| Posted on Thursday, 23 October, 2003 - 08:58 am: |
Hi, I have just been looking around and found (to me) an incredibly confusing formula, eip=-1
now i understand pi to be the number of times the diameter of any circle fits around the circumference, but i have no idea what 'e' and 'i' are.
Anyone who could enlighten me please do so...
any help would be much appreciated...
Thank you
| Posted on Thursday, 23 October, 2003 - 09:40 am: |
e is just a number, and like pi, it has some interesting properties. If you could give us an idea of what level of maths you're at, then hopefully we can explain some of the properties of e to you. i is the square root of -1, i.e. i2 =-1. This might seem like a strange idea - if you think about it, there isn't a real number that you can square to get -1, so mathematicians call this number i, and describe it as an imaginary number, or complex number. Hope this helps, and if you want any more details, just let us know!
Vicky
| Posted on Friday, 24 October, 2003 - 08:59 pm: |
Thanks for the info on i, I am 15, and live in New Zealand.....I dont know the American equivelant (i assume this is an american site)
| Posted on Friday, 24 October, 2003 - 09:41 pm: |
We're based in the UK, but our users are all over the world.
If you'd like to know more about complex numbers, one of our volunteer team wrote a very good introduction .
Do say if you'd like to know more about e. Although the usual definition involves calculus, there are some things we could tell you about it which don't!
| Posted on Wednesday, 29 October, 2003 - 03:33 am: |
thanks for the link...I will read the introduction (just had a quick look at the moment) but i would like to know about e, if it isnt too much trouble i would like to hear the non-calculus explaination...but the calculus one would also be interesting.
Thanks
isp
| Posted on Wednesday, 29 October, 2003 - 10:29 am: |
e is a number with some special properties. The approximate value of e is:
The definition of e is:
(The dots indicate that this sum never ends, if you find this idea strange just ask; someone will be happy to explain further).
There are many curious and interesting facts about e, the formula you found is (I think) one of the most interesting. Just think, it unites possibly the four most important constants in maths; e , pi , i , and 1.
You may ask why anyone would be interested in such a strange number. As Emma said, the usual motivation for e involves calculus. Since it sounds like you haven't yet encountered this, I'll have to think for a bit how to explain it best.
| Posted on Wednesday, 29 October, 2003 - 04:24 pm: |
Hi Steve,
I'll tell you how I was introduced to e:
Consider the graph y=2x . Take some graph paper and draw it. Then, draw tangents at various points along the curve and work out the gradient of the curve at these points. Plot these values on the same graph, and draw a smooth curve through these points ( ie at x=2, let the corresponding y value be the gradient of y=2x , same at x=3 etc, and draw a line through all these points).
If you have done this correctly, then you will see that the curve of the gradient points is quite close to the original curve y=2x , but it "sits just under it."
Now repeat the process, but with y=2.5x . Again, draw tangents, mark these values on the same graph, and draw a curve through them.
If you have done this correctly, you will see that it lies much much closer to y=2.5x , but still sits just below it!
Now do it with y=3x . This time, the curve of the gradients sits just above the original curve.
So this now makes you ask the question:
"Is there a number (call it b for example,) such that when you draw y=bx , and then the curve of gradients, they lie exactly on top of each other?"
And the answer is yes. You know that it lies between 2.5 and 3, and the value Tristan has given is exactly that value (2.71....), which is called e. Who knows why it is called e, but it is.
Try it - draw y=ex , and then repeat the whole tangent process, and you will find that the curve of the gradients lies exactly on top of the curve y=ex .
Well I hope that gives you at least a feel for what this mysterious e is! Please post back if you would like to know anything else.
Matt.
| Posted on Wednesday, 29 October, 2003 - 04:46 pm: |
e is named after Euler
| Posted on Wednesday, 29 October, 2003 - 09:44 pm: |
Steve,
Here's how I first learned about e...
Suppose I have an investment that earns interest at the wonderful rate of 100% annually. So if I invest $100 today, I'll get $200 back a year from now.
Now, suppose I compound that investment, but leave it at the same rate. That is, instead of 100% per year, I get a rate of 50% every half-year. If I compound it every half a year, then I end up with $225, because after half a year it's worth 50% more, or $150, and then after another half a year it's worth 50% more again, or $225.
If I compound it more frequently, I get more money at the end. For example, if I compound it every quarter, I end up with $244.14.
Can you see where this is going? I want to compound it as often as possible -- every day, or better yet, every minute, EVERY MICROSECOND! In fact, I want to compound this investment CONTINUOUSLY!
If I did that, do you know how much money I'd have at the end of a year? That's right. One hundred times e dollars, or just under $271.83.
--Graeme
| Posted on Wednesday, 29 October, 2003 - 10:41 pm: |
I first met e when looking at catenary curves. These are the shapes that a perfectly uniform rope would make when suspended between two points. This was an unsolved problem of considerable interest towards the end of the 17th century. The solution was finally found by Johann Bernoulli and was of the form y=(ex +e-x )/2. This probably isn't the best introduction to e, but it helps to illustrate the diversity of applications of this number.
| Posted on Wednesday, 05 November, 2003 - 05:09 am: |
wow...this really has gained more interest than i imagined. Thank you all for your help. This certainly does help me with a basic understanding of e. I can see how that equation is so fascinating. The only other two confusing numbers there are * and i (I think that stands for the speed of light, but i am unsure).
I have encountered what you call catenaries before, just under the name parabolas. Are these the same thing.
Thanks
| Posted on Wednesday, 05 November, 2003 - 08:11 am: |
The * is just multiply: the equation is really eip=-1. As I said before, i is the square root of -1. (Just for reference, the letter commonly used to denote the speed of light is c). Catenaries and parabolas are not the same thing: although they may look similar, they are actually quite different curves. Vicky