage]

The missing dollar:

Three guests decide to stay the night at a lodge whose rate they are initially told is $30 per night. However, after the guests have each paid $10 and gone to their room, the proprietor discovers that the correct rate should actually be $25. As a result, he gives the bellboy the $5 that was overpaid, together with instructions to return it to the guests. Upon consideration of the fact that $5 will be problematic to split three ways, the bellboy decides to pocket $2 and return $1 each, or a total of $3, to the guests. Upon doing so, the guests have now each paid a total of $9 for the room, for a total of $27, and the bellboy has retained $2. So where has the remaining $1 from the initial $30 paid by the guests gone?!

(Yes, I know the answer, of course; I'm just posting the problem since it was requested).

Now prove that a nilpotent matrix does not have an inverse, because I think I just blew that on my Linear Algebra midterm an hour ago.

Suppose A is a nilpotent matrix, i.e., there is a positive integer n such that Aⁿ=0 (the zero matrix).

Proof by contradiction: Suppose A does have an inverse, denoted A^-1. Then:

0 = AⁿA^-n = I (the identity matrix), a contradiction.

[phildonnia]

Well, then, it should be a simple matter to convert 0.99999.... to a fraction, say, fifths.

(Sorry, I couldn't resist).

[Ladewig]

Originally posted by Iamme
Here is a neat trick I just learned from a caller who ....

Why do you think it is some sort of trick or shortcut? That is the formal method of converting decimals.

[Kaylee]

LostAngeles -- You must go to a really cruel school that gives midterms on weekends, esp. in the spring!

Anyway Phildonnia seems to be doing a great job with the practical jokes. :D

And Cabbage seems to be doing an excellent job handling the genius stuff so I will leave that to him/her (? -- why doesn't JREF have gender icons? I'm a she by the way, not a man as someone typed in this thread..). This way I won't have to admit that I only took one semester of linear algebra eons ago and don't remember WTH you are talking about … oops!

I’ll just stick to the elementary school math for now. ;)

Ah but I think I can handle the missing dollar puzzle.


It's a trick question from Planet X!!! Don't go there! But if you must ---

The guests paid a total of $27.00 (9 * 3)
The hotel manager and busboy took in a total of $27.00 ( $25.00 + $2.00)

Cash In == Cash Out, no problema.

Iamme, glad you brought that puzzle up and that Cabbage remembered it (as a genius should). And hey if you like the Carey Brothers approach -- whatever works! :)

[Dr Adequate]

Originally posted by Cabbage
Suppose A is a nilpotent matrix, i.e., there is a positive integer n such that Aⁿ=0 (the zero matrix).

Proof by contradiction: Suppose A does have an inverse, denoted A^-1. Then:

0 = AⁿA^-n = I (the identity matrix), a contradiction. How about:

Suppose A is a nilpotent matrix, Suppose A is a nilpotent matrix, i.e., there is a positive integer n such that Aⁿ=0 (the zero matrix), so det(Aⁿ)=0 so det(A)ⁿ = 0 so det(A) = 0.

So A has no inverse.

Now, back to the OP.

* sighs, bangs head on desk *

.289 * 64/64 = 18.496/64

But this is not a fraction.

A fraction is a quotient of whole numbers. You still have a decimal there, as in 18.496.

To convert this to an actual fraction, it's enough to remember that .289 = 289/1000. There you go.

It is also possible to convert infinitely repeating fractions. This is more interesting. Suppose our number --- call it x --- is .289289289289...

Then 1000x - x = 289

999x = 289

x = 289/999

The task of proving that every fraction has a terminating or repeating decimal is left as an exercise for the reader.

[Art Vandelay]

Originally posted by Iamme
To change a decimal to *any* desired fraction, this is all you have to do; multiply the decimal by the bottom number in your fraction. Suppose I want to change .289 to 64ths. You multiply .289 X 64. Then you take that number and put *it* over 64. That's it. Also, if you want to know how many times a number goes into another, you should divide.

:confused:

You didn't know this? The number of halves in .5 is equal to .5 divided by1/2. Dividing by one half is the same as multiplying by 2.

LostAngelesNow prove that a nilpotent matrix does not have an inverse, because I think I just blew that on my Linear Algebra midterm an hour ago.Step One: define "nilpotent" to mean "has no inverse".
Step Two: done.

