One divided by 3 equals zero point three recurring

[Stylesjl]

And zero point three recurring times three equals, one!?

Isn't there something inherently illogical in that proof? 0.3 recurring times three should equal 0.9 recurring instead of one. Or is there some funny rule that rules this isn't the case?

Because it is also illogical for a number to divide by three but upon multiplying by three not return to exactly where it started

Isn't mathematics weird?

And another thing: Dviding by Zero, often considered impossible, but could it be considered to be infinity rather then a non-existant answer?

 

[batteryacid]

it is not a funny rule, it is 1
when calculate the difference between 1- 0.999... you will get 0.0000... ergo are both equal

 

[bigfatron]

And zero point three recurring times three equals, one!?

Isn't there something inherently illogical in that proof? 0.3 recurring times three should equal 0.9 recurring instead of one. Or is there some funny rule that rules this isn't the case?

Because it is also illogical for a number to divide by three but upon multiplying by three not return to exactly where it started

Isn't mathematics weird?

And another thing: Dviding by Zero, often considered impossible, but could it be considered to be infinity rather then a non-existant answer?

0.9 recurring is not equal to 1, but it is infinitely close to it as an approximation. To prove this, take any point in the recurring sequence, say 0.9999, and consider the difference between that number and 1, in this case 0.0001. Pick a difference that you consider to be small enough, I can always identify a point in the recurring sequence where the difference between that number and 1 is smaller than your choice.

0.9 recurring is the number where the recurring sequence has infinite terms, therefore it must be infinitely close to 1 and is mathematically equivalent.

For dividing by zero, the issue is that division by zero always results in inifnity as the answer - as a result the usual mathematical rule, that:
a/b = c/b <=> a=c
breaks down where b=0.

Since we rely on the rule above very frequently in mathematical calculations it is very important to ensure in large computations that the number you are dividing by is not 0 or very close to zero, as it may distort the reuslt or even make it completely meaningless.

A long, long time ago my Op Research professor told me that of the £20m spent on mathematical calculations for a nuclear power plant in the early 70's, over 1/3 was spent proving that none of the calculations were invalidated by 'divide by zero' fallacies. I believe commercial number-crunching programmes now do this by default, but I'm way out of date on this stuff.

BTW, maths is great!
BFR

 

[Bill3000]

Isn't there something inherently illogical in that proof? 0.3 recurring times three should equal 0.9 recurring instead of one. Or is there some funny rule that rules this isn't the case?

0.9 recurring equals one. Said funny rule is simply the fact that the limit as n goes to infinity of 1 - 1/10^n = 1. It's the same exact number.

Dviding by Zero, often considered impossible, but could it be considered to be infinity rather then a non-existant answer?

Only in a number system in which -infinity = +infinity. And even then, 0/0 is still undefined.

Isn't mathematics weird?

Nope!

 

[shortguy]

For dividing by zero, the issue is that division by zero always results in inifnity as the answer

It's no more infinity than it's negative infinity, if we're talking non-zero numbers in the numerator. And if we have 0/0, then the answer (such as it is) could be anything.

 

[BasketCase]

0.9 recurring is not equal to 1, but it is infinitely close to it as an approximation. To prove this, take any point in the recurring sequence, say 0.9999, and consider the difference between that number and 1, in this case 0.0001. Pick a difference that you consider to be small enough, I can always identify a point in the recurring sequence where the difference between that number and 1 is smaller than your choice.

The source of the OP's confusion, I believe, parallels this bit by BFR. The problem is that when you divide one by three, the answer ".3 recurring" is itself an approximation. The exact answer is "one-third". But .3 recurring never exactly equals one-third no matter how many 3's you tack on the end.

 

[Truronian]

And zero point three recurring times three equals, one!?

Isn't there something inherently illogical in that proof? 0.3 recurring times three should equal 0.9 recurring instead of one. Or is there some funny rule that rules this isn't the case?

0.9 recurring isn't a decimal number, its a limit written in shorthand... in this case the limit is 10. By your logic 0.3 recurring is never quite equal to a third...

Because it is also illogical for a number to divide by three but upon multiplying by three not return to exactly where it started

Its not illogical, its impossible.

Isn't mathematics weird?

At times, but your examples aren't very brilliant...

And another thing: Dividing by Zero, often considered impossible, but could it be considered to be infinity rather then a non-existant answer?

Its not considered impossible, it is impossible, as infinity is not a real number. In the extended complex plane, diving by zero is perfectly fine...

 

[batteryacid]

it is NOT approximately 1, it is EXACTLY 1 as I posted before the difference between 1-0.9 recurring is 0.0 recurring or simply 0.

0.3 recurring is also EXACTLY 1/3 and not approximately 1/3 . 0.3333 would be one example of being approximately 1/3.

