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Then it should be:
If you pick a ball, record its color, then put it back before picking another time, the result would be 1/36, regarding the sequence.
Randomly pick two, not putting back?
Exactly! The universe is exact, our measurements are approximations. So when we do mathematics, should we be exact, or only close enough that we can base engineering on it?
-Well, I find you are really a Platoist- The real world is imperfect, but the concept of it could be; Every chair is just an approximation of the concept of chair, which is exact.
It is philosophic stands that underly our disagreement.
Hi I am Chinese. Nice meeting you!
I have received tons of junk mails and now nearly every page I visit has a banner " you are the 999,999th visitor of this page~"
Okay another paradox involving infinity or infinitesimal.
One grain of wheat is weightless. (No weight)
Two are also weightless.
...
1 million grains are weightless.
But how can a pile of wheat so heavy as to bend a camel?
The only way to solve this paradox is to admit the premise of a weightless grain is wrong.
Or, as some mathematicians would like, a pile of wheat consists of Infinite grains.
But the new paradox would be-how can they abtain one grain from infinite grains? by reducing it? by getting a portion of?
A rational is an integer divided by a non-zero integer.
3.33333333333...... is the real equivalence of 1/3. Reals could have infinite digits to accomodate both rationals and irrationals at the same time. However, I find contradiction within infinite digits. If you ignore this contradiction you can accept reals and regard 0.333... indeed the equivalence of 1/3.
How do you pick?
Stanely_Marsh uses the standard way.
4) First 6[sup]6[/sup] possible basic outcomes(ways), and each outcome share the same probability of 1/6[sup]6[/sup]. Agree?
Now we need only to count how many basic outcomes coincide with a 6 before a 1.
Excluding the ways without 6 or 1, we can substract this number from the total number of ways:
4[sup]6[/sup]
Because of the symmetry of 6 vs 1, we can predict the ways of 6-1 and 1-6 are the same. Strictly, whenever there is a way of 6-1, there is definately a way of 1-6 created by swapping the two, vise-versa. So they share the same amount of ways.
Thus for 1-6 or 6-1
there are
(6[sup]6[/sup]-4[sup]6[/sup])/2 ways.
And each way has the probability of 1/6[sup]6[/sup]
So the possiblity of all the ways satisfying 1-6 or 6-1 together should be
(6[sup]6[/sup]-4[sup]6[/sup])/2 (1/6[sup]6[/sup])
Binary choices. It can never be binomial probability because it is acctually larger than 1.
Uh! My answer is 11 under 5 steps. But your problem is under 6 steps.
So for 6 steps:
2=4×1+2×(-1)
-4=1×1+5×(-1) (like A-C-B-A-B-A-C, including a circle A-C-B-A (-3) cancelling itself and a repetition A-B-A(0) cancelling itself)
To make 2 displacement (equivalent to -1 displacement), there are M ways. M is the number of combinations to put 2 B's into the row of 4 A's (where A represents 1 and B represents -1)
_A_A_A_A_
Now fill two seperate B's onto the _'s or two adjecent B's (BB) onto the _'s
So
If one combination turns out to be BA_A_ABA_, that means
-1+1+1+1-1+1=2
or A-C-A-B-C-B-C
Now let's deal with the ways to arrive at -4 displacement. 1 1's and 5 (-1)'s mean to put one B among the row of 5 A's. Like this:
_A_A_A_A_A_
six places to choose, thus 6 ways.
Altogether there are 15+6=21 ways.
However, I cannot work out some 2^6/3. Nor can I find the underlying rule for it.
Correction:
specil->special
recent surveys->recent studies
"You already agreed that the length of a particle is not a constant value (uranium is "longer" than hydrogen). Also, the particles are constantly moving and vibrating. Again, you attempt to ignore these very basic facts."
--Initially the defination of One Metre is the length of a bronze ruler. The ruler was kept by the Franchmen in Paris. Every metre defined in other countries was the duplication of this ruler. Do you mean just because microly the particles of the ruler is constantly viberating, the defination of the metre is wrong?
Indeed, microly changing distants do not cause too much macro problems-the particles of the words now you read are moving. However, temperature would be a problem-how did they deal with the expansion subject to temperature? They further defined the length under 20°C is the real length of one metre. Wait, how did they define 20°C? They used mercury. They defined the height of the mercury in a thermometre dipped in a mixture of water and ice to represent 0°C, and the height when it is in boiling water to represent 100°C. Then the differiential of the height is devided by 100 equal intervals-wait, how can we divide such intervals? By compasses, perhaps. Again the intervals are constantly changing.