Are there any singular matrices that aren't nilpotent? Is the determinant defined for all linear operators?

Dr Adequate
289 * 64/64 = 18.496/64

But this is not a fraction.

A fraction is a quotient of whole numbers. You still have a decimal there, as in 18.496.So pi/2 isn't a fraction? Is f(x)=x/2 sometimes a fraction and sometimes not?

dictionary.com
fraction:
Mathematics. An expression that indicates the quotient of two quantities, such as 1/3.

It is also possible to convert infinitely repeating fractions. This is more interesting. Suppose our number --- call it x --- is .289289289289...

Then 1000x - x = 289

999x = 289

x = 289/999
Except that you have to define what ".289289289289..." means before you can do any algebra on it. It seems to me that it is simplified if we simply define it to be equal to 289/999, and then we don't have to deal with all this "Does .99999...=1" stuff, because it is simply defined to be true. Most of the reason why there's such a controversy is that people try to have a discussion about what properties is has without bothering to properly define it beforehand. Arguing about whether .99999...=1 without defining .99999... is like arguing about whether nohigles are kolaq. And no, saying "it's a decimal point followed by an infinite number of nines" is not a valid definition. That's just meaningless.

[Kaylee]

Originally posted by Dr Adequate
289 * 64/64 = 18.496/64

But this is not a fraction.

A fraction is a quotient of whole numbers. You still have a decimal there, as in 18.496.

Originally posted by Art Vandelay
So pi/2 isn't a fraction? Is f(x)=x/2 sometimes a fraction and sometimes not?

dictionary.com
fraction:
Mathematics. An expression that indicates the quotient of two quantities, such as 1/3.


Succinctly put Art. :) Plus I'd just like to add (even though I'm definintely not a math maven) that sometimes one needs to convert numbers to a common denominator so a larger problem can be solved.

ETA: the bad pun just slipped out, I swear.

[Dr Adequate]

Originally posted by Art Vandelay
Step One: define "nilpotent" to mean "has no inverse".
Step Two: done.

Are there any singular matrices that aren't nilpotent?How about

00
01

?

This is idempotent (is equal to itself squared).

(Exercise for the reader: prove that the only idempotent matrix with an inverse is the identity matrix.)So pi/2 isn't a fraction? Is f(x)=x/2 sometimes a fraction and sometimes not?
Pace dictionary.com, yes, I should understand a fraction to mean a quotient of whole numbers: a rational number. (Imagine if someone set you a math puzzle to produce "a fraction" having such-and-such properties, and after an hour or so they revealed that the answer was the square root of two divided by 1.) No pi/2 is not a fraction, and f(x) = x/2 is always a fraction if it's from the integers to the rationals and never a fraction if it's from the reals to the reals, and... well, you get the picture.Except that you have to define what ".289289289289..." means before you can do any algebra on it. On the contrary, saying how to do the algebra defines what it "means". It seems to me that it is simplified if we simply define it to be equal to 289/999 Unfortunately, the only reason that I was able to prove the equivalence was by accepting .289289289... as a number I could perform algebra on.and then we don't have to deal with all this "Does .99999...=1" stuff, because it is simply defined to be true. That is certainly how the decimal system is defined, but it doesn't follow that we have to define every decimal-to-fraction conversion. For one thing, the list of defintions would then become infinitely long. Most of the reason why there's such a controversy is that people try to have a discussion about what properties is has without bothering to properly define it beforehand. Arguing about whether .99999...=1 without defining .99999... is like arguing about whether nohigles are kolaq. Quite so. It "means" whatever it's defined to "mean".

[H3LL]

Shame on many posters here.

Many posts on this board whine about how science and other subjects are not being taught as they should.

Carl Sagan opens his book, "A Demon Haunted World" with how he became fascinated by the wonder of science, in his case Astronomy.

Here we have a person that has demonstrated his wonder and joy at finding out something new in a rational subject.

And many belittle him for doing so

Shame, shame, shame.

It is tiny events such as these that we should grasp and encourage. Tiny events like these that then lead people onto discover great things. It is events like these that gave us Carl Sagan and others.

Iamme, or someone like him could go on to discover a new theory or law.

And you MOCK!?!?!

Pah!

