Country Boy

"pi" is a number and does not change value. I presume what you mean is that in curved space (it doesn't have to be "extreme") the Euclidean relation between circumference and diameter of a circle does not hold so "pi" is irrelevant.

I am confused by your references to "lone pair of electrons". Na, Cl, and H do not have a "pair" of electrons, they have a single valence electron. O has a pair. That is why it requires two H atoms to bond with one O atom to form water, H2O. When NaCl dissociates in water, the Cl ion, Cl- bonds with a H ion, H+ to form the acid HCl, as you say. The Na+ ion bonds with the hydroxide radical, OH-, to form the base NaOH.

A "dimension" just means a number used to specify something of interest. If I am marking points on a straight line, then I can designate one of those points, arbitrarily, and designate every point by its distance along that line from that point, using positive and negative to specify "left" or "right" of that point. One number, dimension one. On a plane there are a whole circle of points at a fixed distance from a fixed point so I would also need to specify a direction- two numbers, dimension two. Physicists are interested in "events"- what happens at a specific position at a specific time. It takes three numbers to specify the position and one to specify time. Therefore "four dimensions".

An airplane can "fly on a diagonal" because the upward force of air on its wings partially cancels the downward force of gravity to keep it from falling. Once you have fallen out of the airplane, you have only the downward force of gravity- your acceleration is -g. At time t later, your velocity is -gt+ v0 and your position is -(g/2)t^2+ v0t+ x0 where vo is your velocity (velocity of the airplane) at the moment you fell out and x0 is your vector position at that time. You will fall in a parabolic arc with only the initial position and slope or that arc determined by the motion of the airplane.

Once, when I was young and foolish (I'm not young any more, just foolish) I decided to wire a light fixture myself. Of course, I turned off the electricity to the fixture- at the wall switch. I was very surprised, when I got a shock, to discover that wasn't enough. Now I throw the circuit breaker- or call an electrician.

Early in the morning it can be difficult to tell the difference between your coffee cup and your doughnut- but it is very important to avoid pouring hot coffee over yourself! So topology is important.

Taking the age of the earth as 100% each of these other ages would be divided by the age of the earth, 4.54, then multiplied by 100 to give a percent. For example, the Archean Era would be 3.8/4.54= 0.837 so 83.7%. Proterozoic would be 2.5/4.54= 0.551 so 55.1%. Phanerozoic would be 0.57/4.54= 0.126 so 12.6%.

1. What do you mean by space-time being "dense"? 2. Since the gravity field of any object extends infinitely, what do you mean by the "edge" of a gravity field?

You titled this "light". Are you referring specifically to light waves rather than "any waves"? Do you know what "polarize" means?

The purpose of life is to produce more gametes.

Your double negative is a little confusing! Also, by "(-1/n, 1/n)" so you mean the interval or the pair of numbers. If the first, what "product topology" are you talking about? If the latter, that's a single point, not a set.

"Science proceeds by proving itself wrong. " I wouldn't put it that way- science does not prove that science is wrong, it proves that certain hypotheses are wrong. The "scientific method" works by: 1. Observe and gather evidence. 2. Find as many different theories and hypotheses that fit that evidence as possible. 3. Extrapolate from those theories to find situations where different theories would give different results. 4. Run experiments that emulate those situations to see whether or not you get the appropriate results. Discard those theories that do not give the expected results. 5. Repeat! No, science will never give 100% assurance that a theory is correct. It works by successive approximations to reality. If you know a better way, please tell us!

A proton has a mass of 1.6726210 x 10-27 kg. At 99.99% the speed of light that mass will be multiplied by 1/\sqrt{1- .999992}= 1/0.00019999 which is about 5000. That is, the proton whizzing around CERN at 99.99% the speed of light will have a mass of about 5.33 x 10-24 kg. No, that's not going to cause very much damage!

Predating the actual development of "topology" but considered an application of basic topology: Euler's "Seven Bridges of Konigsberg'. https://en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg

I'm not sure what you mean by 'verify' here. Given a region bounded by $y= x^4$, $x= 0$, and $x= 1$, it is easy to find the area, $\int_0^1 x^4 dx= \left[\frac{x^5}{5}\right]_0^1= \frac{1}{5}$. Also $\int_0^1 x(x^4)dx= \int_0^1 x^5dx= \left[\frac{x^6}{6}\right]_0^1= \frac{1}{6}$ and $\int_0^1 y(x^4)dx= \int_0^1 x^8 dx= \left[\frac{x^9}{9}\right]_0^1= \frac{1}{9}$. The centroid is the point $\left(\frac{\frac{1}{6}}{\frac{1}{5}}, \frac{\frac{1}{9}}{\frac{1}{5}}\right)= \left(\frac{5}{6}, \frac{5}{9}\right)$. You could also physically test a centroid- cut the figure from some thin material with constant density and "glue" a string to the point you believe to be the centroid. If it actually Is a centroid then the figure will "balance" there, without any side going up or down. Or- draw two perpendicular lines through the centroid. If it is the centroid then the figure will balance on a knife edge along both of those lines.

This is certainly the place to come for irrational help!

druS said "Out of interest, can you name a root of a prime that is rational? Genuine interest if the answer is yes". No, the roots of every prime number are all irrational. But that is not exactly what you said before. You said, before, "I thought any root of a prime number is pretty much a definition of an irrational number". Now, I don't know what "pretty much" is intended to mean here but while it is true that "the root of every prime number is irrational", the converse "every irrational is the root of a prime number" is false. $\pi$ and e are irrationals that are not roots of a prime number. You might be interested in this: Any number, rational or irrational, that satisfies an "n"th degree polynomial equation is "algebraic of degree n" (the rationals are all "algebraic of degree 1"). Numbers such as $\pi$ or e are NOT "algebraic" of any degree. They are "transcendental" numbers.

