Math Is Fun Forum

Err, I like the cancelling out theory.

I don't understand the one photon at one time experiment. Really weird. The Photon splits into two parts of itself before it sees two slits?? Really weird.

But can anyone explain why it won't work when two slits are wide or distant from each other?

No, after one year or two, Ben's is still better than Simon's-that's why I said it depends.
Ben's curve is a linear one with a good beginning, while Simon's is an exponential one with a poor beginning. In the long run, first interest of the latter will meet the former or even overnumber the former and then keep increasing the gap, after a while the total sum of money of the latter will drawf the latter.

However, "the total sum" does not count in reality. Today's one dollar is different than that after a year. Because bank and interest rate exist.
100today=100(1+interest rate) next year
100/(1+interest rate)=100 next year.
Under this consideration, we can transfer the cash gains of respective years all to equivalent gains today and sum them up, to so called Present Value. And if the compound rate 6.5% is less than the bank interest rate, there is no guarantee for it to beat the annual earnings.

Well the infinite series thing is applied by taking a finite part of it. So Maclaurin series is very useful in getting any digit we want.

Basically yes, but 12 should be replaced by 4 because of the word "quaterly".

I think maybe at last Harry and Vordermolt just cancel out each other, as Neo and Mr.Smith did in Matrix.

Very passionate indeed.
Zach, would you mind checking up post 735 too? A little far, though.

This post is on what can we find only by looking at the LL scale

Here is a photo of a slide rule
click while pressing "shift" to get the pic in a new window

So you can observe the first side of the slide rule.
Note the bottom and the upper part. LL3 LL2 LL1 LL01 LL02 LL03
A little bit confusing.
But just drag the slide bar at the bottom of the window and check the right end of the slide rule.
The mark

And notice the x mark to the right of D scale.
Yes, exactly LL scales are exponent scales of D, note that D is already exponential. So e^D=e^(10^L). Note that LL3 scale increase dramatically.

Other than telling exponent of e and ln backwards,
having these 6 scales of exponent brings about two benefits.

One is that you can easily get the power of 10th or 1/10th easily.
Because e[sup]x[/sup]=e[sup]0.1x×10[/sup]= (e[sup]0.1x[/sup])[sup]10[/sup]
Check 2 on LL2 and the value on the scale above, the LL3 scale. 2^10≈1.03×10[sup]3[/sup]
Yes, if you are familiar with computer, 1KB=2^10Bytes=1024Bytes. 1030 is close to 1024, as far as the slide rule can tell.

Another benefit is the inversions
Because e^x=1/e^(-x) and so on, you can check inversions by two corresponding LL scales. LL3-LL03 LL2-LL02 LL1-LL01. The small scales of LL1 and LL01 can give very accurate inversions of numbers near 1. Suppose you are calculating how much you need to save under the interest rate at 3.5 to get 1000 next year. 1/1.035 is needed. By checking I can see it's 0.9656.

Additionally, just check the positions of 1.01 1.012 1.02 1.03 1.04 on LL1 and 1 1.2 2 3 4 on D. You will notice their positions are near, horizontally.
This is because e^x ≈ 1+x when x is small.
So e^.01x ≈1+.01x
The formula can be derived by derivative:
(e^x)'=e^x
e^(0+h) ≈e^0+(e^x)|'[sub]0[/sub]h- h is small.

I see, the power law is something like a shortcut.

Standard proof:

When f(x)->a and g(x)->b and b≠0
lim[f(x)/g(x)]=a/b

Yes, let's just ignore the controversial infinite digits and have simple but elegant finite assumption.

e is definitely another approximation of finite things

The bacteria multiplication is
1 2 4 8 16...
not
1 e e^2 e^3...
Asking the problem of having all the digits( an infinite amount) of e has no meaning.
You will never calculate that out either. (you and a computer can only calculate an Finite amount after a finite time)
e^x if fitting the data, ok. 2^x also. If you are convenient with e^x, fine. but keep it in mind actulaly no spontanous growth rate or multiplication thing in reality, only in human's mind for convenience.