:(

[CurtC]

Originally posted by phildonnia
Well, then, it should be a simple matter to convert 0.99999.... to a fraction, say, fifths.That's easy - it's 4.999999.../5.

Iamme - here is a cool shortcut for you. Let's say you want to multiply two numbers that have a difference of two, like, say, 19 and 21. You can take the number in the middle (20), square it, and subtract one, so 19*21=399. Now, what's 24 times 26, quick? And 17 times 15?

The elegant answer to Cabbage's question above (spoiler alert), which I came up with myself, is that it was asked wrong - you took $27 and added $2 to attempt to get $30, but what you should have done was to take $27 and sutract $2 to get the $25 net that each paid. My uncle asked me this question about 35 years ago, and it bothered me for several more before I figured it out.

[PogoPedant]

Originally posted by H3LL
Shame on many posters here.

Many posts on this board whine about how science and other subjects are not being taught as they should.

Carl Sagan opens his book, "A Demon Haunted World" with how he became fascinated by the wonder of science, in his case Astronomy.

Here we have a person that has demonstrated his wonder and joy at finding out something new in a rational subject.

And many belittle him for doing so

Shame, shame, shame.

It is tiny events such as these that we should grasp and encourage. Tiny events like these that then lead people onto discover great things. It is events like these that gave us Carl Sagan and others.

Iamme, or someone like him could go on to discover a new theory or law.

And you MOCK!?!?!

Pah!

:(
Just thought this needed to be repeated.

Another neat trick:

To see if a (whole) number is divisible by three, simply check to see if the sum of it's digits is divisible by three.

Example: 6187621388721 is divisible by three:
6 + 1 + 8 + 7 + 6 + 2 + 1 + 3 + 8 + 8 + 7 + 2 + 1 = 60, and sixty is divisible by three.

On the other hand 6187621388722 is not divisible by three:
6 + 1 + 8 + 7 + 6 + 2 + 1 + 3 + 8 + 8 + 7 + 2 + 2 = 61,
6 + 1 = 7, and seven is not divisible by three.

The same trick works if you want to find out if a number is divisible by nine.

To see if something is divisible by eleven, there's a slightly more complicated trick, but still mental arithmetic level. Alternatly add and subtract each digit. If you end up with the sum zero or a number divisible by eleven, then the original number is divisible by eleven.

Example: 6187621388718 is divisible by eleven:
6 - 1 + 8 - 7 + 6 - 2 + 1 - 3 + 8 - 8 + 7 - 1 + 8 = 22,
2 - 2 = 0.

While 6187621388723 is not visible by eleven:
6 - 1 + 8 - 7 + 6 - 2 + 1 - 3 + 8 - 8 + 7 - 2 + 3 = 16,
1 - 6 = -5, not divisible by eleven and not zero.

Any errors in the above post is fully attributable to the Demon known as NoCoffeeYet

[LW]

Originally posted by PogoPedant

While 6187621388723 is not visible by eleven:
6 - 1 + 8 - 7 + 6 - 2 + 1 - 3 + 8 - 8 + 7 - 2 + 3 = 16,
1 - 6 = -5, not divisible by eleven and not zero.

Any errors in the above post is fully attributable to the Demon known as NoCoffeeYet

Hey, you are right. I didn't see it before 11:30 AM.

But continuing with tricks with numbers, there is the old favorite where you sum a multiple digit number with its reverse, as in:

3654 + 4563 = 8217

No matter what numbers you chose, the end result is divisible by 9. So, if someone does that and them wipes out one digit from the answer, you can always find out what the digit was by looking what number should be added to the sum of the remaining digits to make it divisible by 9.

Suppose that someone had wiped out the '1' in the above example. The remaining digits sum to 8 + 2 + 7 = 17. The least multiple of 9 greater than that is 18, and 18-17 = 1.

[Kaylee]

Thanks to all who posted the "tricks." Playing around with them made me look at our numeric system with a
new appreciation. (http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_numerals.html) Who knew that something we all use everyday without a thought (well I never thought about it anyway :) ) was so slick?

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.

