Does calling 2/3, .67 or 67%, bother you? It’s just rounding to how many decimal places you want. Once a number is repeated in a quotient, it means to it will extend to an infinite number of decimal places.
I agree with you mathematically. I don't agree linguistically. There is a difference.
No there's not.
Not trying to sound snarky but you obviously never took calculus because under your logic you would "disagree" with all of it.
No, I literally said I agreed mathematically. I did take calculus and understand that in that context "infinitely close to 1" is the same as 1.
My point was that in human language that is not how we see things. I wanted to point out that logic in language and logic in math do not always overlap...
Does calling 2/3, .67 or 67%, bother you? It’s just rounding to how many decimal places you want. Once a number is repeated in a quotient, it means to it will extend to an infinite number of decimal places.
Again..
2/3 is
--
.6
Sixes to the heavens and beyond
.67 is overestimating 2/3 because it's based on 33/34/33 which is not an equal third
Not trying to sound snarky but you obviously never took calculus because under your logic you would "disagree" with all of it.
No, I literally said I agreed mathematically. I did take calculus and understand that in that context "infinitely close to 1" is the same as 1.
My point was that in human language that is not how we see things. I wanted to point out that logic in language and logic in math do not always overlap...
Then you should understand asymptotic math . 1/3 and 2/3 are similar to asymptotes to the axis. A parabola or hyperbola can breeze by the axis but never touch it.
Does calling 2/3, .67 or 67%, bother you? It’s just rounding to how many decimal places you want. Once a number is repeated in a quotient, it means to it will extend to an infinite number of decimal places.
Again..
2/3 is
--
.6
Sixes to the heavens and beyond
.67 is overestimating 2/3 because it's based on 33/34/33 which is not an equal third
.3 bar and .6 bar are asymptotic values...my other two posts disappeared...is asymptote a bad word?
No, I literally said I agreed mathematically. I did take calculus and understand that in that context "infinitely close to 1" is the same as 1.
My point was that in human language that is not how we see things. I wanted to point out that logic in language and logic in math do not always overlap...
And the point is that .999999... isn't "infinitely close to 1". It's literally, exactly, completely the same as 1. There is no difference between .9999999.... and 1, not mathematically and not linguistically. The fact may be surprising or irritating, but it is still a fact.
I love this proof. I don't really agree with it, but it is right, nonetheless.
OK, then look at the number ...333333333.0 (infinitely many 3s on the left of the decimal point rather than the right).
Eq #1: x = ...33333333.0
Eq #2: 10x = ...333333330.0
Subtract Eq #1 from Eq #2.
9x = -3. Divide both sides by 9.
x = -3/9 ----> x = -1/3
That seems odd.
Haha fun counter point, however.. the 3s on the right of the decimal point converge on a real number (adding exponentially smaller numbers to a number) whereas on the left to infinity (adding exponentially larger numbers to a number). Basic arithmetic doesn't really work with infinity.
...3333.0 is infinity.
As is:
...3330.0
Or
...1111.0
It's all the same value, doesn't matter how you get there.
Not trying to sound snarky but you obviously never took calculus because under your logic you would "disagree" with all of it.
No, I literally said I agreed mathematically. I did take calculus and understand that in that context "infinitely close to 1" is the same as 1.
My point was that in human language that is not how we see things. I wanted to point out that logic in language and logic in math do not always overlap...
I love this proof. I don't really agree with it, but it is right, nonetheless.
OK, then look at the number ...333333333.0 (infinitely many 3s on the left of the decimal point rather than the right).
Eq #1: x = ...33333333.0
Eq #2: 10x = ...333333330.0
Subtract Eq #1 from Eq #2.
9x = -3. Divide both sides by 9.
x = -3/9 ----> x = -1/3
That seems odd.
It does seem odd. But odd things sometimes happen. In this case, you are doing infinity-infinity.
Another well known example of mathematical oddities is the sum of all natural numbers. What would you guess 1+2+3+4+5+6+7+8+9+10+11... =? I bet you never would have guessed -1/12. But its true, and is even in some physics textbooks.
OK, then look at the number ...333333333.0 (infinitely many 3s on the left of the decimal point rather than the right).
