IsaacAsimov

What does the \mathop command do?

I've been trying to learn Latex all day on Sat. May 12, sometimes using a guide I got from the Net, and a lot of trial and error. I figured out most of it, but I haven't figured out what the \mathop command does, or how to separate the formulas on different lines on the screen. Can anybody help?

I wasn't aware that we did.

I have tried to use integrals to compute pi:<br><br>[math]\mathop A(n) \rightarrow{\int_a^b} f(x) dx[/math] as [math] n\mathop\rightarrow \infty[/math]<br>[math] = 2{\int\limits_{-1}^{1}}\sqrt{1-x^2}dx[/math]<br>From p. 209 of Freshman Calculus:<br><br>[math]\mathop{\int}\sqrt{a^2-x^2}dx = \frac{a^2}{2}\arcsin{\frac{x}{a}}+\frac{x}{2}\sqrt{a^2-x^2}+C [/math]<br>a=1: [math]=\frac{1}{2}\arcsin{x}+\frac{x}{2}\sqrt{1-x^2}[/math]<br>[math] = \left. 2 \bigg(\right|_{-1}^{1}\bigg) \frac{1}{2}\arcsin x + \frac{x}{2}\sqrt{1-x^2}[/math]<br>[math] = 2 \bigg[\bigg(\frac{1}{2}\arcsin1+\frac{1}{2}\sqrt{1-1}\bigg)-\bigg(\frac{1}{2}\arcsin-1-\frac{1}{2}\sqrt{1-1}\bigg)\bigg] [/math]<br>[math] = 2 [((\frac{1}{2})(\frac{\pi}{2})+\frac{1}{2}(0))-((\frac{1}{2})(\frac{-\pi}{2})-(\frac{1}{2})(0))] [/math]<br><br>

We design cars with computers, and as a result we get better cars. We design computers with computers, and we get better computers. Why don't we start designing robots with computers? We should get better robots, which could use computers to design even better robots, which could use computers to design even better robots, etc., etc., etc. It's a neat idea, don't you think?

I think that dreams could be hallucinations because when you're dreaming, you don't experience with your body any of the 5 senses. The brain could start making up things. People that were placed in isolation chambers had hallucinations for the same reason. What do you think?

Here is a sneaky method of estimating pi: Draw a circle, with a radius of 1, let a be a small angle, and let x be the opposite side inscribed in the circle. sin a = x, and pi is approximately (360 sin a)/(2a), or pi=(180 sin a)/a, as a gets small. Angle a is measured in degrees. Try the formula on a few values of a, using a calculator: a............................pi .01.........................3.14154029392740 .0001......................3.14159264835381 .000001...................3.14159265358927 Question: Why is the formula so accurate? Answer: My calculator converts the angle to radians, then uses several terms of an infinite series to estimate the sin. Here is the equation in radians: pi=(pi sin a)/a. We are using pi to estimate pi. Pi=3.14159265358979 is hard coded into my calculator. We are using pi with 15 digits of accuracy to estimate pi to 13 digits of accuracy. At least it works!

That is an interesting formula. I'm trying to figure out how to insert special scientific symbols such as square root, right arrow, infinity symbol, etc. I can see the notation you used in your post, but it looks sort of like a programming language, and I'm not familiar with that language. Is there a list of commands that I could get so I could type up a post using that notation? Marlon S.

This is something I got out of Freshman Calculus, p. 255 An=delta x(f(x0)+f(x1)+f(xn-1)) Let delta x=(b-a)/n, x0=a, x1=a+delta x, xk=a+k delta x for k=1,2,...,n I will be using the formula for a circle, x2+y2=1, or y=sqrt(1-x2), evaluated between 0 and 1, so it will be a quarter of a circle. Let a=0, b=1, n=10 delta x = (b-a)/n=(1-0)/10=1/10=0.1 x0=a=0, x1=a+delta x=0+0.1=0.1, x2=a+k delta x=0+2(0.1)=0.2,... Using a calculator: An=0.1(f(0)+f(1)+f(2)+...+f(9)) =0.1(sqrt(1-02)+sqrt(1-0.12)+sqrt(1-0.22)+sqrt(1-0.32)+sqrt(1-0.42)+sqrt(1-0.52)+sqrt(1-0.62)+sqrt(1-0.72)+sqrt(1-0.82)+sqrt(1-0.92)) =0.1(1+0.99499+0.9798+0.95394+0.9165+0.8660+0.8+0.71414+0.6+0.43589) =0.1(8.26126) =0.826126 x4 = 3.30 which is close to pi, but not very close

The whole purpose of the exercise was to calculate pi using circumscribed polygons. Using a calculator for the tan(x) function makes it a little easier. I'm tired of getting negative feedback from you people! Don't you think I deserve a little praise for my mathematical efforts?

