How do we arrive at whole numbers?

Question

for example: consider a random number, 2.15. now one can add infinite numbers after the decimal point. like u can add to 2.15 and make it 2.157. then how do we have number 3, if we never reach it?

maybe i am making some fundamental mistake. but it just occured to me , consider 2.999, you can keep adding 9's to this , and though tending to 3 , one does not actually reach it. can you point out the mistake i am making, because i agree i am making little sense cause i am not able to put this in words. thanks for ur answers

Answer 1

I'm not sure what you are asking, because you can't add any digits to 2.15 to get it to equal 3.

Why?

2.15abcdefg . . . ≤ 2.1599999999999. . . = 2.16 < 3

Likewise
2.157abcd . . . ≤ 2.1579999 . . . = 2.158 < 3


I get what you are talking about now. There are a number of ways of seeing that 0.999999 . . . =1 (thus 2+0.99999. . . =2.99999 . . . =3). First of all, you can't stop putting the 9's at the end of the number, that's what the ". . ." means, if you do, then it is not equal.

Here is why (one reason):

first notice (take my word for this) that (x-1)(x^n+x^(n-1)+ . . .+x^2+x+1)= x^(n+1)-1. Therefore 1+x+x^2+ . . . + x^n =(x^(n+1)-1)/(x-1).

Now take the limit as n goes to infinity (where 0≤x<1):

1+x+x^2+x^3+ . . . = lim (1+ . . . x^n)=lim (x^(n+1)-1)/(x-1) = 1/(1-x). This happens because lim x^(n+1) = 0 since 0≤x<1:

Ok, now if you didn't understand all that, just skip to this part (of course you are just reading this now, so you wouldn't have known to skip, so that's bad advice :))

0.99999 . . . = 0+9/10+9/100+9/1000+9/10000+ . . . = 9/10 (1+1/10+1/100+1/1000 . . .) =9/10(1+1/10+(1/10)^2+(1/10)^3+ . . .)

Let x=1/10, then we have 0.999999 . . . = 9/10(1+x+x^2+x^3+ . . .) = 9/10(1/(1-x)) =9/10(1/(1-1/10)=9/10(1/(9/10) =9/10(10/9)=1

There is a really easy way to see this though (and when stated formally would be a proof in topology):

There is no number between 1 and 0.9999 . . . Since there is no number between them, they must be the same number.

Answer 2

We started with whole numbers. We were given 10 fingers not 10.12 fingers. So it wasn't until the caveman from Geico commercials got part of his finger caught under a boulder that we got fractions. He then had 9 2/3 fingers.

Answer 3

I'm not sure if this will completely anwser your question but it's rather neat.

1/9 = .111111....
2/9 = .222222....
.
.
.
8/9 = .888888...


8/9 + 1/9 = 9/9
or
.88888.... + .111111 = .99999999

So 9/9 = .99999999...

We also know that 9/9 = 1

So 1 = .9999999....


The answers you seek are in calculus. There is a concept called a limit, and that's how this stuff works out.