Math Is Fun Forum

Tricky!

6x may do.

Since you cannot accept verbal logic, I can just provide some proof with reference.

"The division algorithm requires you to keep going until you reach 0.  Since you will never reach 0, you will always keep going.  Just because I didn't take the next step doesn't mean that it isn't flat out obvious I will get a 1, it is.  In fact, I don't keep going because I always know what I will be getting, more 1's."

So are you using a guess as your proof? A guess cannot make up a proof no matter how obvious it looks, as you can consult to your math teacher.

If you are not using a guess, I assume you were using a deduction proof, agree?

A deduction proof can reach any step that you can can assign a sequence number. Step 1000, Step 198745 for instance. However, it does not give crevidence to Step Infinity, as you could consult to any textbook introducing deduction.

Then how is the infinity situation been proved? I'm afraid to say it is not proved unless you assume something first.

My points are:
A)
0.999...=1 is not proved unless a new assumption is added, such as one that infinite digits can make up a number while this number share more or less the same +-*/ rule with integers and rationals or one that 1/9 is assumed as 0.111... with infinite ones.
B)
You can reject this kind of assumption when you find it is of little use in reality or it contradicts reality when you believe you study math for its usefulness rather than its complexity. Accepting the assumption is not a must.

I agree with you that math is not alway a reflection of reality. I propose math is made of assumptions and their logical conclusions. That is why I've spent so much on figuring out an assumption within the proof.

Good question.

First they assume a polygone with infinite sides and call it circle. Then they calculate polygones with finite sides and try to approach the circle, and state that they can write out as many digits of Pi as long as they've measured enough polygons. At last they say it's a state of Pi with infinite digits, which is the ratio for a circle.

I doubt such state do exists, and their logic from the 2nd sentence to the 3rd.

In real life, your cup is not round but a polylinder with many sides. But Pi is a good approximation for a multi-polygon's length and area.

Another example might be we use e to approximate reproduction process.
1 -> 2 -> 4 -> 8... =2^t=e^(tln2)
We write the way on the end with e and ln cancelling each other out simply for the simplification reason.

I am sorry to remind you that "your proof" was first proposed by me, in an answer to your induction request in the other thread discussing 0.999...

Today's Base systems are built coherently infinite digits belief and real system background such as 0.999... So you provided a in-built proof.

My proof was stated clearly at the begining, yet I can show it more obviously here:
Cannot be devided at any digits=>Can be devided or expressed at infinite level
--false logic. You need to assume that there is indeed an infinite level first, and assume the result.

Since when is math ever based on what we see in the real world, George?
Since ancient times, Ricky. Since we humans first identified 2 apples.

Let us see how maths interpret the real world and practice. When you divide a pizza by nine pieces, you can say you get 9 more or less same ninths. You do get the chance of getting them same when the total amount of particles is devided by 9. But let's say what happens when you state 0.1111... first you divide it by 10, get 1 tenth, since 1 ninth is larger than 1 tenth but less than 2 tenths. So you pick up another 1 tenth, and devide it smaller by 10. Here you find that 1 ninth is larger that 1 tenth plus 1 hundredth but less than 1 tenth plus 2 hundredths and you scrible down 0.11 and go on. These process are what 1/9 and 0.111...work in practice.  The difference between the former and the latter is that the latter implicitly holds a forever divisiblity assumption, that a pizza can be devided smaller and smaller without limitation, which has been tested wrong ever since the invention of microscope.

"If that isn't an infinite number of 1's, I don't know what is."

Yep, you match what I said-you just proved as many 1's so long as many steps. What I disagree is the infinite digits proposal, that because you cannot express it in finite digits, you guess that there is a state when infinite digits do exist, and make some seemingly reasonable rules to it.

The reason why I am against 0.999...=1 is that it's trying to equate some entity human never being able to testify on the left with something out of common experience on the right which is touchable in day-to-day life , on an assumption-hidden means. The main purpose, I guess, is to stablish absolute correctness for  results by infinite ways.

Do you remember the exact defination of a limit near some points? Note the difference in the domains of x and y- y can reach y0, while x is not able to reach x0. Mr Leibniz's intention was very clear, as he tried to avoid the logic flaw of the reached point. (Search for "Berkley vs Newton" for more details)

Modern Mathematicians are so confident that they provide such kinda false proof, where  they embeded reached infinity assumption such as infinit digits implicitly, trying to verify maths' correctness and accurateness, and its ability to gain a stable result, although none of them is able to write out 3.14159... or 2.7818... wholy.

Thanks for your advice, I know base systems, and I agree with you that is artificial. Human ideas are indeed artificial, some of which belong to religion, some belonging to arts, and some belonging to science. I guess the unique charactor of the last one mentioned is its courage to risk enough to be tested wrong  but not yet.

Poor Pluto! He no longer has a name!!

I've got some progress, and I wish you can put the finishing touch.

Firstly, notice 1-cos(2t)=2sin[sup]2[/sup](t)

Thus you get the chance of cancelling sin[sup]2[/sup](t) in the denominator and the numerator together.

And the crucial step is to express the denominator as some product with sin[sup]2[/sup](t).

the second and the last part are ok, but what about the first one?

use a[sup]3[/sup]-b[sup]3[/sup]=...

sin(3t)-sin(t)=2sin(t)cos(2t) -- one sin(t)

sin(3t)^2=?

Use Euler formula and find sin(3t) expressed as trig of t. In this way you can complete another sin(t)

sin(3t)=3cos[sup]2[/sup](t)sin(t)-sin[sup]3[/sup](t)

Euler Formula

Not necessarily. You have to take both the revenue and the cost into account and find the profit peak.

yeah mathsyperson is very thoughtful, I used to judge it should be all.

Still unavailable. I don't know what goes wrong- whether it be the internet jam or my government monitoring. Can you recommand some good proxy softwares ( free better) ?

Besides, I never managed to enter  wikipedia, but I heard it has a secure access starting with https://, yet  don't know the detail.

I still wonder how and why...

Social scientists as well as economists tend to categorize things into several groups according to their measurability. Here are measurability standard:
1. Can be categorized or can be classed . That means you can categorize some of them as As and some as Bs and so on. for example, cats and dogs. A and B is enough to simblize them, 1 and 2 are ok but brings on ambiguity.
2 Can be compared which means the categories are grouped according the extent of one property. for example, a person's height, experts grading.
3 Categories can be added freely and the added result is the simple sum of the added two.
4 Have an absolute zero value meaning null or nothing.

A variable satisfying 1 and 2 is called a cardinal variable or grade variable.
A variable satisfying all of the 5 is ideal, but rarely available in social science .

Natural numbers satisfy all 5 and definitely has absolute grading. 5 is larger and extremer than 3 for sure.