.999... =/ 1
I dont understand what you mean to say with that. Can you explain?
Also, aren't decimal numbers more accurately represented as a fraction of some multiple of 10?
Not really. 1/100 or 0.01 has the same accuracy. If you want to avoid rounding numbers you often keep the fractions. I higher math we usually keep all the fractions and rarely use a decimal writing style, it helps you see things.
For the upper one, the odd expression in the middle is an ascii representation for the 'does not equal' sign. So just...
.999... does not equal 1
1/100 DOES express the decimal as a fraction of a multiple of 10. Isn't that more accurate (since decimal numbers can ONLY represent multiples of 10 by their nature) than saying .333... = 1/3?
No, there isn't such a thing as more accurate. In math you are either accurate or you are not and as long as you dont round your decimals, decimals are accurate. 1/3 is exactly 0.333... and 0.999... is exactly 1. 100% accuracy on both.
Well, I decided it was time to do some much deeper reading on this subject, and quite honestly I'm REALLY surprised I haven't stumbled into this conundrum before.
My personal opinion is that it is more of a proof that the number .9... is an imaginary number. I'm also of the opinion that .3... isn't the same as 1/3, it's actually 1.../3..., and likewise .9... would be 9.../10... I didn't read far enough to find out if there's more elegant notation for this.
My argument is this: These proofs are crossing an 'event horizon' between imaginary and real numbers in order to work. I don't know how the math or physics communities feel about calling 'infinite' numbers 'imaginary' numbers, but I don't suppose either community will be swayed much by my opinion about it. 😁
I don't think they would feel great about it since we already have a fixed meaning when we talk about imaginary numbers it is the number i for which i²=-1 is true. Imaginary numbers can also be multiples of i but it is mostly i.
Periodic numbers like 0.3... are actually rational numbers - a really tame beast. You would have to go two steps above to come to the imaginary numbers. In between are irrational numbers like square root 2 and pi.
Rats, I forgot all about irrational numbers!
Infinity is a tricky concept, like zero. I suppose people will always have arguments about ideas that human heads don't comprehend well.
Thank you for the education, and re-education, no need to educate me further on this, I won't remember it anyway! Thanks for the engaging and humbling discussion!
you are welcome. It is always hard to break certain walls in math, once you are through, you dont even know what kept you on the other side.
No, this is the kind of wall that I would never break through, I would just accept that it was there and work around it, or I would get lucky enough to come up with the math to express it more perfectly. Imaginary numbers were easy for me to accept, for some reason. Irrational numbers... let's just say there's a reason I forgot about them!
I would use a the geometrical series to proof the equality to 1
...and now the language is getting outside of my knowledge. This is why I don't post in STEM communities 😂