EPTThe value of e as a fraction (ratio of two whole numbers)
e = 2.718
Math help question
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LonelyIslands
Canada590 Posts
The value of e as a fraction (ratio of two whole numbers) e = 2.718 | ||
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WhatisProtoss
Korea (South)2325 Posts
Anyways, I know that Euler's number = Sum of the Sequence: 1/x!, where x: 0 to infinity A good approximation that I learned in high school was the 2 + 1/(1+1/(2+2/(3+3/(4+4/ ... )))) This sequence (in whole numbers) comes out to be: 2, 2.6666, 2.7083, 2.7167, 2.7167, 2.7181, 2.7183 You can see that only after 7 sequence places, E is approximated to 5 decimal places already. Just draw out the 2 + 1/(1+1/(2+2/(3+3/(4+4/ ... )))) that I mentioned and that should be your answer.  | ||
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LonelyIslands
Canada590 Posts
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LonelyIslands
Canada590 Posts
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WhatisProtoss
Korea (South)2325 Posts
6th sum of sequence = 1957/720 ~ 2.7181 That should be your answer, I'm just trying to think of all the possible mind-games a teacher can try on students... I think that's all you need. | ||
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LonelyIslands
Canada590 Posts
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WhatisProtoss
Korea (South)2325 Posts
1/0!, 1/1!, 1/2!, 1/3! ..... 1/(infinity)! So, E = the sum of ALL those terms, 1/0! to 1/(infinity)! And when I'm saying "Nth sum of sequence", I mean 1/0! + 1/1! + 1/2! + ... + 1/N! = Nth Sum of Sequence So, the 6th Sum of Sequence is your approximation for E=2.718: 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! = 1957/720 ~ 2.718 | ||
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WhatisProtoss
Korea (South)2325 Posts
Sigma(1/N!), N = 0 to infinity. OH SWEET! Wiki came up on image search nicely: ![[image loading]](http://upload.wikimedia.org/math/a/1/0/a10a05335ccb3b560a678ed2dd287fdb.png) | ||
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LonelyIslands
Canada590 Posts
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WhatisProtoss
Korea (South)2325 Posts
[edit] Oh, tell me what math class this is... I just want to make sure my answer is not too complex. It is Algebra? High school analysis? | ||
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LonelyIslands
Canada590 Posts
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WhatisProtoss
Korea (South)2325 Posts
On October 03 2006 13:09 Canuck wrote: Well I have like 5 more questions and I have no clue how to do them; prof never taught us :S Oh, professor? You're in college, huk. Just post all of them, I'll answer them if it doesn't take too much time. | ||
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LonelyIslands
Canada590 Posts
- The result of (3+i)*(2+3i) as magnatude and phase - The angle of (1+i) displayed in degrees - 3+4j/sqroot-6.5 - Real, imaginary parts, magnitude and angle of 3+4i | ||
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LonelyIslands
Canada590 Posts
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LonelyIslands
Canada590 Posts
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WhatisProtoss
Korea (South)2325 Posts
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WhatisProtoss
Korea (South)2325 Posts
On October 03 2006 13:12 Canuck wrote: - The result of (3+i)(1+i) in rectangular coordinates - The result of (3+i)*(2+3i) as magnatude and phase - The angle of (1+i) displayed in degrees - 3+4j/sqroot-6.5 - Real, imaginary parts, magnitude and angle of 3+4i Ohhh, before I do these problems.... I will tell you that graphs with i's in them are exactly the same as y's. You do graphs of x vs. i, instead of x vs. y. So, (3+i)(1+i) = (3, 1)(1, i) - > when (x,y). Get it? If you do not understand, post that you do not and I shall clarify. Things you need to know (1) Also, i*i = -1, so when you multiply these out, make sure you change the i^2 to -1s. (2) What is Phase? Let's say you have x + yi. x + yi = M * e^(i*PHASE). Phase is usually denoted as a phi symbol. (This is a conversion from Rectangular to Polar coordinates) -------------------- On to the questions: (1) The result of (3+i)(1+i) in rectangular coordinates: (3+i)(1+i) = (2+4i), Answer: Rectangular coordinates (Cartesian) : (2, 4) (2) The result of (3+i)*(2+3i) as magnatude and phase (3+i)*(2+3i) = (3+11i) Answer: Magnitude = sqrt(3^2 + 11^2) ~ 11.402 In order to find the Phase, first find the angle of when x=3 and y=11. Angle = arctan(11/3) <- this is in radians, ALWAYS use radians Phase = arctan(11/3) (3) The angle of (1+i) displayed in degrees Like said above: Angle = arctan(1/1) = pi/4 Answer: But that is in radians, so use degrees. pi/4 = 90 degrees (4) 3+4j/sqroot(-6.5) What is that J???? 3+4i/sqrt(-6.5) = 3 + (4*i)/(sqrt(6.5) * i). i's cancel: Answer: 3+4/sqrt(6.5) ~ 4.5689 (5) Real, imaginary parts, magnitude and angle of 3+4i 3 + 4i = (3, 4) Answer: Magnitude = sqrt(3^2 + 4^2) = 5 (easy, 3-4-5 triangle) Angle = arctan(4/3) ~ 53.13 degrees (or 0.927 radians) | ||
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Polemarch
Canada1564 Posts
On October 03 2006 12:44 Canuck wrote: I have a math question I need help with as I never did this before: The value of e as a fraction (ratio of two whole numbers) e = 2.718 it's impossible to do that, because e is an irrational number, which means it's impossible to do that, hehe. are you sure that's what the question is asking, or are you allowed to say it's impossible? details in links http://en.wikipedia.org/wiki/Irrational_number http://en.wikipedia.org/wiki/Proof_that_e_is_irrational edit: WhatIsProtoss is giving you some approximations for e as a ratio of two whole numbers. i don't think that's what they're asking for, but if so, you might as well just do something like 2718/1000 (take some more decimal points if you like), which will be easier | ||
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WhatisProtoss
Korea (South)2325 Posts
On October 03 2006 14:06 Polemarch wrote: [removed quote within quote] edit: WhatIsProtoss is giving you some approximations for e as a ratio of two whole numbers. i don't think that's what they're asking for, but if so, you might as well just do something like 2718/1000 (take some more decimal points if you like), which will be easier The value of e as a fraction (ratio of two whole numbers) It explicitly says that it should be a ratio of two whole numbers. You cannot just say 2718/1000, you want an approximation with a smaller ratio... As such, 1957/720 is 30% smaller than the ratio that you gave. | ||
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Polemarch
Canada1564 Posts
but let's not get sidetracked... because nor does it say it wants an approximation. Canuck, i think you're probably misinterpreting your question, because it's an interesting property of e that it's impossible to write it (exactly) as the ratio of two whole numbers. | ||
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