Then technology improved. We have got laser beams.Now one metre is defined according to the length a specil laser beam could travel within a given time. The time is defined by radioactive decay. The metre, however, is still not accurate. Because 1) the light does not always travel in the same speed, according to recent surveys 2) the light does not travel in absolute line due to the gravity, according to Einstein.
Perhaps the most ratinal thinking about measure is that it cannot be absolute exact. The exactness only exists in some mathematicians' imagination.
It is not that I don't like irrationals or reals, but that I believe their existence is inferior to logic consistency.
My solution:
Define the original point as 0, a clockwise movement as 1, and a counter-clockwise movement as -1.
Apparently we need to move 5 times and reach C, which one movement -1 can reach.
Just by adding each movements up we get the destination, where we should notice an arbitary times of 3 could be substracted or added to find the equivalent destination. For example, 1+1+1+1+1=5, 5-3×2=-1, so the destination is C.
Generally, we can get these possiblilities of a sum of 5 movements that can be represented as 3n-1.
5=1×5;
-1=1×2-1×3
The first possiblity, of course, involves only one way. But the second allows many combinations of 1's and -1's. Moreover, we should realise that consecutive the same 1's or -1's have no order. For example, 1 1 is the same to the swap of it, and there is only one way.
To count the combinations of 2 A's and 3 B's (A=1,B=-1), we only need this method:
fill out 2 A's on the _ below-
_B_B_B_
Four blanks to choose 2 from, or 1 from(AA put together).
The number of combinations of the second case thus is:
Sorry it is not correct yet.
Why mine is always 1.6seconds??? cannot accept it...
Ok, so back to the original question. Show me that the length of your desk is not irrational.
Simple, because the length is basically an integar, the length of any line made of particles is rational, or put another way, discrete. You may compose a right triangle by actually two line segments of 1,000,000,000 particles, then you wanna find the longest side length. You may use the ruler made of the same material, then find the side of 1,414,213,562 or 1,414,213,563 particles, to be the most precise.
This would, for sure, cause some problem of imperfection in regard of the length. But it would be a lesser problem than regarding 1,000,000,000(√2) particles composing the side.
Um, why is it that you are raising 10 to the infinity?
Again you don't understand what you mean. Well, first I would like to explain that writing 0.000... is not just a game. Nor does it simply mean zero. It infact represents 0(1)+0(1/10)+0(1/100)+...+0(1/10^∞). The simble of infinity may scare you, but it is true when you say infinite digits, and if you like, we can replace it by the super quantity q, as what I used in a previous post. Then how can I add them together? By unifying the denominator. So the addition simply turns to 0/10^∞, here I could add 538/100 in, making the notation of the one you don't like.
Or simply put, the infiniteth digit means a time of 1/10^infinity. Astounded? You creat it.
Initially people had integers. Then they had rationals, litterally one integer devided by another. Later decimals was used. Decimals are indeed helpful to guage a complex rational, but imediately people found lots of times a rational cannot be turned into finite decimals. Then some smart guy had the idea of recurring to trace the pattern for roungding conveniency. But the further idea of infinite decimals is misleading. Infinite, litterally means endless and irreachable came into use.
It is very funny to accept an no-end as an end. How many digits does an infinite decimal contain? Larger than any integer.
What is larger than any integer? Cannot figure, but is no integer , no limitation.
But how does an infinite decimal stand static rather than forever growing (1 decimal, 2 decimals, 3 decimals and so on)?
By simply an amount of digits larger than any amount.
The game is fun!
Right, currently what we could do is reduce the use of energy as we can and opt fuel-efficient or non-gasoline vehicles.
Interesting. Because all particles, even in solids, are moving. Also, the volume (and thus, length) of a solid depends on the temperature. While these are minute differences, you still must take them into account.
It is not simply that atoms are balls laying there. They are very active things, moving around. Your calculations do not take this into effect.
Yes, I admit that would be a problem. But things in the world may be not static as we used to think-we used to think distance is distance, time is time-but now we no longer think so.
I argue that there is no such thing as finite decimals. That is, 5.38 is really 5.380000000...
Then, there is no "going" to infinite decimals, since everything is. There is no gap because there is no difference.
It is really interesting to see how you have distorted a rational
into the "number" you like:
And you may call it evolution.
