Going all the way back to the beginning on this one, phyllo, but your post inspired me to consider something:
What would this question look like in base 9?
In base 9, 1/9 isn’t 0.11111…, it’s 0.1.
Which means in base 9, your equation 1/9+1/9+1/9+1/9+1/9+1/9+1/9+1/9+1/9 would just be .1 + .1 + .1 + .1 + .1 + .1 + .1 + .1 + .1
And, in base 9, 0.1[size=50]9[/size] * 10[size=50]9[/size] (where 10[size=50]9[/size] = 9[size=50]10[/size]) obviously equals one.
So we have a situation where, in order to deny your proof of .111_ + .111_ ... = .999_ and also = 1, someone has to maintain the position that a value that is trivial to represent in a base 9 decimal number is impossible to represent in a base 10 decimal number. I’m not saying here that I think that’s a clearly false position, but I am saying it’s a very interesting position. That changing which base you’re in changes which values are fundementally possible to represent. It’s a position everyone who voted ‘no’ on the OP is forced into taking.
Yes exactly, in base 9 the value we NORMALLY in base 10 represent as 1/9 (or .111…) would then be represented as 1/10 or 0.1
So the people who voted “no” have to take the position that that value is not possible to represent in a base 10 decimal, but it is possible to represent in a base 9 decimal.
That’s all I was pointing out. I think that’s a position worth making explicit.
That’s correct. And it shouldn’t be seen as a strange position in the least given that “base-n system” refers to a narrow set of mathematical expressions.
Since not two things can be absolutely identical, no one thing can be the same as any other one.
1=1 can only work in abstract terms, and for each time we nominate an orange or a coin as ONE, their actual value can only be abstract.
Maths is not discovered as if it were lying under a rock, it is a system of description by which we measure and quantify the world.
SO when the question come up is ONE equal to 0.9999999999999999999999999999999999999 recurring, we should not be surprised that a abstract system we choose to impose on reality can end up asking odd and insurmountable questions, such as how many decimal places has PI, or what is the suqare root of minus ONE.
You are confusing two different statements. You are confusing “Every physical object is identical to every other physical object in every aspect” with “One is equal to one”. The truth value of the former does not affect the truth value of the latter.
There is no such thing as “working in abstract terms”. At best, you’re saying that the statement is false and only approxiamtely true (close to being true without being true.) But that’s not the case.
“Their actual value”? We’re asking how many oranges/coins are out there. Such questions cannot always be answered using reason alone. How many white socks are there in Xi Jinping’s drawer? You have to make some observations. Ideally, you should go his room and count his socks. That’s an empirical process.
That’s true but it’s completely irrelevant.
We invent and then use language in order to decribe reality. We come up with sentences that aren’t reality itself but that nonetheless describe some aspect of it. Even “2 + 2 = 4”, which is a highly abstract statement, does exactly that. It describes a portion of reality (and it does so accurately.) It’s basically saying “The expression 2+2 and the symbol 4 have one and the same meaning within the modern day language of mathematics.” Demonstrating this for statements such as “There are 3 bottles of wine on Ecmandu’s table” is even easier. The fact that every human language is a human fabrication does not mean that statements employing those languages have nothing to do with reality. It’s like claiming that humans invented the word “tree” so any statement employing that word has nothing to do with reality.
wtf has come closest to defending the idea that the two are equal albeit without success in the end. His argument rests on the premise that the idea of “the result of an addition of an infinite number of terms” is undefined and thus in need of being defined.
When you say 1 apple = 1 apple, you are not claiming that the apples are identical. The statement 1 = 1 is a statement about quantity, and that the quantity of apples on the left side of the table (1) is equal to the quantity of apples on the right side of the table (1). The quantity of apples is EQUAL, both have the SAME quantity of apples.
If there were 3 apples on the left side, and 4 apples on the right side, then the quantities are NOT EQUAL.
Nobody ever claimed that 1 = 1 means that both apples are the same exact color, or size, or flavor. What they are claiming is that a quantity of 1 apple is equal to another quantity of 1 apple.
1 is a NUMBER to represent a QUANTITY.
The unit of measure “apple” is a loose term, meant to cover a variety of different objects that we call “apples.” Nobody ever claimed all apples are equal.
If you want to get more technical about it, then it’s the UNIT OF MEASURE that needs to be clarified. Red apples, or green apples. Granny Smith Apples…
1 Inch
1 Mile
1 Meter
1 Foot
1 Shoe
1 House
It’s the unit of measures that the quantities are referring to. Some units are meant to be loose terms, and some units are very specific.
1 Red Apple is not equal to 1 Green Apple, because the unit is more specific than just the generic unit “apple.” The quantities are equal, the units are not.
Don’t be surprised if you have to repeat that another 50 times with no one changing their mind. The only thing that changes in this thread is the vote count.
I observe the same people coming back weeks, months, years later making the same claims regardless of how many times their arguments have been clearly shown to be false.
Learning is a funny thing. Sometimes you can explain it 9,000 different ways, and they don’t get it. Then all of a sudden a light bulb moment happens, and BAM, they are overjoyed with their newfound knowledge.
I used to teach brand new mechanics in the Army. I can see it in their eyes when they finally get it. A good teacher can get people to learn. A bad teacher blames the student for not learning. A teacher’s job is to teach! That means conveying knowledge until the student “gets it.”
There are tons of barriers to learning. It is very very rare that someone doesn’t learn because they are too stupid. Most times students don’t learn because the teacher sucks!
