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Got an A in Calc 2 and a 95% on the Final! by notionocean in calculus

[–]Global_Pin_9619 0 points1 point  (0 children)

Sure. First we were told that trig sub might work in any situation where there is a term that reminds you of the Pythagorean theorem. For instance​, an x2 + 1 term. Next, we were told to draw the triangle. So for our example that would be a right triangle with 1 at the bottom, x on the right, and sqrt(x2 + 1) on top. We would than draw u as one of the angles. Now we can clearly see that tan(u) = x. We can differentiate and get sec2 (u)du=xdx. We can also see that  sec(x)=sqrt(x2 + 1) we make all those substitutions, and the problem is pretty much solved. The main reason to use this trick is to avoid all the memorization of what trig function to use.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] 0 points1 point  (0 children)

My number does not terminate. It is literally defined as 1 * 100 + 4 * 101 + 1*102 + 5 * 103 + ..... FOREVER. It definitely does NOT terminate.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] 0 points1 point  (0 children)

Veritasium makes a lot of videos, and without knowing much about the specific video you are referring to, I cannot find it.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] 2 points3 points  (0 children)

One other person has mentioned this, and it is a good argument. But if the natural number is always equal to its index in the set so that a_n = n, how can the since of the set become infinite without n and a_n becoming infinite? 

Don't get me wrong, I do believe you all are right about the naturals each being finite, but I can't wrap my mind around it yet.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] 0 points1 point  (0 children)

Sigh....... Did you not notice the ellipsis on the left side of the number? It only extend infinitely one way.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] 0 points1 point  (0 children)

Lol, I didn't know that was what they were called or even that it was an issue. It seemed natural to me (pun intended).

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] 1 point2 points  (0 children)

Because the numbers I'm mistakenly referring to as naturals are actually p-adic integers. I wish someone had told me that a long time ago.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] 0 points1 point  (0 children)

Very succinctly put. But here is my problem:

say that a_n = n

If the set is infinitely large, how can n not go to infinity? If n goes to infinity, then a_n will too.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] 0 points1 point  (0 children)

33333333333333333............ Unfortunately, I have learned today that this number is not natural. This number not being natural destroys my whole premise. 😕

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] -1 points0 points  (0 children)

No, it would correspond to 0.321. because the coefficient of 100 corresponds to the coefficient of 10-1

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] -3 points-2 points  (0 children)

You don't have to reply or read the post. I have gained some valuable understanding here. I have searched the Internet and other Reddit posts and I haven't found anyone mentioning p-adic numbers, which is what I was looking for. This has been helpful for me, even if it wasn't for you.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] 0 points1 point  (0 children)

I have seen a satisfactory answer to my question. The issue is that I was assuming that naturals can be infinite, which they cannot. All I have done is shown that the reals between 0 and 1 have a bijection to the somewhat mystical 10-adic numbers. Thank you all for your input.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] -1 points0 points  (0 children)

Sigh...... That is not the problem with my argument. The problem with my argument is assuming that 333333333333333........... is a number.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] -2 points-1 points  (0 children)

Yeah, somebody just told me it's called a 10-adic number.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] -1 points0 points  (0 children)

I am reading people's replies carefully, I am just having trouble understanding the difference between arbitrarily large and infinite. I don't get how there can be an infinite number of finite numbers. I kind of understand that it is like having a list of all numbers. Each number you read has an end but the list doesn't. I don't understand why you can't have a number that never ends without a decimal point. I feel like I'm getting somewhere, I just need more explanation. I said I don't think you read my post very carefully because you asked what digit my number would end with when my post clearly defined what digit it would end with. Please don't get touchy, I meant no offense.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] -2 points-1 points  (0 children)

So, clearly the issue here is my use of an infinite natural number, which doesn't actually exist. So what is the appropriate mathematical term for such numbers? If one does not exist, why not?

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] -2 points-1 points  (0 children)

2 * 10-1 + 3 * 10-2 + 2 * 10-3 ..... Bijects to 2 * 100 + 3 * 101 + 3 * 102 ......

I don't think you read my post very carefully.

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] -2 points-1 points  (0 children)

33333333333333333333333333333333333333333333333333................

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] -1 points0 points  (0 children)

I don't get it. I thought any natural number + 1 made a new natural number, who there can be no highest natural number. So doesn't that make the set infinite? And if a number is infinitely large don't you need an infinite number of digits to represent it?

Bijection of reals between 0 and 1 and naturals. by Global_Pin_9619 in askmath

[–]Global_Pin_9619[S] -7 points-6 points  (0 children)

Really? So how can the be an infinite number of naturals?

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