Every time someone shows me a game or puzzle the first thing I try to do is break it. I liked this one:
there is the old favorite where you sum a multiple digit number with its reverse, as in:

3654 + 4563 = 8217

No matter what numbers you chose, the end result is divisible by 9. So, if someone does that and them wipes out one digit from the answer, you can always find out what the digit was by looking what number should be added to the sum of the remaining digits to make it divisible by 9.

But found that the individual numbers (of the numbers added to get the sum) must sum up to a number divisible by 9 for it to work. So for example I got it to work for 423 but not for 8,976.
4+2+3 = 9
8+9+7+6 = 30 ( not divisible by 9.)

423 +324 = 747 (which equals 9 * 83)
8976 + 6798 = 15,774 (which is not divisible evenly by 9)

Edited to fix link.

[Dr Adequate]

there is the old favorite where you sum a multiple digit number with its reverse, as in:

3654 + 4563 = 8217

No matter what numbers you chose, the end result is divisible by 9. So, if someone does that and them wipes out one digit from the answer, you can always find out what the digit was by looking what number should be added to the sum of the remaining digits to make it divisible by 9. You're right, this doesn't work. On the other hand, if the instructions were to reverse the number and subtract the smaller of the two from the larger, then that works. I think the trick's got garbled in the telling.

[LostAngeles]

Originally posted by Art Vandelay
LostAngelesStep One: define "nilpotent" to mean "has no inverse".
Step Two: done.


BWHAHAHAHAHAHAHAHA!!!!!!!!!!!!!:roll:

OK, *wipes tears from eyes* I did apparently do that right, though I didn't take Dr. A's approach with the determinant and I should have. I like that a lot.

Shera, seriously. My school offered Linear Algebra on Saturdays and on Tuesday nights this semester.

That's it.

It's because Adminsitration Hates Us. Not the people who work in Adminsitration, Adminsitration itself, as in, the entity known as Adminisitration.

H3LL's right. We shouldn't have mocked lamme. I'm sorry.

[Kaylee]

Originally posted by Dr Adequate
You're right, this doesn't work. On the other hand, if the instructions were to reverse the number and subtract the smaller of the two from the larger, then that works. I think the trick's got garbled in the telling.
Yes you are right. And for no better reason than the mundane fact that the difference between 10 and 1 is 9 and the difference between 110 and 11 is 99. But the way LW presented it, it does not sound mundane at all. The same is true for all the other tricks as well.

Presentation IS everything. Even in arithmetic.

[CurtC]

Yeah, that trick works because if you take a general four-digit number, abcd, that is really:

1000a + 100b + 10c + d

Now rearrange the digits, for example:
1000c + 100a + 10d + b

And subtract to give this:
900a + 99b - 990c - 9d

This last number is obviously divisible by 9, but it's not obvious upon hearing the problem. I've also seen one of those "magic" web pages using this principle as the key.

[LostAngeles]

Originally posted by Dr Adequate

(Exercise for the reader: prove that the only idempotent matrix with an inverse is the identity matrix.)

Seems straightforward. Proof by contradiction?

A² = A

Suppose A^-1 exists and A != I.

A² = A
AA = A
A^-1(AA) = A^-1(A)
(A^-1A)A = (A^-1A)
(I)A = I
A = I because inverses are unique.

[Iamme]

(Ladewig)

Why do you think it is some sort of trick or shortcut? That is the formal method of converting decimals.

----------------------------------------------------

Gee...everyone here is making this out to be common knowledge. Hmmm. Well, I just couldn't remember learning this. Maybe I did..but after all these years, I am lucky if I still remember my name. The talk show hosts are are quite sharp when it comes to diagnosing and fixing everything, and they weren't familiar with this 'trick' either.

Oh well. I still remember a few things though.

I have found it very usefull to know the volume of a circle and a cylinder, for figuring out the amount of liquid in a given length of pipe, for example. I also remember how to figure how much water pressure is exerted down on a column (water column, relative to psi).

See if you can figure out the exact psi. If the water height in a column is 16 feet, what will the psi be at the bottom. Hint: it's *about* 1 psi for every 2+ feet of water height.] Do you know that the principle of water column is what keeps water from leaking past the toilet flaper in a toilet tank?

