Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

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Agnishom wrote:

Our whole universe was in a hot dense state,

Then nearly fourteen billion years ago expansion started —Wait...

The Earth began to cool;

The autotrophs began to drool;

Neanderthals developed tools;

We built a wall (we built the pyramids).

Math, science, history, unraveling the mystery,

That all started with a Big Bang!

Bᴀɴɢ!

Ah, one of my favorite comedy shows ever.

For some reason Geogebra disagrees with the fact that AF is perpendicular to BE...

Still...

People all know my email (and country of origin) so they could potentially spam email me and cause a hazard to my spam folder.

Just a thought, you didn't have to implement it.

**rileywkong**- Replies: 3

Hi all,

I'd like to suggest an anonymous/guest feature for *ahem* sensitive or otherwise *anonymous* topics.

An alternate solution would be to consider triangles QAP and RDC.

(AoPS community please don't hate on me )

Calligar wrote:

As you pointed out, if you divide 1 / 0, you can not reach a number. There isn't a number in existence (that I'm aware of) that can answer that question. Some may argue the answer is ∞. But then comes an issue with using ∞ as a number, which it isn't (or at least not exactly in that way). Plus there are issues to that. If say 1 / 0 = ∞, what does 2 / 0 = ? Does it also equal ∞, does it equal 2∞? How do you even begin to make sense of it? You can make sense of it, but I don't know anything in mathematics that will answer it except for maybe...1 / 0 = undefined (or something of that nature). Probably already realized all this, but at least interesting to point out.

As far as I know, ∞ = 2∞. Think of it this way:

*Say we have a hotel that has ∞ floors, with 1 room on each floor. Let's say that the hotel is full. Then, a party of ∞ comes along. How do they make space for them? By moving the person in floor #1 to floor #2 and fitting 1 person from the party in it. Same for the person in #2, #3, so on. *

*Or, in math language (aka Engrish), if we have a set that has ∞ terms, we can multiply each term by 2 to fit in an additional ∞ terms.*

So, writing that out in mathematical terms, ∞ = 2∞.

@bob Oops, I meant a 0.

@anon I guess, if you want to think about it that way...

**rileywkong**- Replies: 3

Hi all,

Reading the many "0.999... = 1" topics, I suddenly realize that really, infinity is hard to grasp.

Laymen think of infinity as the "largest number in the number system". Meaning, if you count from 1, 2, 3, 4.... the last number you count to is infinity.

Of course, you can't count that high. No one can.

So, our friend Cantor devised a clever notation: aleph-null, or ℵ₀ (the first letter of the Semitic alphabet, btw), denoting the cardinality of the natural numbers (what I said above). And, this extends to 0.999..., for example. 0.999... has aleph-null trailing 9's behind it.

What happens when we multiply 0.999... by 10? We get 9.999... But Wait! It's actually 9.999...8! After aleph-null 9's, there's an 8! (edit: I meant 0.999...0 . )

You may think that spells the end of the commonly-used proof of 0.999... = 1. But no. Aleph-null + 1 = aleph-null.

Hence, the proof is sound.

Now what if we extend aleph-null to include pi, e, 1/2, and so on? We get aleph-one, or ℵ₁.

I'll leave that to you to muddle over.

Cheers!

What about both?

Moreover the Lord answered Job, and said,

Shall he that contendeth the Almighty instruct him? he that reproveth God, let him answer it.

Then Job answered the Lord, and said,

Behold, I am vile; what shall I answer thee? I will lay mine hand upon my mouth.

Once I have spoken; but I will not answer: yea, twice; but I will proceed no further.

For classical probability:

2 fair dice are rolled. If the sum of the 2 dice is called x, what is the probability that x is

1) prime

2) x mod 3 = 1

?

Hmm. I can't change it.

I mean for you to

1. disable BBCode

2. remove the quote marks surrounding the brackets.

**rileywkong**- Replies: 6

Hi all,

This is my first topic, having lurked here for about 4 months.

I'd like to share things I find on the Net here and just math(s) in general.

Have a great day!

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