Sunday, December 25, 2005

Irrational Entertainment

First, I’d like to wish all of our Christian readers a very merry Christmas!

For those of us enjoying a quiet Sunday before the start of Chanukah tonight, I offer a bit of mathematical diversion. Please simplify the following infinite fraction. Explain your work.

Comments:

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Thanks for the good wishes, Dr. Bean. Likewise, Happy Chanukah to you.

As for the math, I'm too busy counting how many Turtles I can eat a one sitting.

Q

As for the math, I'm too busy counting how many Turtles I can eat a one sitting.

Q

Q: Thanks. It's important to keep pushing yourself for new personal records!

Irina: Thanks! What do you mean what is it? It's one plus a big gnarly fraction.

Cruisin-mom: My thoughts exactly.

Og: If you can solve the problem you posed, then it's a fair trade. (I'll put up not just the answer but the solution to the problem I posed if no one else does.) But if you just found an interesting identity that you can't demonstrate yourself, then no fair. Off the top of my head it looks like x might be transcendental (meaning not rational and not the root of a polynomial with rational coefficients, pi and e are examples of transcendental numbers). It looks like a familiar series, and off the top of my head I'd guess that x = pi, but I'll be darned if I can prove it. I only tried for a few minutes and got nowhere. I think you have to take a polynomial expansion of some trig function or something, which is something that I've forgotten about at least a decade ago.

Hang on a second. Let me cheat and try to solve it numerically… (I agree that this is cheating, but it's the best I got.) I made an Excel spreadsheet and took the calculation out to 40 terms. I got 3.11659… so I think pi is right. Again, a proof would be nicer, and if you can provide one, I'll be much obliged.

The problem I offered is much less susceptible to numerical solutions, but a little algebra will crack it.

Irina: Thanks! What do you mean what is it? It's one plus a big gnarly fraction.

Cruisin-mom: My thoughts exactly.

Og: If you can solve the problem you posed, then it's a fair trade. (I'll put up not just the answer but the solution to the problem I posed if no one else does.) But if you just found an interesting identity that you can't demonstrate yourself, then no fair. Off the top of my head it looks like x might be transcendental (meaning not rational and not the root of a polynomial with rational coefficients, pi and e are examples of transcendental numbers). It looks like a familiar series, and off the top of my head I'd guess that x = pi, but I'll be darned if I can prove it. I only tried for a few minutes and got nowhere. I think you have to take a polynomial expansion of some trig function or something, which is something that I've forgotten about at least a decade ago.

Hang on a second. Let me cheat and try to solve it numerically… (I agree that this is cheating, but it's the best I got.) I made an Excel spreadsheet and took the calculation out to 40 terms. I got 3.11659… so I think pi is right. Again, a proof would be nicer, and if you can provide one, I'll be much obliged.

The problem I offered is much less susceptible to numerical solutions, but a little algebra will crack it.

bean: x = pi. I figured you would recognize gregory's series right away. Go get gregory;s series and plug in the variables.

When i was in Seminary, we had a Jesuit physics teacher. IQ in the over-200 category (this was why I wanted so badly to be a Jesuit) he used this simple formula to remind us that Natural Law was written by the Hand of G-d. He said, if a number as complex as pi can be represented by as simple a series of numbers as the gregory series, and the answer is so plainly functional, and so elegant in base ten, how could it be else? He also felt that Science was the Window into the Mind of the Creator, and all we had to do was look in. I tend to agree.

When i was in Seminary, we had a Jesuit physics teacher. IQ in the over-200 category (this was why I wanted so badly to be a Jesuit) he used this simple formula to remind us that Natural Law was written by the Hand of G-d. He said, if a number as complex as pi can be represented by as simple a series of numbers as the gregory series, and the answer is so plainly functional, and so elegant in base ten, how could it be else? He also felt that Science was the Window into the Mind of the Creator, and all we had to do was look in. I tend to agree.