Recurring means "infinitely repeating" and not "stopping after a finite distance after the decimal point" and is thus ALWAYS an exact expression for such fractions

 

[TheBladeRoden]

Decimals are flawed

 

[Lone Wolf]

it is NOT approximately 1, it is EXACTLY 1 equal to one as I posted before the difference between 1-0.9 recurring is 0.0 recurring or simply 0.

0.3 recurring is also EXACTLY 1/3 and not approximately 1/3 . 0.3333 would be one example of being approximately 1/3.

Recurring means "infinitely repeating" and not "stopping after a finite distance after the decimal point" and is thus ALWAYS an exact expression for such fractions

He's right. 0.9 recurring = 1. Enough said.

 

[Leifmk]

0.9 recurring is not equal to 1, but it is infinitely close to it as an approximation.

Incorrect, "0.9 recurring" is exactly, identically, equal to 1. There is no approximation.

"Infinitely close" does not make sense in the standard real number system, by the way.

 

[Leifmk]

The source of the OP's confusion, I believe, parallels this bit by BFR. The problem is that when you divide one by three, the answer ".3 recurring" is itself an approximation. The exact answer is "one-third". But .3 recurring never exactly equals one-third no matter how many 3's you tack on the end.

You are utterly and completely wrong; it is not an approximation, it is shorthand for a limit notation, and ".3 recurring" does exactly equal one-third.

 

[StarWorms]

1/3 = 0.3333...
0.3333...*3= 0.9999....
1/3 * 3 = 3/3 = 1
0.3333... * 3 = 1

0.999 recurring is 1.

 

[Leifmk]

1/3 = 0.3333...

Given this (and ignoring your typo where you omitted the ... the last time), the rest follows trivially.

The problem, for those who do not understand the argument, is that they do not understand that that is what the notation "0.333..." actually means, or indeed how the real number system works in general (and with limits in particular).

 

[Atticus]

The whole thing is about definition of real numbers. More educated people here know the real definition of real numbers and see presentation 0.999... as a limit of series 0.9+0.09+0.009+.... But on the other hand the less educated don't understand it, because they think that 0.999.. is a real number, not a limit.

Most people seem to have implicit definition for real numbers: "they are all the decimal representations, such as 34.882 , 0.333.... &c". They rarely even doubt validity of that 'definition' becuse they are never asked what the (real) numbers are, and they take it for granted that there is no confusion about it.

While there is a good definition for real numers, it takes time to explain and some familarity with maths. For people who aren't that interested in maths it suffices just to make two corrections to that naive definition above:

1. Two decimal presentations can represent the same real number. For any positive number x holds
-x< 1-0.999... <x, (see previous posts)
so the number 1-0.999... can't be negative nor positive, hence it must be 0. (Note that only numbers that can have two presentations are those who'se other presentation ends with recurring 9).

2. We must allow presentations which have infinitely many decimals and doesn't end with dots. Otherwise there wouldn't be such real numbers as pi or squareroot 2. Many find this condition disturbing because you can't write down such numbers. Partial comfort is that if you are assured that there is a infinetely long decimal presentation for squareroot 2, you can just give new name to that presentation and note it with [squareroot sign] 2.

Some people are surely irritated by this new 'definition', and many for good reason. They should just study more maths: if you want good explanation, you must be sometimes ready to accept long explanation. The above mentioned 'new definition' is only a definition for those who don't want to throw away all of their initial naive definition.

 

[nihilistic]

The source of the OP's confusion, I believe, parallels this bit by BFR. The problem is that when you divide one by three, the answer ".3 recurring" is itself an approximation. The exact answer is "one-third". But .3 recurring never exactly equals one-third no matter how many 3's you tack on the end.

You see, .3 recurring implies that there is an infinite number of 3 tacked behind that decimal point. Incidentally, it is 1/3.

Decimals are flawed

No, it's just that not every number has a unique decimal expression. Most numbers do, as most numbers are transcendental.

 

[Nylan]

This is what the use of significant digits is for.

 

[Gogf]

(1) .9999... = x (Given)
(2) 9.9999... = 10x (Multiply by 10)
(3) 9 = 9x (Subtract equation #1)
(4) x = 1 (Divide by 9)

 

[warpus]

And zero point three recurring times three equals, one!?

Isn't there something inherently illogical in that proof? 0.3 recurring times three should equal 0.9 recurring instead of one. Or is there some funny rule that rules this isn't the case?

Because it is also illogical for a number to divide by three but upon multiplying by three not return to exactly where it started

Isn't mathematics weird?

And another thing: Dviding by Zero, often considered impossible, but could it be considered to be infinity rather then a non-existant answer?

You don't understand irrational numbers, which is why this seems weird and illogical to you.

 

[Fugitive Sisyphus]

1. For every two distinct numbers A and B, there is a number C such that A<C<B.

2. There is no number such number C, such that .999(repeating)<C<1.

3. Therefore .999(repeating)=1