Eq #1: x = ...33333333.0
Eq #2: 10x = ...333333330.0
Subtract Eq #1 from Eq #2.
9x = -3. Divide both sides by 9.
x = -3/9 ----> x = -1/3
That seems odd.
It does seem odd. But odd things sometimes happen. In this case, you are doing infinity-infinity.
Another well known example of mathematical oddities is the sum of all natural numbers. What would you guess 1+2+3+4+5+6+7+8+9+10+11... =? I bet you never would have guessed -1/12. But its true, and is even in some physics textbooks.
What do you mean by equal? Consult Srinivasa Ramanujan Aiyangar, if you dare.
For those wanting "linguistic" precision: The elements of the sequence {0.3, 0.33, 0.333, 0.3333, 0.33333, ...} become infinitely close to the number 1/3. None of these elements equals 1/3, but the elements get closer and closer to 1/3 the further out we go in the sequence.
The decimal 0.3333333... is used to represent the *limit* of this sequence, which is equal to the number 1/3. This is an elementary mathematical fact.
For those wanting "linguistic" precision: The elements of the sequence {0.3, 0.33, 0.333, 0.3333, 0.33333, ...} become infinitely close to the number 1/3. None of these elements equals 1/3, but the elements get closer and closer to 1/3 the further out we go in the sequence.
The decimal 0.3333333... is used to represent the *limit* of this sequence, which is equal to the number 1/3. This is an elementary mathematical fact.
Finally the one completely correct statement in this entire thread...
No, I literally said I agreed mathematically. I did take calculus and understand that in that context "infinitely close to 1" is the same as 1.
My point was that in human language that is not how we see things. I wanted to point out that logic in language and logic in math do not always overlap...
And the point is that .999999... isn't "infinitely close to 1". It's literally, exactly, completely the same as 1. There is no difference between .9999999.... and 1, not mathematically and not linguistically. The fact may be surprising or irritating, but it is still a fact.
Well, I do understand the math (1 - .9999... = 0.0000...) but that doesn't work in language. Luckily the philosophers and linguists are not captives of the mathematicians.
Remember, Ludwig Wittgenstein was King and Ernst Mach was only his acolyte.
p.s. I am loving this thread. My roommate in college was a math major and we went over this many, many times. He was a philosophy minor (and 3:54 1500m runner), so we had a lot of running miles to talk about this stuff.
This post was edited 4 minutes after it was posted.
OP wants a formal sense in which 0.333... and 1/3 are equal. Here is one.
One definition for equality between x and y is as follows: for all e greater than zero, |x - y| < e. Why is this a definition? (1) If x and y are equal, e.g., x=1 and y=1, this clearly holds. (2) If this holds, note that if x =\= y, there would be some e > 0 where the statement failed, so x must equal y. Thus if x and y are equal, this statement holds (1); and if the statement holds, x and y are equal (2).
Now we'll show x = 0.333... and y = 1/3 satisfy this definition. Pick any e > 0. A little informally, it's clear that for some decimal place, z = 0.3333...333000... has |z - 1/3| < e. Because z < x, |x - 1/3| < |z - 1/3|. Thus, |x - 1/3| < e.
(The answer earlier using a geometric series works too, but that the series equals a/(1-r) actually relies on the logic expressed above.)
The key word is "equal." No, it does not equal .3333...
The reason is that "infinitely close" is the same same as "exactly."
It has, as usual, to do with definitions.
If you were an ant marching down a number-line towards the number 1 and you were at .999, you'd never "get there" if you could only add more 9s to the end of the decimal.
That is what 1/3 is, so sorry, it is not the "exact same thing."
But you (or your teacher) might define "equal" differently than I do.
1/3 does equal exactly 0.3333...
If you aren't sure about this, I suggest you check out some real analysis.
Sort of makes sense to me but I'm not a mathematician.
How does one show this?
Not an answer to your question, but it's interesting because the three dots or ellipsis doesn't mean 'recurring' where I'm from. I'm assuming that's what your question is about
We use a dot above whatever digits are recurring - ie 3.34 with dots over both 3 and 4 is 3.34343434 etc.
But I have never seen ellipses used to show recurring digits before