Anthony, the key is to deny infinitesimals, because of the 10×0.999...=9.99... "proof".
Note the maths symbols on top of this page, copy them to your post
Simple: If we define one metre on the desk. Then we can examine how many particles the 1 metre has passed through. Let's denote the amount of particles as N. Then how much length is a given distance on the desk? Just count again, and get the number M. Therefore the length we want to know is M/N metres.
Yes, I do have problems with Real as well, which defines numbers as infinite sets, and the defination of many infinite decimals, the root of 2, e and pi, to name a few.
Before this sentence-
"It is with this in mind that we make the following definition"
you do the proof within a finite framework
To be precise, you proved the Series r is always increasing, which is r[sub]n+1[/sub]>r[sub]n[/sub]
After the sentence and the defination, you use the concept of infinity (or infinite decimals).
Once you use the concept of infinite decimals, you cannot explain how the finitth decimals goes to infiniteth decimals. That is quite simple- between finite quantities and infinite "quantities" is an untrancendable gap.
George, as promised, I will be answering your posts either tomorrow or Saturday, so don't worry, they won't go ignored. I ask you (as a friend in a debate, not as a moderator) not to post until I have, otherwise I fear there will just be utter confusion.
Till now, you have disagreed on decimal expansion on the real numbers, and I always figured the problem was higher up, for example definition of the convergence of an infinite series, which would intuitively come after a consensus of real numbers and decimal expansions have been reached. Let me address this now.
I believe we all agree on the following things:
1. A decimal expansion of any real number r takes the form of:
2. For any real number r, given an epsilon (e), we can find a finite number of decimal digits d such that:
But we have problems with this. Specifically we can't write all real numbers or even all rational numbers with a finite number of digits. So what do we do about this?
One thing is to do nothing. Accept this failure of decimals. Personally, I feel unsatisfied with this. Is there no way we can save decimal expansion of infinite decimals? Certainly, we can't "see" infinitely many decimals.
But wait! We have this really cool method of dealing with infinite things invented/discovered by Cauchy et. al. Limits. They allow us to handle infinite sequences of numbers. And in the end, isn't this all we're dealing with?
So let's just explore this possibility for the time being. How can we use limits? Certainly it may not be possible, but lets just give it a try.
For any real number r, let us define a sequence of real numbers (using the same notation as before):
It should be clear that:
For any integer n. So this sequence is monotonely increasing, as well as bounded. By the monotone-bounded convergence theorem, this sequence must converge. As we (hopefully) agreed in #2 (see above), it does in fact converge to r.
So where are we at? Given any real number, we can use the decimal expansion to approach it, and get arbitrarily close, a limit per say.
It is with this in mind that we make the following definition:
Note that the notation above simply means an infinite amount of decimal points.
The results of this definition are the following:
1. No contradiction with any math that I'm aware of
2. Every real number has at least one decimal expansion equal to it.
3. Some decimal expansions are not unique.3 can be considered a problem. I certainly do. But I argue (with only opinion, not pure logic) that having multiple decimal expansions are a lesser problem than not being able to represent some real numbers with a decimal expansion.
An immediate consequence of this definition is that 0.999... = 1.
Most people accept this definition by intuition alone. You're the first person I've ever seen which did not.
Edit: I don't think my signature has every applied to any one of my posts greater than this one.
Specifically we can't write all real numbers or even all rational numbers with a finite number of digits.
-glad you admit this
we can't "see" infinitely many decimals
-the problem I proposed in this thread is that infinite decimals, if existed, would have contravasory among themselves. Not as simple as we cannot see.
it does in fact converge to r.
- do you interpret "converge" or abitarily proximate as "reach"? The same old topic again.
an infinite amount of decimal points.
-again contravasory and no sense.
The most important thing I insist is that an amount that you can tell as a determined number, such 2, 3.5, has no way to transform into either infinitesimal or infinity. Infinite sequences are a dilema as well. Infinite sequences mean only you can add in new entries without a stop.
If you mean an infinite series itself can be defined as a decimal expansion, then infinite digit numbers are at least non-static.
George, with my definition on what an decimal expansion is, it does in fact make sense.
You first assumed r to be a decimal expansion that can has infinite decimals.
As I had disproved, the concept of infinite digits itself is plausible, but you assumed it.
Your defination just hided what you cannot prove inside the premise and then you "proved" the result you want.