Yes and no! I wrote a program on a C64 to try and see if there was a pattern that could be used to solve Fermat's Last Theorem. I got Honorable Mention which is behind 3rd place. LOL Math was one of my strongest subjects. When I applied myself, I did reasonably well in college. I am slow and sloppy and that is part of what did me in, in engineering. I have the hours and grades for minors in Electrical Engineering and Math. My official minor is English. My degree is in Computer Science under the science track. The other option was business but you might as well have taken MIS rather than that option. Math is just a tool. I neither love or hate it.
This guy is great. He is a mathematics professor at Cal-Berkeley and I always enjoy listening to him.
I am giving you the real number, and trying to explain it in a way that helps you understand. The real number between 0.9.. and 1 is 0.9.. It is all the real numbers between itself and the next number. It holds that internally. And the way you can see it is by use of precision, if you can’t see that it is itself the number between itself and the next number.
This makes zero sense. How can a number be between itself and another number? And just to be clear, you are aware that .999... or .9~ are different notations for the same thing, and that .9~ is the same as .99999~ and is the same thing as .9999999999999~? It is simply a short hand notation that .9~ is the decimal followed by all nines, in all decimal places. We cannot write it out because there are an infinite number of them, so we just short hand it with .9~. I don't mean to be condescending here, but I think we have a definition problem or a problem what infinite number of 9s means, especially when you use words like 'precision'. And if this is the case, we can bark at each other all day and never come to any kind of true argument because we are talking about two different rule sets. Do you not believe that .9~ is . followed by an infinite number of 9s? Or is there some ambiguity past however many 9s I manually write out?
By occupying all available decimal points between itself and the next closest number, is how a number can be between itself and another number. Any decimal point infinitely repeating is a special number that is both between itself and another number, because it is infinitely repeating. That's what the repeating means, it's repeating the decimal. And no, 0.9.. and 0.9999.. are not the same thing. They have the same value. They are not the same thing. One is more precise than the other. And precision matters, especially here. I absolutely believe that 0.9~ or 0.9.. or 0.9 bar, or whatever notation we're going to use today, is 0 and then a decimal point, and then an infinite number of 9s. Absolutely. And I absolutely believe that since you can't write out an infinite number of 9s, that you use 1 as the place, because 1 is the closest value you have to work with, and will give you the most accurate answer. But it is not the same. Just very, very close.
I was answering it in the manner in which it was asked. A quick search and it looks like in 2010 it was out to 2.7 trillion places.
One is not more precise than the other. It is just a notational difference. It is like saying pi and π have the same value but are different, and some how arguing that you cannot replace one with the other. .9~ and .999~ are identical in every way, one is just easier to write and the other just gives it a bit more eyeball candy.
I can't discuss what is "standard thinking" vs "correct thinking." You are right, everything you are saying is standard convention. That doesn't make it correct.
Pi has an infinite number of decimal places, but since it is irrational, it never repeats. Thus, for pi, there is a level of precision necessary. However, the point on the Real number line for Pi is static, never moving. Our precision as to where it is exactly can improve, however, but never fully tack it down. .9~, however, is rational and repeats ad infinitum. It is precise because I declare to you that every place after the decimal has a 9 in it. Every. Single. One. By definition, I am telling you what it is. Pi I cannot do that with as it is non-repeating.
They are not identical in every way. One has three precise decimal points, and the other has one precise decimal point. They have the same value, they are not the same. Is 0.9~ rational or irrational?
