The Man Who Knew Infinity (2015)

The Man Who Knew Infinity (2015)

Baekhyun // 160917 Infinity Challenge #4

**Taylor:**I've figured out why you're such a jerk. You have updog.**T2:**What's "updog?"**Taylor [yelling out the door]:**Arika, get in here! I told you I could do it!

Listen

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**hollandville** on instagram recorded the part where tom basically confirms that he’s filming for infinity war !!!!

kagome “i’m tired” higurashi

**MC:**Why is there a shopping cart in our front yard?**Saeyoung:**You said that your heels were killing you but you didn't want to take them off because that would make your feet dirty... so I agreed to push you home~**MC:**Let's get married at the space station!

inktober days 30-31

Admiralpotato, “Q*bert Club”, Utah Museum Of Contemporary Art (UMOCA).

**T2:**You’re stupid.**Taylor:**That’s it?**T2:**Give it time. It’ll eat at you.- [later]
**Taylor:**Am I stupid?**Lifeline:**Yeah, a little bit.**Taylor [mentally]:**Damn them!

Hum Hallelujah // Fall Out Boy

escort and hwan busts

hi hi hi ! it’s em but hm i’ve never done one of these before haha so imma give it a shot. but i’m here to do a follow forever !

wowow i’ve only had my blog since november 2015 and i already have almost 300 followers hehe thank you ! when i first joined tumblr,,, i had an emo blog and i was a sad smol but it had turned into full kpop by december and now i’m a very happy carat lol. hm its been a fun year for me on tumblr, i’ve made so many new friendos and its just been a good time hehe. everyone is so nice it just makes me so happy :’)) anyways, , , here’s my follow forever….

(also watch out bc i might start posting my fanart soon :)

*italicized* - mutuals

**bold** - blogs that i really enjoy

♥ - some really good friendos that i lov

[#]

*@05-lunar* || *@6yl *|| * @8xu *||

[A-D]

*@adoree-choihanso*l || * @adorexuminghao* ♥||

[E-H]

*@everyday-boom-boo* || *@glasseswonwoo* || *@hcshi *|| * @heoni* ♥♥♥♥♥||

[I-L]

*@iaminlovewithiwa-chan* || *@ileanaaxc* || *@ilovewoozi* || *@iluboo ♥♥*|| *@impaledbyfeels *|| *@inchanyeolspanties* || *@jeonghangel1004 *|| *@jeonghanie* || *@jeonnguks* || *@jhopefull-y* || *@jinhong* || * @jisoosmeoli* ||

[M-P]

*@mainlyawkward* || * @manuty *||

[Q-T]

*@red-jeonghannie* || *@santahosh* || *@seokmywoozi *|| *@seoulleater* || *@seungcoupie *|| *@seventeenjoshuatrash *|| *@seventeensgf* || *@seventeenruinedmylife* || *@sicat *|| *@simply-jihoon* || *@singercheol *|| *@spacebinch* || *@spooklezi* || * @soonhosh* ||

[U-Z]

* @up10chen* ||

anyways, , , thank you for a great year with this blog :’)) i lov u guys mucho

“same-sized” infinities

The idea of “two sets are the same size iff there is a bijection between them” leads to the interesting property that you can extend the idea of size to infinite sets.

Here are illustrations of how two infinite size sets can have the same size, some of which might seem surprising if you are only used to finite sizes.

Some sets I will use as examples are integers Z={…,-2,-1,0,1,2,…}, positive integers Z^{+}={1,2,3…}, non-negative integers Z^{*}={0,1,2,3…}, the non-negative even integers 2Z^{*}={0,2,4,6,…}, the positive rational numbers Q={p/q : p,q are positive integers and q is not 0}, and positive real numbers R^{+}, which include the irrationals.

**Z ^{+} and Z* are the same size**

The function is f:Z*->Z^{+}, f(x)=x+1

**Z ^{*} and 2Z* are the same size**

The function is f:Z*->2Z^{*}, f(x)=2x

**Z ^{*} and Z are the same size**

The function is one where we split Z* into evens and odds, and then take the evens to the negatives and the odds to the positives

**Q on the interval [1,infinity) and Q on the interval (0,1] are the same size**

The function is f:Q[1,infinity)->Q(0,1], f(x)=1/x. (We can also use a similar function to show any interval of real numbers has the same size as the entire real number line)

**Z ^{*} and Q* are the same size**

This one is a bit trickier. We will show that Z* is the same size as the cartesian product Z*×Z*, and notice that Q* is a subset of Z*×Z*, so cannot be larger than it, but Q* contains Z*, so it cannot be smaller.

First, we split the cartesian product into lines based on the sum of the two numbers in each pair

Let t(n) be the n^{th} triangular number, or the sum of all non-negative integers below and including it. Notice that t(p+q) is the number of pairs with sum below p+q, because each row where the sum is m has m-1 elements.

We construct a function f:Z*×Z*->Z*, f(p,q)=t(p+q)+p

**R and R×R have the same size**

First, we restrict R to the interval [0,1), which has the same size as R. Then, we realize that we can represent each real number by its decimal expansion, which has a Z^{+} sized expansion. We can split every expansion into even and odd number places, and make a new number in R×R restricted to [0,1)×[0,1) using the two expansions to make its two numbers.

**T2:**Congratulations! You won a lifetime supply of tangerines!**Taylor:**But you only gave me one tangerine.**T2 [pulling out an Occupier]:**Yep!

women’s appreciation week day six: unfairly treated female character

korra;

i’ve realized that even though we should learn from those who came before us, we must also forge our own path

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