To say that something has a "half life" of, say, T means that half of it will be gone (so half is still left) after time T. Since half is still left, if we start with amount A, after time T, we will have (1/2)A. After another time T, we will have half of that, (1/2)(1/2)A= (1/2)^2A. After still another time T, we will have half of that, (1/2)((1/2)^2A)= (1/2)^3A. Continuing that, after "n" periods of T, will we have left (1/2)^n A left. But, since this is a continuous process, n does not have to be an integer. Given any time, t, we can think of that as t/T periods of t whether that is an integer or not. After time t, we will have left (1/2)^(t/T)A. You are told "it takes 24 days for 75% of an isotope to undergo a first order decay reaction". So (1/2)^(24/T)A= (3/4)A. Obviously "A" cancels leaving (1/2)^(24/T)= 3/4. Since the unknown, T, is an exponent, use logarithms- (24/T) log(1/2)= log(3/4). (T/24)log(3/4)= log(1/2), T/24= (log(1/2))/(log(3/4)), and, finally, T= (24log(1/2))/(log(3/4)) days.

I really don't understand what you are trying to do. You have three percentages given. What do they have to do with x and y? What are x and y supposed to mean in your equation?

Using "variation of parameters": Given that y'+ ay= f(x), the "associated homogeneous equation" is y'+ ay= 0. y'= dy/dx= -ay so dy/y= -a dx. Integrating ln(y)= -ax+ C, y= C'e-ax Now we look for a solution to the entire equation of the form y(x)= u(x)e-ax Then y'= u'e-ax- aue-ax. Putting that into the equation, u'e-ax- aue-ax+ aue-ax= u'e-ax= f(x). u'(x)= eaxf(x) so u(x)= integral eax f(x) dx. The general solution to the given differential equation is y(x)= C'e-ax+ e-ax(integral eax f(x)dx).

For the first, you are given the graph of y= x4+ (1/32)x-2. The derivative is y'= 4x3- (1/16)x-3. Notice the "3" and "-3" powers. When you square that, something nice happens: y'2= (4x3- (1/16)x-3)2= 16x6- 2(4x3)(1/16)(x-3)+ (1/256)x^-6. I wrote that middle term out In detail to make clear that the "x" terms cancel leaving 16x6- (1/2)+ (1/256)x-6. When we add 1 that "-1/2" becomes "+1/2" so that is simply (4x3+ (1/16)x-3)2 , a "perfect square". sqrt{1+ (dy/dx)2}= 4x3+ (1/16)x-3. Integrate that from 1 to 2.

"General relativity describes gravity as a curve in space-time. In a curved space-time, an object moving in a straight line and at constant speed will, in normal space, appear to accelerate along a curved path. So curved space-time is equivalent to the accelerated shrinking of normal space over time. " You are misunderstanding "accelerate". An object that is either speeding up or slowing down is "accelerating". Even an object moving around a curve at constant speed is "accelerating". It does NOT follow that "normal space" is shrinking.

"When we detonate a nuke, it releases way more energy then what it took to detonate it" That argument would apply to non-nuclear bombs and even gun powder! An explosive device that took more energy to detonate it than you got out of it wouldn't be much use would it? With gunpowder or "chemical bombs" it is the energy holding the molecules together that is released. Nuclear bombs release the energy holding nuclei together.

It has been pointed out to me that -x= x^3+ x- 3 becomes x^2+ 3x- 3= 0, not "+3" as I had. That was a typo. And neither of those factors as (x+2)(x+ 1)! The solution to that quadratic equation, using the quadratic formula, is x= (-3+/- sqrt{9+ 12}/2= (-3+/- sqrt{21}/2.

If there are 40 cards in the deck and 4 of them are aces, the probability the first card drawn is an ace is 4/40= 1/10. There are then 39 cards left in the deck, 3 of them aces. The probability the second card drawn is not an ace is (39- 3)/39= 36/39= 12/13. The probability of drawing "ace, non-ace" in that order is (1/10)(12/13)= 12/130= 6/65. Similarly, the probability the first card drawn is not an ace is 36/40= 9/10 and the probability the second card is an ace is 4/39 so the probability of drawing "non-ace, ace" in that orderis (9/10)(4/39)= (3/5)(2/13)= 12/130= 6/65. The probability of drawing one ace out of the two cards is 2(6/65)= 12/65. I am not sure that using your other method, while correct, would be any simpler. If there are 40 cards in the deck and four of them are aces, there is 4/40= 1/10 probability the first card is an ace. There are then 39 cards in the deck, 3 of them aces. The probability the second card drawn is an ace is 3/39= 1/13. The probability both cards are aces is (1/10)(1/13)= 1/130. If there are 40 cards in the deck and four of them are aces, 36 cards are not aces so there is 36/40= 9/10 probability the first card is not an ace. There are then 39 cards in the deck, 35 of then not aces. The probability the second card drawn is not an ace is 35/39. The probability the cards are two non-aces is (9/10)(35/39)= 21/26. The probability the are not "two aces" nor "two non-aces" is 1/130+ 21/26= 3/390+315/390= 318/390= 159/195= 53/65. The probability the cards drawn are one ace and one non-ace is 1- 53/65= (65- 53)/65= 12/65 as before. I found the second method much harder than the first!