Seeking a reasonable mathematical explanation to a simple mathematical conundrum

[cheenusj]

Hi everyone,

I came upon a simple mathematical conundrum a few decades ago that has irked me for a while now and for which I don't have a reasonable mathematical explanation.

The only reasonable "explanation" I currently have is that 'certain mysterious things happen when tending towards infinity', whether infinitely big or infinitesimally small, say for example triangle angles summing to more than 180 degrees in infinitely large non-Euclidean triangles.

The problem is just one equation and I am hoping you have either heard of this "problem" or have a reasonable or better explanation than "mysterious things happen towards infinity".

Say, x = 0.9... (tending towards infinite, i.e. I don't have the "bar" to go on top of the "9" for mathematical notation).

Multiply both sides of the equation by 10 and you obtain 10x = 9.9... (tending towards infinity).

Subtract the latter from the former and since the ".9..." tending towards infinity knock themselves out, you have 9x = 9 (10x-x = 9.9...-0.9...), therefore x = 1.

However, "x" started off at x=.9 tending towards infinity, not x = 1.

Since multiplication is a series of additions and substraction being a negative addition, the process of multiplying something out, then doing a substraction and then simplifying it should always leave everything as it first started, without changing anything, i.e. you should end up with what you started with.

Taking a set of 'simpler' examples, this is easily observable and the above example seems "idiotic" (pointless), because the mathematical operators used will always result in ending with the same x-value that one started off with.

I also seem to find a "pedestrian" explanation that seems to illustrate the head-scratcher that this is for me.
One-ninth is 0.1 tending towards infinity, i.e. 1s going on forever.
Two-ninths is 2 times one-ninth, therefore I can easily derive that each "1 digit" behind the decimal point can be multiplied by 2. Therefore, two-ninths is 0.2 tending towards infinity.
I can keep incrementing and this is always correct until eight-ninths which is once again 0.8... (tending towards infinity). But then what is nine-ninths? Obviously it's "one", but if I used the logic above, one could easily argue that each 1-digit after the decimal is multiplied by 9, resulting in 0.9 tending towards infinity, which is wrong.

Since this "issue" has been bugging me for a while and I don't have any other contcts in academia or the field of mathematics, I was wondering what explanation and what "mistake" is held in the logic of the above "equation" and operations?

Many thanks for your thoughts.

Kind regards,

Cheenu
London, U.K.

 

[Greg]

9 * 1/9 = 1. If we have 0.999... the number 0.000...1 can be added to 0.999... to obtain 1. What happens to 0.000...1 as the number of places between the 1 and the decimal point increases without bound? One could say that decimal expansions of rational numbers sometimes leave something to be desired.

 

[MountEvariste]

However, "x" started off at x=.9 tending towards infinity, not x = 1.

Yes, but isn't that what you were trying to prove? You wouldn't have achieved anything if you ended up with $x= 0. \overline{9} $.

Since multiplication is a series of additions and substraction being a negative addition, the process of multiplying something out, then doing a substraction and then simplifying it should always leave everything as it first started, without changing anything, i.e. you should end up with what you started with.

I don't understand this logic. If I start with $x = \cos(0)$ is it wrong to end up with $x=1$?

Or let's say if I start with $x= \sqrt[3]{7+5\sqrt{2}}+ \sqrt[3]{7-5\sqrt{2}}$ is it wrong to end up with $x = 2$?

As long as you follow the rules of whatever field you're working with, you can end up with a different forms of the same thing.

 

[tkhunny]

Say, x = 0.9... (tending towards infinite, i.e. I don't have the "bar" to go on top of the "9" for mathematical notation).

0.999... does not mean "tending towards" anything It means there ARE infinitely many. It NEVER quits. Your entire dilemma stems from your assumption that it is somehow finite. It isn't.

I always find it useful to ask, If 0.9999... is NOT equal to 1, then how far from 1 is it? You cannot answer this question. Anything you pick, another can show how it is not that far.