Calculating tan(x) is easy - just use a calculator or a computer.

Area of Circumscribed (around a circle) n-gon = n/tan[(n-2)(180)/(2n)] square: Find: As, pi Given: r=1, n=4 As=n/tan[(n-2)(180)/(2n)]=4/tan[(4-2)(180)/(2x4)]=4/tan[2(180)/8]=4/tan45 = 4 hexagon: Find: Ah, pi Given: r=1, n=6 Ah=6/tan[(6-2)(180)/(2X6)]=6/tan[(4)(180)/12]=6/tan60 = 3.464 decagon: Find: Ad, pi Given: r=1, n=10 Ad=10/tan[(10-2)(180)/(2x10)]=10/tan[8(180)/20]=10/tan72 = 3.249 hectagon: Find: Ah Given: r=1, n=100 Ah=100/tan[(100-2)(180)/(2x100)]=100/tan[98(180)/200]=100/tan88.2 = 3.14263 chiliagon: Find: Ac Given: n=1000 Ac=1000/tan[(1000-2)(180)/(2x1000)]=1000/tan[998(180)/2000]=1000/tan89.82 = 3.1416 megagon: Find: Am Given: n=1E6 Am=1E6/tan[(1E6-2)(180)/(2x1E6)]=1E6/tan[(999998)(180)/2E6]=1E6/tan89.99982 = 3.141592654 = pi

Here is a problem I thought of that I found a solution for: Problem: How long does it take light to travel 1/2 way around the Earth using 6 communications satellites that form a circumscribed hexagon? Solution: Find: t in s Given: distance from center of Earth to a side =a=rE=6.37E6 m, distance from center of Earth to a vertex =r, n=6 sides, c=3E8 m/s r=a/[cos(pi/n)]=rE/cos(pi/6)=6.37E6 m/0.866=7.36E6 m 3r=2.21E7 m s=vt t=s/v=2.21E7 m/3E8 m/s=0.07 s = 7/100 s

Here's a problem I thought of that I found a solution for: How long does it take 2 people to communicate with each other at any 2 points on the Earth using 4 communications satellites? Or: How long does it take light to travel 1/2 way around a square circumscribed around a circle (Earth)? Solution: Find: t in s Given: c=3E8 m/s, mean radius of Earth rE=6.37E6 m Let a=1/8 distance around square=rE s=vt, t=s/v circumference of square Cs=8a, 1/2 Cs=(1/2)(8a)=4a=4rE t=s/v=4rE/c=4(6.37E6 m)/3E8 m/s=0.08 s = 8/100 s

Thank you for correcting me, however I was just giving the general solution. If you like, try inputting your values for the speed of light and see what the travel time would be. Isaac

Here's a problem I thought of that I found a solution for: Problem: How long does it take for a person on the surface of the Earth to communicate with someone else on a different point on the Earth, assuming a fibre optic link and using circles with circumscribed polygons? Or: How long does it take light to travel 1/2 way around the Earth? Solution: Find: t in s Given: c=3E8 m/s, mean rE=6.37E6 m s=vt, t=s/v CE=2(pi)rE, 1/2 CE=1/2(2)pi(rE) t=s/v=(1/2)CE/c=pi(rE)/c=3.14(6.37E6 m)/3.00E8 m/s=0.07 s = 7/100 s

Problem: How many times has the Sun revolved around the Galaxy in the last 1 billion years? Solution: Find: n Given: Total time taken t2=1 billion y=1E9 y (365 d/y)(24 h/d)=8.76E12 h Time to travel once around Galaxy t1=2.19E12 h n=t2/t1=8.76E12 h/2.19E12 h = 4 times exactly

Hi, my name is Marlon Schmitt. I like physics and mathematics, and I can do some computer programming. I like doing calculations on calculators and computers, especially if it involves the number pi. I like Rubik's Cube, playing ping pong and playing computer games.

Here's a problem that I found a solution for: How fast does the Sun travel around the Galaxy (Milky Way)? Find: v Given: radius from center of Galaxy to Sun, r (CS)=2.5E17 km, time it takes Sun to travel around Galaxy, t=2.5E8 y s=2(pi) x r(CS)=2pi(2.5E17 km)=1.57E18 km t=2.5E8 y (365 d/y)(24 h/d)=2.19E12 h v=s/t=1.57E18 km/2.19E12 h =717 258.6 km/h