I leave you to ponder this quote:

Suppose you wish to walk across the street. To get to the other side, you must go half of the way across the street. But first, you must walk half-way to the point that is half of the way across the street. But before that, you must walk half-way to the point that is a quarter of the way across the street. And before that, you must walk half-way to the point that is one-eighth of the way across the street. And so on forever. Because there are an infinite number of steps you must take to cross the street, you cannot cross the street.

It makes sense when you think about it!

tanmai79 Nov 24, 2018

#1**+2 **

Zeno missed a chance to make a profound (for the time) discovery about infinite series.

\(\text{let the width of the street be }L\\ \text{via Zeno's semi correct reasoning }\\ L = \dfrac{L}{2} + \dfrac{L}{4} + \dots = L \sum \limits_{k=1}^\infty~\left(\dfrac 1 2\right)^k \\ \text{i.e. } \sum \limits_{k=1}^\infty \left(\dfrac 1 2\right)^k = 1\)

.Rom Nov 24, 2018

#2

#3**+1 **

it turns out that if I take some number, call it x, and that number is between 0 and 1, then

\(1 + x + x^2 + x^3 + \dots = \dfrac{1}{1-x}\)

if as in this case x=1/2 we have

\(1 + \dfrac 1 2 + \dfrac 1 4 + \dfrac 1 8 + \dots = \dfrac{1}{1-\dfrac 1 2}= 2\)

and subtracting 1 from both sides

\(\dfrac 1 2 + \dfrac 1 4 + \dfrac 1 8 + \dots = 1\)

Now you know some math I bet your friends don't!

Rom
Nov 24, 2018

#4**-1 **

Most of my friends are doing 6th grade math, like is whats supposed to happen for me. But i'm in single subject acceleration, which put me a grade ahead in math. At my school, the double up program lets you do math for 2 periods, and I'm also in that, so we're almost done with 7th grade.

(I just said that to say that I already know a lot more math than my friends. It's an interesting math, uh, thing, though!(I don't know what else to call this other than a math thing))

tanmai79
Nov 24, 2018

#5**+1 **

here's an even more interesting fact!!!

when substituting x=1 in rom's equation, we get:

one plus one plus one.... an infinite amount of times=one divided by zero!!!!!

this equation is very important, because it proves that one divided by zero is in fact defineable, and that it's value is exactly 1+1+1+1....!!!! or in other words infinity!!

Guest Nov 24, 2018

#9**0 **

tanmai, here's a cool trick: go to the google calculator, and calculate "1/0". The calculator will give the answer "infinity"!!!

Guest Nov 24, 2018

#11**-1 **

that is technically impossible to solve, though. Google's calculator must be wrong!

tanmai79
Nov 24, 2018

#12**+1 **

how is this impossible??

here's an example:

person A: "hey listen, I was wondering what 1/0 is"

person B: "I'm glad you asked! 1/0 is in fact infinity!"

person A: "ah cool thanks man"

see?? solveable!!

this is the beauty of math!

Guest Nov 24, 2018

#13**-1 **

Wait, I just saw the guest response from earlier. It makes sense... I'll have to remember to tell that to my AoPS class (my teacher says it's possible to divide by 0 in calculus, but I'm only in an Algebra 1 class)

tanmai79
Nov 24, 2018

#15**+1 **

You can't ever divide by zero. The operation is undefined.

There are things called limits in calculus. You can calculate the limit as something you are dividing by goes to zero but you cannot ever actually divide by zero and get a usable answer, again the operation is undefined.

Hate to be mr. pedantic but this is misused too often to just let pass.

Rom
Nov 24, 2018

#16**+1 **

god, tan! I can't believe you fell for this guest's tricks! he clearly knows nothing about math!

Guest Nov 24, 2018