Or, the neat trick where you can tell how tall a tower is if you have a few other mneasurements and can figure angles...you don't have to drop a tape measure down from the top to know how tall the tower is (or tree, building, etc.). Or, the necessary math to know how to lay out stairs on a sloped slab or ground so that you can divide up the rise of each stair and have it come out perfect, at the base of the first step. (i.e., it is 8 feet from slab to landing directly below the landing, but the ground pitches downhill from that point by an inch per foot, let's say. You can figure everything out mathematically so that, let's say, each stair rise is 6.81 inches, rather than having every stair be an even 7 inches (standard), while the bottom step is 10 inches or something.

[Iamme]

If any of you posters feel like you are math wizzes and would liked to be challenged to see if anyone can knock you off your pedestal, I suggest you subscribe to Discover magazine. In the back of each magazine are geometric, math and other puzzles that are mind boggling. I am actually more curious about the individual who is able to think them up rather than knowing the solution. Sort of like the Rubic's Cube. You'd be smart to be able to actually figure out the cube. But consider the guy who came up with it in the first place!

[Iamme]

Thanks Cabbage for the missing dollar puzzle. There are variation sof the same theme but the jist of it is the same.

(CurtC)

Iamme - here is a cool shortcut for you. Let's say you want to multiply two numbers that have a difference of two, like, say, 19 and 21. You can take the number in the middle (20), square it, and subtract one, so 19*21=399. Now, what's 24 times 26, quick? And 17 times 15?

(Iamme)

624; 255. Neat trick. You outta tell the Carey Bros. Maybe they will send you something.

(CurtC)

The elegant answer to Cabbage's question above (spoiler alert), which I came up with myself, is that it was asked wrong - you took $27 and added $2 to attempt to get $30, but what you should have done was to take $27 and sutract $2 to get the $25 net that each paid. My uncle asked me this question about 35 years ago, and it bothered me for several more before I figured it out.

(Iamme)

The story teller can make or break this 'puzzler'. You are right though. It indeed is a case of saying what was done, with twisted logic.

[Alkatran]

Why the subtract 1 rule works (I figured that one out in grade 10 through inference I think...)
(x+y)*(x-y) = x^2 - y^2
So if x is 16 and y is 1:
(16-1)*(16+1) = 15*17 = 16^2 - 1^2 = 256 - 1 = 255



Want an even more obvious one?

To multiply any number by 9, multiply it by 10 and subtract the number.
9x = 10x - x

This actually makes mental calculations much faster because multiplying by 10 is so easy.
245*9 = 2450-245 = 2205


In grade 12 math, when the teacher was slacking off (aka, all year) we would sometimes see who could multiply two digit numbers the quickest.
After a few games the rules started getting slightly complicated for picking the number. No multiples of 5, 11, 9, no numbers too close together (in which case the x^2 - y^2 rule worked quickly), etc...

[Art Vandelay]

Originally posted by Dr Adequate
Pace dictionary.com, yes, I should understand a fraction to mean a quotient of whole numbers: a rational number. That's an odd definition of fraction. According to this definition, it would be invalid to say that "a large fraction of US oil comes from Saudi Arabia". It would also be invalid to use the term "fraction" for any abstract space. If you have two numbers separated by a fraction bar, what is it if not a fraction? To me, "fraction" is simply a synonym for "quotient". If I want to say that something is rational, I'll say it's rational.

(Imagine if someone set you a math puzzle to produce "a fraction" having such-and-such properties, and after an hour or so they revealed that the answer was the square root of two divided by 1.)Depends on the context. If you sent it, knowing your definition of "fraction", I'd be annoyed. But in general, I would interpret that as a request for a particular fraction, so if they want root two over one, theywouldn't take two over root two (it's the same number, but not the same fraction).

On the contrary, saying how to do the algebra defines what it "means". I suppose you could look at that way, but it doesn't invalidate what I said. You would still have to say how to do the algebra before you actually do the algebra, which means that you would have to define what it means before you use it. Plus it would be rather trivial; your "proof" would then consist of merely stating the rules ex nihilo, then immediately using them. Sort of like my response to Lost Angleles, except I was kidding.

That is certainly how the decimal system is defined, but it doesn't follow that we have to define every decimal-to-fraction conversion. Decimal to fraction conversion is already defined; the problem is that ".289289289289..." is not a decimal, so the standard rules do not apply.

For one thing, the list of defintions would then become infinitely long.
One can define what putting a bar over a decimal means, and that would cover all well defined repeating decimals. Any other decimal would not be well defined, so no attempt at definition should be made.