Actually, here's an interesting explanation. And no, I can't provide a "proof", only the supposition that every iteration brings x closer and closer to pi.

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpi.html

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpi.html

I know Dr. Bean is good at these kinds of puzzles. I'm not. But I am a practical scientist and once in a while a puzzle comes along that I can solve one way or another.

I think the answer is 2.732050808. If I'm correct, I will tell you how I figured it out.

I think the answer is 2.732050808. If I'm correct, I will tell you how I figured it out.

Og: Fun stuff. Thanks.

Lord Emsworth: Get a blogger account. (I can help you if you like.) Nope. That's not it.

Lord Emsworth: Get a blogger account. (I can help you if you like.) Nope. That's not it.

OK my next guess is 3.

I tried to get a blogger account but my ^)(&)(**&T%( connection is so variable I gave up waiting.

I will shortly get a blogger account.

I tried to get a blogger account but my ^)(&)(**&T%( connection is so variable I gave up waiting.

I will shortly get a blogger account.

Lord Emsworth: Um. No. Guessing is not a reasonable solution strategy when your answer can be anywhere in the set of real numbers. Please cut that out.

Let x = the denominator of your gnarly fraction.

So your fraction is 1 + 2/x. It is also x - 1.

1 + 2/x = x - 1

Add 1 to both sides: 2 + 2/x = x

Subtract x from both sides: -x + 2 + 2/x = 0

Multiply both sides by -x: x^2 - 2x - 2 = 0

Solving the quadratic equation gives you x = 1 plus or minus sqrt(3).

Your fraction x - 1 cannot be negative, so your fraction is sqrt(3).

Happy Chanukah.

So your fraction is 1 + 2/x. It is also x - 1.

1 + 2/x = x - 1

Add 1 to both sides: 2 + 2/x = x

Subtract x from both sides: -x + 2 + 2/x = 0

Multiply both sides by -x: x^2 - 2x - 2 = 0

Solving the quadratic equation gives you x = 1 plus or minus sqrt(3).

Your fraction x - 1 cannot be negative, so your fraction is sqrt(3).

Happy Chanukah.

Oven: [standing up and clapping enthusiastically] Very nice! Fun, no? I always thought it was cool that you could express an irrational number as an infite series of fractions of integers. But then again, I'm easily amused.

Merry Christmas, Oven.

Merry Christmas, Oven.

Sorry, but feh on this post. Feh, feh, feh. Ralphie, I think that I am sharing the same lump of coal as you. ;)

bean: Try rollerblading, or something. Pure mathematics is the sole geekiest hobby on earth. Even the people who excel and make a living at it do less well than physicians. Or, macDonald's employees. Hell, Estes model Rockets are more interesting, and you can (in complete secrecy) compute the trajectory, etc. to satisfy the geek drive.

Happy Hanukah!

Happy Hanukah!

Oooh, I am late to the party, but this is the best Chanukah present! Math is better than candy and almost better than...oh, never mind.

Oven and Dr. Bean: I had this right!!!! If Oven is right, the actual answer is 1 + sqrt(3). According to my calculator the square root of 3 = 1.7320508075688772935274463415059 So the answer is

2.7320508075688772935274463415059

which is the answer I gave you (rounded off, of course).

Why does the gnarly fraction also equal x - 1?

Happy Chanukah and Merry Christmas, as appropriate.

2.7320508075688772935274463415059

which is the answer I gave you (rounded off, of course).

Why does the gnarly fraction also equal x - 1?

Happy Chanukah and Merry Christmas, as appropriate.

Jack: [with index fingers in ears] LA LA LA I CAN'T HEAR YOU

Og: I'm not quitting my day job. This

Stacey: Happy Channuka! I thought you'd like it. But are you actually gonna do it?

Do you have your husband whisper e^(i*pi) = -1 to you? On second thought, I might not want to know.

