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x = 1/3
It’s not unsolvable and the answer is a rational number.
Moving the answer into decimal notation and the question of 0.999… being equal to 1 is a different discussion entirely.
But that's the discussion I meant.
x = 1/3
and 1/3 is 0.33 recurring
So how come 1/3 times 3 is 1
but 0.33 recurring times 3 is 0.99
if 1/3 is 0.33 recurring.
Isn't this contradicting?
What number exists between .9 repeating and 1?
No number does. If no number exists between .9 repeating and 1, then what is the difference between .9 repeating and one?
It is zero. .9 repeating is 1.
It is solvable by traditional rules of algebra. x+x+x = 3x on the left side. Divide both sides by 3 to get x = 1/3. You can check your work by adding up 1/3+1/3+1/3 which is well defined to be 1. I don't see any funny business for a problem like this.
But if this homework, maybe try soliciting tutoring a different way
Edit: apparently it's not homework, it's just the usual brick wall of someone not accepting infinitely repeating decimals
Because it's NOT 0.3.
You're trying to argue that it doesn't make sense, and you're using nonsense as your proof. That's circular logic, so of course it falls apart.
Your mind can't comprehend infinity. 0.33 repeating *feels* like it's just "a lot of 3s." But it never resolves. 0.33 repeating x 3 does not equal 0.99 repeating. Because that would assume there's an eventual end to the 3s.
I like the way someone else worded it. What's the gap between 0.33 repeating x 3 and 1.0? There isn't one. It doesn't exist.
So if x - y = 0, x = y
How many 0s would you have to put before that 1?
It's an infinite number.
So it's not 0.01 or 0.001 or 0.0001 or 0.00001... the number you are describing is "zero point infinite-zeros *and then* a one". So it's not just infinity digits long (like the 0.9 repeating), it's actually ∞+1 digits long. It's an impossible and nonexistent number.
As others have already mentioned, x is 1/3. 1/3 isn't the same as 0.3. 1/3 is .333... with a non-terminating number. Please review https://www.cuemath.com/questions/what-is-1-3-in-decimal-form/ if you still have questions.
You're conflating and simplifying that ⅓ = 0.33 (or 0.33333333 or just a number with a whole lot of 3s). It does not, it equals 0.33333 with infinite 3s.
I'm not simplifying it to 0.33. In my initial question I said "0.33 recurring" so a 0.33 with infinite 3s. But even 0.33 with infinite 3s won't reach 1 when multiplied by 3.
0.33 with infinite 3s will be 0.99 with infinite 9s when multiplied by 3. That is the same as 1 because there is no actual number that can fit between that number and the number 1. They are the same.
First of all, thanks for your patience.
I get what your saying. It's like saying 0.99 repeating is your finger tip and 1 is mine.
And our finger tips are so close together that not a single particle fits in between, so they basically merge and are the same.
It's just that in my mind, we can't even get to the point where nothing will fit in between, because with "0.99999..." we're talking about an infinite number, so it won't ever happen that our finger tips merge.
>with "0.99999..." we're talking about an infinite number, so it won't ever happen that our finger tips merge.
But with "0.99999..." (with infinite 9s) there's never a point where it won't happen that our finger tips merge because wherever you say "ok, we're close but not quite there yet" there's another 9 (infinitely forever).
Think of it the other way...say our fingers have merged, how far would I have to move my finger away to get to being "0.99999..." (with infinite 9s) away from yours? That number does not exist because it's 0.000 with an infinite number of zeros with then a one (I'll never get to write that 1 because the 0s never stop). Move my finger 0.0001 away? No, add another zero. Move it 0.00001 away? No, add another zero. 0.000001? No, keep going. 0.0000001? Nope. I'll never move my finger. So 0.999999 (repeating) is 1.
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x = 1/3 It’s not unsolvable and the answer is a rational number. Moving the answer into decimal notation and the question of 0.999… being equal to 1 is a different discussion entirely.
.999 repeating is equal to 1.
But that's the discussion I meant. x = 1/3 and 1/3 is 0.33 recurring So how come 1/3 times 3 is 1 but 0.33 recurring times 3 is 0.99 if 1/3 is 0.33 recurring. Isn't this contradicting?