BTW, did you mean to say that idempotent matrices are only ones which are equal to their square? (Another word for such matrices is "projective". Once you multiply a vector by a projective matrix, it gets "projected" onto the column space of the matrix, and further multiplications by the matrix will not change it; it's now an eigenvector. So everything in the column space is an eigenvector, which means that each column is an eigenvector. All of them have eigenvalue 1, so if they're independent, the matrix is degenerate.)

Here's another trick: say you want to square a number halfway between two integers, say 7.5. Round it up (8), round it down (7), then multiply the two together (56), then add .25 (56.25).

[LW]

Originally posted by Dr Adequate
You're right, this doesn't work. On the other hand, if the instructions were to reverse the number and subtract the smaller of the two from the larger, then that works. I think the trick's got garbled in the telling.

Doh. I have an excuse, I wasn't awake yesterday (had 2 hour sleep).

[LW]

Speaking of cute tricks, 10 years ago I teached myself to find integral cube roots for numbers smaller than 1,000,000. Unfortunately, I didn't have an use for it and forgot the details so I can't do it anymore. But putting "cube root trick" into google resulted in this page (http://mathforum.org/library/drmath/view/65174.html).

[Kaylee]

Originally posted by Iamme
I have found it very usefull to know the volume of a circle and a cylinder, for figuring out the amount of liquid in a given length of pipe, for example. I also remember how to figure how much water pressure is exerted down on a column (water column, relative to psi).

See if you can figure out the exact psi. If the water height in a column is 16 feet, what will the psi be at the bottom. Hint: it's *about* 1 psi for every 2+ feet of water height.] Do you know that the principle of water column is what keeps water from leaking past the toilet flaper in a toilet tank?

I don't recall what psi in mathematics means, and I'm having an unsuccessful day with the search engines. I don't think we are talking about anomalous cognition here.... ;)

Thanks!

[drkitten]

Originally posted by Shera
I don't recall what psi in mathematics means, and I'm having an unsuccessful day with the search engines. I don't think we are talking about anomalous cognition here.... ;)


Pounds per Square Inch.

For those people who measure the water height in feet.

[Kaylee]

Originally posted by new drkitten
Pounds per Square Inch.

For those people who measure the water height in feet.

Thanks! :)

[69dodge]

Originally posted by LostAngeles
Seems straightforward. Proof by contradiction?

A² = A

Suppose A^-1 exists and A != I.

A² = A
AA = A
A^-1(AA) = A^-1(A)
(A^-1A)A = (A^-1A)
(I)A = I
A = I because inverses are unique.Yes, that's right.

No reason to call it a proof by contradiction, though, because the proof never uses the unnecessary assumption that A isn't equal to I. Rather, you've simply proven directly that A is equal to I.

Also (I'm such a nitpicker...), the step to the final line from the previous one doesn't rely on the uniqueness of inverses (BA = I and CA = I imply B = C), but on the definition of the identity (IA = A).

[LostAngeles]

Originally posted by 69dodge
Yes, that's right.

No reason to call it a proof by contradiction, though, because the proof never uses the unnecessary assumption that A isn't equal to I. Rather, you've simply proven directly that A is equal to I.

Also (I'm such a nitpicker...), the step to the final line from the previous one doesn't rely on the uniqueness of inverses (BA = I and CA = I imply B = C), but on the definition of the identity (IA = A).

No, go ahead and nitpick. The more clear and correct I can make my proofs, the better. :) I appreciated it.

[Iamme]

Originally posted by LW
Speaking of cute tricks, 10 years ago I teached myself to find integral cube roots for numbers smaller than 1,000,000. Unfortunately, I didn't have an use for it and forgot the details so I can't do it anymore. But putting "cube root trick" into google resulted in this page (http://mathforum.org/library/drmath/view/65174.html).

------------------------------------------

Likewise, 20-23 years ago an angel told me how gravity worked, but I forgot what he said. I figured he must have been an angel as he pulled up next to me while I was working on the water system on my island. I knew everyone there. I never saw this blonde haired blue eyed guy before who rolled down his car window next to me, and we somehow (can't remember that either) got into a religious debate. During the debate he pointed out how man doesn't know anything even though he thinks he has all the answers...and at that point he told me how gravity could work. Honestly. I'm not making this up.

