Recent content by Hyrum

I'm going to start by explaining what exactly induction is. Induction is the process of proving a condition to be true for every element in a given set by showing that if it's true for one, it's true for all. To answer your question "So am I allowed to use the assumption as a given to make the...

All the nth complex roots of any number z are z1/n times the complex roots of 1. All of the complex roots of 1 are solutions to this equation: $$x^n - 1 = 0$$ Recall that given a polynomial equation such as ax^3 + bx^2 + cx + d = 0 with roots z_0, z_1, z_2, then $$z_0z_1z_2 =...

You're welcome! Glad I could help :)

If you go far enough ahead you'll come across the Fundamental Theorem of Calculus which states an intimate relationship between Integration and Differentiation. It basically says that given a function f(x) for which an antiderivative F(x) is known, then $$ \int_a^b f(x) = F(b) - F(a)$$ A basic...

Yes, he was saying the result of the limit could not depend on a variable defined by the limit. He used this as an example $$\lim_{n \to 2} n x^n = n x^2$$ And was saying that it didn't work because the n in nx2 is defined in the limit.

Right. So it should 1/r2 so it expands outward as a sphere, shouldn't it?

To clarify: The random process doesn't necessarily follow exponential distribution. All it follows is the probability of a given atom decaying in a certain time-frame. The exponential distribution is an effect caused by this probability and the fact that atoms are disappearing and will no...

Don't you mean $$\frac{1}{r^{d-1}}$$ where d is the dimension of space (3 in our case)?

Limits are used to get as close to a "breaking point" as possible to observe what happens there without actually hitting it. so you have your definition of e: $$e = \lim_{n \to \infty} (1 + \frac{1}{n})^n$$ Since this definition would break down at n is infinity you approach it from as...

Alright, thanks again! This was really confusing me.

Awesome, Thanks! Just to check one thing, cases of gravitational waves are an exception to the object being static, right? I know that these are energy being emitted from the system as waves, just making sure it is the actual objects in the system emitting it and not an effect on the field...

It doesn't seem like Quantum Mechanics would allow it because it would require being certain of the position of the object when the energy is transferred. What I got from your answer is that Space-time is bent by the presence of Energy and this is what causes the change in motion. But a...

Thanks! This is what I thought first, but wouldn't this require that only the necessary amount of energy is being transferred that will be transferred to all other objects, which I can't see quantum mechanics allowing? Or does it require constantly giving off energy in order to maintain the...

In a Newtonian Universe where gravitational and electric forces are instantly experienced, Energy is exchanged between 2 objects acting on each other in order to affect their motion. However in a relativistic universe, these forces travel at the speed of light and so given distance d between...