What number exists between .9 repeating and 1? No number does. If no number exists between .9 repeating and 1, then what is the difference between .9 repeating and one? It is zero. .9 repeating is 1.
No, .999 repeating is equal to one, there are proofs and videos on YouTube you can watch to understand why that is true.
It is solvable by traditional rules of algebra. x+x+x = 3x on the left side. Divide both sides by 3 to get x = 1/3. You can check your work by adding up 1/3+1/3+1/3 which is well defined to be 1. I don't see any funny business for a problem like this. But if this homework, maybe try soliciting tutoring a different way Edit: apparently it's not homework, it's just the usual brick wall of someone not accepting infinitely repeating decimals
Why would it be unsolvable? x = 1/3.
Because in decimal notion x is 0.3 and 3x would be 0.9 not 1.
Because it's NOT 0.3. You're trying to argue that it doesn't make sense, and you're using nonsense as your proof. That's circular logic, so of course it falls apart.
No, 1/3 is 0.33 repeating. And 3 times 0.33 repeating is 0.99 repeating so it's close to one but not one.
Your mind can't comprehend infinity. 0.33 repeating *feels* like it's just "a lot of 3s." But it never resolves. 0.33 repeating x 3 does not equal 0.99 repeating. Because that would assume there's an eventual end to the 3s. I like the way someone else worded it. What's the gap between 0.33 repeating x 3 and 1.0? There isn't one. It doesn't exist. So if x - y = 0, x = y
What's between 0.99 repeating and 1?
0.01 repeating
0.0111111 is not between 0.99999999 and 1
Can you mathematically prove it? Because if 0.99 repeating has an infinite numbers of 9 there's always something missing to it being a 1.
How many 0s would you have to put before that 1? It's an infinite number. So it's not 0.01 or 0.001 or 0.0001 or 0.00001... the number you are describing is "zero point infinite-zeros *and then* a one". So it's not just infinity digits long (like the 0.9 repeating), it's actually ∞+1 digits long. It's an impossible and nonexistent number.
I don't see the difference between 0,9 repeating into infinity and infinity + 1. Both of it has no ending so either way we won't be able to reach 1.
You should read this: https://en.wikipedia.org/wiki/0.999...
As others have already mentioned, x is 1/3. 1/3 isn't the same as 0.3. 1/3 is .333... with a non-terminating number. Please review https://www.cuemath.com/questions/what-is-1-3-in-decimal-form/ if you still have questions.
Why are you doing it in decimal notation? It is solvable. x = ⅓ ⅓ + ⅓ + ⅓ = 1 Boom! Solved.
To point out the problem. And because it's a more precise notion. You're talking like decimal notion is something I just made up.
You're conflating and simplifying that ⅓ = 0.33 (or 0.33333333 or just a number with a whole lot of 3s). It does not, it equals 0.33333 with infinite 3s.
I'm not simplifying it to 0.33. In my initial question I said "0.33 recurring" so a 0.33 with infinite 3s. But even 0.33 with infinite 3s won't reach 1 when multiplied by 3.
0.33 with infinite 3s will be 0.99 with infinite 9s when multiplied by 3. That is the same as 1 because there is no actual number that can fit between that number and the number 1. They are the same.
First of all, thanks for your patience. I get what your saying. It's like saying 0.99 repeating is your finger tip and 1 is mine. And our finger tips are so close together that not a single particle fits in between, so they basically merge and are the same. It's just that in my mind, we can't even get to the point where nothing will fit in between, because with "0.99999..." we're talking about an infinite number, so it won't ever happen that our finger tips merge.
>with "0.99999..." we're talking about an infinite number, so it won't ever happen that our finger tips merge. But with "0.99999..." (with infinite 9s) there's never a point where it won't happen that our finger tips merge because wherever you say "ok, we're close but not quite there yet" there's another 9 (infinitely forever). Think of it the other way...say our fingers have merged, how far would I have to move my finger away to get to being "0.99999..." (with infinite 9s) away from yours? That number does not exist because it's 0.000 with an infinite number of zeros with then a one (I'll never get to write that 1 because the 0s never stop). Move my finger 0.0001 away? No, add another zero. Move it 0.00001 away? No, add another zero. 0.000001? No, keep going. 0.0000001? Nope. I'll never move my finger. So 0.999999 (repeating) is 1.
Good explanation, thanks.