Just like how I wasn't making up the post (some other thread) where I said I had read chapter 12 in the Bible's Old Testament..the Book of Daniel, and the next day, in the only cirrus clouds in the sky, straight out in front of me down the highway were the Roman numerals XII printed by the cloud formations! You might think I'm being silly, on drugs, or whatever, but I'm not making this stuff up. It must be God. The other day I needed money. I went to the mailbox and got this huge refund from the IRS even though I really think I owed, and the agent told me that I should accept it and not question it!!!!!!!

[CurtC]

My cute trick is that I can figure the day of the week, for any date from 1900 to 2020 or so, in my head in a few seconds (3-10 seconds).

[Cabbage]

Here's another trick for you all, maybe not quite as well known as some of the others mentioned here:

Pick any four digits, not all the same (for example, your four digits could be 6,6,6, and 3, but not 8,8,8, and 8).

Arrange the digits in ascending order. Now arrange the digits in descending order. Subtract the larger from the smaller.
For example, say we pick 8,7,3, and 4. Arrange the digits and subtract:

8743 - 3478 = 5265

Now do the same with the digits of this new number: Arrange the digits in ascending order (2556), descending order (6552), and subtract:

6552 - 2556 = 3996

Now do the same with the digits for this new number (3996).

Repeat this process 6 or 7 times.

Your final result is 6174, known as Kaprekar's constant.

[Dr. Imago]

Originally posted by Cabbage
The missing dollar:

Three guests decide to stay the night at a lodge whose rate they are initially told is $30 per night. However, after the guests have each paid $10 and gone to their room, the proprietor discovers that the correct rate should actually be $25. As a result, he gives the bellboy the $5 that was overpaid, together with instructions to return it to the guests. Upon consideration of the fact that $5 will be problematic to split three ways, the bellboy decides to pocket $2 and return $1 each, or a total of $3, to the guests. Upon doing so, the guests have now each paid a total of $9 for the room, for a total of $27, and the bellboy has retained $2. So where has the remaining $1 from the initial $30 paid by the guests gone?!

(Yes, I know the answer, of course; I'm just posting the problem since it was requested).

This was posted on the Puzzles forum some time back, but I can't seem to find it now. Basically, it points out what happens when you try to intuitively solve a mathematics problem that is presented in a tricky linguistic way.

You have to think of each dollar paid in terms of "place holders". The cost of the room was reduced from $30 to $25 dollars. So, you have to subtract from $30, not add-up from $25, which is what one intuitively wants to do based on the way the problem is presented. Therefore, the 30th, 29th, and 28th dollar went back to each of the guests. After the bellboy returned the three singles to the guests, one is left with $27 dollars being paid out by the guests. In other words, after the bellboy returned the money to each guest, the net result is that, at least to them, they still overpaid for the room, i.e. $27 for a $25 room. Collectively among the three of them, this $2 dollars left over (or roughly $0.6667 each) is the fraction that the bellboy couldn't easily divide, which also adds-up to the amount he kept.

There is no "missing dollar".

-TT

[Iamme]

"Missing dollar puzzle" Google ccp..a simple version:

Three tourists rented a room for $30. The next morning, the manager realized that the room was only $25, so he sent his dishonest son to deliver the $5 difference. But the son only gave each of the three in the room $1, keeping $2 for himself.
This means that each tourist only paid $9 each for the room, rather than $10. But 3 times $9 is $27, and the dishonest son only kept $2, making only $29 all together, not $30. Where did the remaining dollar go?

--------------------------------------

You can't possibly be right, Third Twin. I mean, right on the internet it has it that the dollar is mising. It's got to be one of those rare mathematical anomolies.:D

[Dr. Imago]

Originally posted by Iamme
You can't possibly be right, Third Twin. I mean, right on the internet it has it that the dollar is mising. It's got to be one of those rare mathematical anomolies.:D

How about an easier explanation then?

The totals have to add up to $25 dollars (the actual cost of the room after the rebate), not $30 (the original charge).

$25 cost of room = ($10 original charge * 3 guests) - ($1 dollar refund * 3 guests) - $2 kept by dishonest son/bellboy

;)

-TT