How can .3333333333 X 3 = 1?

Question

If you weren't to round it up to 1, you might say that the number 1 is impossible to reach and does not really exist. Maybe I'm overthinking it but I thought it was an interesting question.

Answer 1

Math professors will tell you that .333... x 3 = 1 because they claim that 0.333... = 1/3 at infinity. Isn't it quite strange how they can claim anything about infinity when they have no idea what it is eh? Okay, let's see what is correct about what math professors tell you:

It is true that the limit of .333... is 1/3. However, .333.... is not equal to 1/3. 1/3 cannot be represented exactly in base 10. Some professors will tell you that mathematics falls apart if 0.999... is not equal to 1. This is untrue because 9/10 + 9/100 + 9/1000 + ... is an infinite sum whose limit is 1. The actual sum is never 1, it is always strictly less than 1. What about the integral of f(x) = 1 from 0 to 1? Well, this is the area under the curve of f(x)=1 but here again, most math professors do not understand the fundamental theorem of calculus. The integral is in fact the limit of an infinite average sum which is quite different to the limit of an infinite sum which is not an average. There is no contradiction and no mathematics falls apart. All that falls apart is the nonsense they teach - Real Analysis. You can blame this topic which is mostly nonsense for incoherent claims such as .999... = 1. This subject came into existence long after calculus had been discovered. It was not around in the time of Archimedes, Newton, Gregory or any of the other great mathematicians. The father of real analysis is Weierstrass - a fool who contributed very little to Mathematics.

The subject of real analysis should be completely revised and until such time, it should not be taught at any learning institution for examination purposes. Even those who support it, do not demonstrate concrete knowledge and ability to support all their arguments and methods. There are definite contradictions and problems with the claims made in real analysis. Finally, you should remember that this topic was invented to silence the critics of calculus. It has failed and continues to fail. Good mathematicians do not need real analysis.

If anyone is talking crap, then it must be 'Mathematician':

It is never possible for one number to have two decimal expansions, unless you are referring to the zeroes that follow a number in which case there would be an infinite number of possibilities. Every representation in any radix system is unique and was designed to be unique. What sense would it make to have duplicate representation. Only A-h oles like mathematician will make claims like this because he knows nothing.

Limits have nothing to do with the definition of a number. Mathematician uses Cauchy sequences to define a number. Only problem is this definition uses numbers which have still not been defined. What does 0.9 mean? what does 0.99 mean?

A number is a concept based on the definition of a 'whole' or 'unit'. All numbers are represented in terms of this whole. How? Every number is a multiple of this whole or part of the whole or both. Yes, you i d i o t Mathematician - pay attention, you might learn something. Limits are a concept that arose from the upper bound property that sets of real numbers have. It started with Archimedes. We do not define a number by its upper bound but by a radix representation. This is how we have always used numbers - as approximations in base 10. Mathematician talks about a limit being the number but when asked what is the limit of a pi or e sequence, he sits with his thumb up his rear-end.

Answer 2

In most math there is negliible deviation from integers that are dropped.

The fact of the matter is that 99.999999% (joke) percent of the time you see a repeating .3, it is the decimal form of 1/3. In THIS case, it is not the 1/3 that is in error, but the .3333333 itself.

Think of it this way. Take a pie and divide it into three equal pieces. Each piece is 1/3. The best and most accurate way to describe it in decimal form is .3333 repeating, but that number is still flawed, it is a decimal interpretation of 1/3. You then have 3 of those .3333s and you put them together to get .9999, but that is just the original flaw in the decimal form being multiplied through. The fact is that 3 times 1/3 IS 1 -- no dispute is arguable there.

Rounding decimals then tries to fix this error, by now saying when you finish and round that last 9 up, it will change the entire problem back to 1.

Answer 3

First, it is important to understand what an infinite decimal expression means. When the expression
.3333...
is written, it means the limit of a certain sequence. Which sequence?
.3, .33, .333, .3333, .33333, .....
What does 'limit' mean? It means the number that the individual terms of the sequence are getting closer and closer to.

The problem is that there can be two different decimal expressions for the same number. This happens because of the way that limits work. For example, the sequence
.9, .99, .999, .9999, ....
clearly gets closer and closer to 1. The limit of this sequence is then 1, so
.99999....
equals 1. That is just how the expression is interpreted.

In the same way, .33333.... is the same as 1/3. In this way, 1/3 *does* have a decimal expansion (all real numbers do), but it doesn't have a *finite* decimal expansion. All that is meant by an infinite expansion, though, is the limit of the finite ones.

The limit concept is at the base of most modern analysis and of all of the calculus. There are no contradictions. The previous poster is just talking through his ***.

Answer 4

Yes 3 times .333 is .999 just like 3 times .33333333333333333333333 is .99999999999999999999999. 3 times 1/3 is 1. However 1/3 does not equal .333, 1/3 equals .333333333333333333333333333333333333333... The reason I put the ...... on there is because the 3s keep repeating. There is an infinite amount of 3s after the decimal point.

Answer 5

You have it backwards. Many numbers cannot be expressed in decimal form. 1/3 is such a number. So, .3333333333...(repeating) is the number that is rounded, not 1.

Now, since you actually asked about .3333333333 x 3, the answer is that .3333333333 x 3 = .9999999999. However, .3333333333 is not 1/3.

Answer 6

Do you know Limit?

If so, it will be easier to me to explain you this question.

Let, x = .33333333.........(upto infinity) X 3
Here, we can say that x approaches to 1(x->1). But x cannot be equals to 1. Okay.

At the same way .33333...... is not equals to 1/3. It is also approaches to 1/3.

Answer 7

.333333333333333333333 = 3/9, ok
and so you are basicly saying why is 3/9x3 = 1
so when you do the math you get 9/9=1
that becomes 1=1 because 9 divided by 9 is equal to 1 which is equal to one
hope that helped

Answer 8

So I think Tom and other "critics of calculus" should learn about "non-standard analysis" and the "hyperreal numbers". There *are* models for analysis in which you have infinitely small, infinitely large numbers. And I think this matches well what you have in mind. Then, in the future, you can tell us, for example, that our use of the real numbers is rubbish and that only the hyperreals make sense. And we will know exactly what you mean.

I've included what I found on wikipedia; I hope you can find better at your local library.

Answer 9

change 0.333333333 to a fraction

let x = 0.33333333.... ----- equation 1
therefore 10x = 3.333333333..... equation 2

equation 2 minus equation 1 :
9x = 3
x = 1/3

hence 1/3 times 3 = 1....
There you have it... the power of mathematics..

this is very right and i agree

Answer 10

change 0.333333333 to a fraction

let x = 0.33333333.... ----- equation 1
therefore 10x = 3.333333333..... equation 2

equation 2 minus equation 1 :
9x = 3
x = 1/3

hence 1/3 times 3 = 1....
There you have it... the power of mathematics..