Lord Emsworth: The actual answer is sqrt(3). I thought Oven's work was great, but let me explain it this way:

Let's call the whole expression y. (I'm not using x just 'cause Oven did; I don't want to confuse.)

So y = 1 + 2

Add 1 to both sides:

y + 1 = 2 + 2

Now look at the denominator under the first fraction bar (the one I made bold). It's identical to the whole expression to the right of the equal sign. That's the important key to get. So under that bold fraction sign we can just put y + 1

y + 1 = 2 + 2

subtract 2 from both sides

y – 1 = 2/(y+1)

multiply both sides by (y+1)

(y – 1)(y +1) = 2

y^2 – 1 = 2

y^2 = 3

So y = plus or minus sqrt(3) but the negative solution doesn't work and is discarded.

y = sqrt(3)

Makes sense?

Og: I'm not quitting my day job. This

**is**just a hobby.Stacey: Happy Channuka! I thought you'd like it. But are you actually gonna do it?

Do you have your husband whisper e^(i*pi) = -1 to you? On second thought, I might not want to know.

Lord Emsworth: The actual answer is sqrt(3). I thought Oven's work was great, but let me explain it this way:

Let's call the whole expression y. (I'm not using x just 'cause Oven did; I don't want to confuse.)

So y = 1 + 2

**/**(2 + 2 / (2 + 2 / (2 + 2 / (2 + 2 / (…Add 1 to both sides:

y + 1 = 2 + 2

**/**(2 + 2 / (2 + 2 / (2 + 2 / (2 + 2 / (…Now look at the denominator under the first fraction bar (the one I made bold). It's identical to the whole expression to the right of the equal sign. That's the important key to get. So under that bold fraction sign we can just put y + 1

y + 1 = 2 + 2

**/**(y + 1)subtract 2 from both sides

y – 1 = 2/(y+1)

multiply both sides by (y+1)

(y – 1)(y +1) = 2

y^2 – 1 = 2

y^2 = 3

So y = plus or minus sqrt(3) but the negative solution doesn't work and is discarded.

y = sqrt(3)

Makes sense?

Actually, I was on the right track. I forgot to do the last (initial) calculation. The answer I gave you was actaully the denominator of the gnarly fraction. If I had divided that into 2 and added 1 the actual answer would have been 1.73205080756888000000 or square root 3.

It's been a long time since I was in school. Do I get partial credit? Your explanation made sense. As B&C said you are a great teacher.

It's been a long time since I was in school. Do I get partial credit? Your explanation made sense. As B&C said you are a great teacher.

Lord Emsworth: To get partial credit, you have to show your work. How'd you manage to do it numerically? Excel? OK. You get a B+.

Cro Magnon Man not able to solve problem. What Doctor Bean want from Cro Magnon? Not even discover wheel yet!

CMM: Don't feel bad. You make the best with what you've got. Re: wheel. Let me try to get you started. You know that big wood club you carry around to whallop food animals or to whallop other males when they make big eyes at your female? Put it down. Step on it. With both feet. Trust me. See how it rolls out from under you and you slip off and land on the back of your head? That's a wheel.

Stacey: Yay! (I hope I'm not making your husband jealous...)

Stacey: Yay! (I hope I'm not making your husband jealous...)

Doc Bean: His minor was in math, so he's working on it, too! Just wish we had more time. I am swamped here. It may get pushed back on the list.

Hope you all had a great Chanukah!

Hope you all had a great Chanukah!

I still say feh, but if it makes you happy I'll send you a slide rule and a pocket protector that I won with Blue Chip stamps. It is going out C.O.D.

Jack: That

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*would*make me happy. It would make me even happier if you explained to the rest of the class how it is that one can do a multiplication problem with a slide rule. (Addition is easy: you just take any two rulers and use them to add the two distances. For example to do 3+5 you put the zero of the second ruler on the 3 of the first. Looking at the 5 of the second ruler, the number next to it on the first is the answer: 8. But how would you do 3x5?)<< Home