A rational number is one that must show p/q, with a non-zero denominator. All repeating decimals are not rational. 0.00000000000000000 by the definition above is non-rational, and it is still represented by repeated digits of zero. 0 is an integer, so 0/0, or 0.0.. should be rational. But it isn’t, holding that the denominator q must be non-zero. There you have a special case. You can have more. Finish the definition, show p/q that can produce the digits .9999999.. for as many 9s as I want. You can’t. It cannot be done. You can do any number of other things. You cannot do this one thing. Demonstrate where this is true for any other rational number that doesn’t exhibit this same trait. Any single one.
Hi, I graduated from UT with a degree in mathematics, and I also have a Master's Degree from UF (and most of a Ph.D). .9 repeating is the same as 1. The confusion likely comes from the unstated assumption that all real numbers have a *unique* decimal representation, which is just not the case. As fl0at pointed out, even non-pathological numbers like 1 can be written in any number of ways, from 1, to 1.0, and so on. Two decimal representations are equivalent if they represent the same real number. Suppose I have two real numbers, a, and b. By definition of subtraction, a = b if and only if a - b = 0. Let a and b be decimal representations of said real numbers a and b. Then the same follows: a = b if and only if a - b = 0. To claim otherwise would be equivalent to saying that the - we use for real numbers and the - we use for decimal representations of real numbers must produce separate results, which is quite absurd. So, what exactly is 1 - 0.999... ? It is 0. This is not just an assertion, or a convention, it is a logical consequence of the way real numbers work, specifically the trichotomy property, which states that for any two real numbers a and b, either a<b, b<a, or a=b. By similar reasoning above, the < and = comparisons must produce identical results for real numbers and their decimal representations. Let's call the quantity 1 - 0.999... = D, for convenience. Is D > 0? If so, there would have to be another number, say D/2, which lies between D and 0. This is clearly impossible, regardless of how you might interpret the quantity D (an infinite string of 0s and then a 1, etc). Is D < 0? Clearly not. So we must conclude that D = 0. This is just one way of proving the equivalence, and there are many others. But note that none of what I used above relies on the calculus, or any advanced concepts. We are simply using basic properties of the real number system to infer properties about the decimal representations that we commonly refer to as "real numbers". Real numbers themselves are *not* their decimal counterparts, but something much deeper. A mathematician would refer to them as "*The* Complete Ordered Field", because they are exactly the only number system that satisfies all of the properties we want out of our numbers (availability of arithmetic operations, an ordering which respects those operations, and completeness, meaning that arithmetic operations only produce other numbers in the same system).
So do I. I am working on an iterative method for float to get .999... but it is difficult on a computer to write it out. I will say Complex Numbers are the more complete field for me over Reals. It allows for the Fundamental Theorem of Algebra.
They are algebraically complete, which is indeed nice. The thing that they lack is an ordering which is compatible with the field operations.
The number that lies between is itself, as there is no decimal greater than 9, and it occupies all decimal places. 0.9 > 0.99 > 0.999 > 0.9999 At each iteration, you are declaring the value of one more decimal place greater than the previous. If you do it infinite times, you are STILL declaring that the decimal place up to the infinite iteration is greater than the iteration at n-1 AND simultaneously looking at the number 0.9.. The number between 0.9.. and 1 is 0.9.. It is itself the number D/2, because it is INIFINITE in decimal places, and INFINITY / 2 is still INFINITY. In short, your degrees are worthless, and you also fail at critical thinking.
There is no notation to say what the value is of 1-0.9.. but the answer, if such existed would be 0...0..1 And anyone that knows regular expressions could probably show a simple one to answer it.
Alan, before you scratch out 1/1 in long division, shifting a decimal place and using 9 instead of 1, even though 1 is clearly the number you should use, do it with 8. And then tell me why doing it with 8 is wrong, but doing it with 9 is acceptable, when both are simply smaller numbers than 1.
Your relatives can wait. Entertain me! Kidding, obviously. No worries. I’m not looking at it til tomorrow.
It’s all convention. You can’t make .99.. equal 1 other than by saying it simply is, using convention. If it actually is, why bother with the decimals and notation and such? I’m with you float, but you’re arguing established convention.
I know. But they can still be good at math, love math, and have nothing change, AND accept that what they were taught isn't perfect. Everyone else does. It's time for mathematics to do the same.
Ok, last two logical ones. @TennTradition, since you inhabit both worlds, tell me: do you believe that monkey's typing on a type writer for infinite time will produce a work by Shakespeare? Of course you do. If I take one of those monkeys, and give him a piece of paper and a pencil and say, write 9s after this decimal place. And after each iteration of a written 9, which we will cause step n, stop and compare to n-1 for equality. The probability that the monkey will conclude that n-1 < n is 1, or 100%, correct? For time of 1, or time of infinity. If it goes to infinity, the number the monkey has written on paper is 0.9.. correct? Thus, 0.9.. has a number that is greater than itself and less than itself, and its itself. This is true for all terminal infinitely repeating 9s. It is true of 0.129999999.. and 87.36928999999 It is special in that it is the max decimal point, infinitly repeated, but everything else holds true. The decimal place at n - 1 or ..(-1) < n ..(0)
The second of the logical explanations. Suppose that we have a decimal system of 0123456789C, where after 9, C. The number then between 0.9.. and 1 would be 0.C Thus 0.9.. is now no longer special. However, 0.C.. is, because the number 0.C.. is now the terminal decimal, and now 0.C.. is now the number between itself and 1. It's that simple.
