## Where Algebra Fails

Paidion
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Location: Back Woods of North-Western Ontario

### Where Algebra Fails

1) Let n = 3/2 + 9/4 + 27/8 + ...

Multiply each side of 1) by 2/3
2) (2/3)n = 1 + 3/2 + 9/4 + 27/8 + ...

Subtract n or its equivalent from each side of 2)
3) (-1/3)n = 1

Multiply each side of 3) by -3
4) n = -3

Therefore 3/2 + 9/4 + 27/8 + ... = -3

Where is the error?
Paidion

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willowtree
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Joined: Sun Nov 02, 2014 1:56 pm

### Re: Where Algebra Fails

Paidion wrote:1) Let n = 3/2 + 9/4 + 27/8 + ...
Multiply each side of 1) by 2/3
2) (2/3)n = 1 + 3/2 + 9/4 + 27/8 + ...

Subtract n or its equivalent from each side of 2)
3) (-1/3)n = 1

Multiply each side of 3) by -3
4) n = -3

Therefore 3/2 + 9/4 + 27/8 + ... = -3

Where is the error?

I don't think the rules of algebra are the problem.
It seems to me that n is an ongoing progression, the next one in line 1) would be 81/16, etc.
In order to keep the equation balanced you must determine how many progressions you want to solve for the value of n and not keep adding more as you go along.

Let n = 3/2 + 9/4 + 27/8 + ... (That is, after three progressions, n = 1.50 +2.15 +3.375 = 7.125)

So for the location of the error we must stop after 27/8 ( the third progression: you cannot just keep adding to one side of the equation and not the other).

Multiply each side of 1) by 2/3
2) (2/3)n = 1 + 3/2 + 9/4 + 27/8 + ... WRONG. (2/3)n = (only) 1 + 3/2 + 9/4
(limit to 3 progressions +27/8 adds another progression on the right and upsets the equation. )

Wish all theological problems were this easy.

Graeme
If you find yourself between a rock and a hard place, always head for the rock. Ps 62..

steve
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### Re: Where Algebra Fails

I knew that!

Paidion
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Location: Back Woods of North-Western Ontario

### Re: Where Algebra Fails

Willow, nothing has been added to one side and not the other.

Here is a valid proof based on the same algebraic principles:

Proof that 1/2 + 1/4 +1/8 + 1/16 + ... = 1

1) Let n = 1/2 + 1/4 +1/8 + 1/16 + ...

Multiply each side of 1) by 2
2) 2n = 1 + 1/2 + 1/4 +1/8 + 1/16 + ...

Subtract n or its equivalent from each side of 2)
3) n=1

Therefore 1/2 + 1/4 +1/8 + 1/16 + ... = 1

The series is an infinite sum, but it is still exactly equal to 1 and not a little bit less as our intuition tells us.
Any competent mathematician will tell you that this proof is valid.

Similarly, we shall prove that .999... =1 (and not a little bit less as our intuition tells us)

1) Let n = .999..

Multiply each side of 1) by 10
2) 10n = 9.999...

Subtract n or its equivalent from each side of 2)
3) 9n = 9

Divide each side of 3) by 9
4) n=1

Therefore .999... = 1
This is another valid proof. So why is the original "proof" that I offered NOT valid?
Paidion

Man judges a person by his past deeds, and administers penalties for his wrongdoing. God judges a person by his present character, and disciplines him that he may become righteous.

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willowtree
Posts: 100
Joined: Sun Nov 02, 2014 1:56 pm

### Re: Where Algebra Fails

Paidion wrote:Willow, nothing has been added to one side and not the other.

Here is a valid proof based on the same algebraic principles:

Proof that 1/2 + 1/4 +1/8 + 1/16 + ... = 1

1) Let n = 1/2 + 1/4 +1/8 + 1/16 + ...

Multiply each side of 1) by 2
2) 2n = 1 + 1/2 + 1/4 +1/8 + 1/16 + ...

Subtract n or its equivalent from each side of 2)
3) n=1

Therefore 1/2 + 1/4 +1/8 + 1/16 + ... = 1

The series is an infinite sum, but it is still exactly equal to 1 and not a little bit less as our intuition tells us.
Any competent mathematician will tell you that this proof is valid.

So why is the original "proof" that I offered NOT valid?

I give in, but just to give me some comfort, please show me how it works with the following:

Since it is is well established in mathematics that 1 +2 + 3 + 4 is equal to 4 + 3 + 2 + 1 would you please, using the same logic as above, convince me that n = 1 in this series.

let n = 1/16 + 1/8 + 1/4 + 1/2 + ...

Graeme
If you find yourself between a rock and a hard place, always head for the rock. Ps 62..

Paidion
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Location: Back Woods of North-Western Ontario

### Re: Where Algebra Fails

You haven't actually reversed the series, Graeme. The original series had 1/2 as the first element, and there is no element in the series larger than 1/2
In the series you offered, there are an infinite number of elements which are larger than 1/2. Also, the smallest element in your series is 1/16, whereas there are an infinite number of elements in the original series which are less than 1/16. However, you could reverse the series by putting the three dots at the beginning, if that will give you any comfort:

n= ...1/16 + 1/8 + 1/4 + 1/2

I think the algebra would still work fine with that one.

But I will point the following out to any who doubt that .999... =1

We have all learned in elementary school that 1/3 = .333... and 2/3 = .666...

Add 1/3 and 2/3 and you have 3/3 or 1.

.333...
.666...
.999...

Therefore .999... = 1

By the way, this has some interesting implications 8 = 7.999..., 13 = 12.999..., etc.

If no one figures out why algebra doesn't work (fails) in the original post, I'll give a reason in my next post which was given to me by some expert mathematicians.
Paidion

Man judges a person by his past deeds, and administers penalties for his wrongdoing. God judges a person by his present character, and disciplines him that he may become righteous.

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dizerner
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Joined: Fri Jun 27, 2014 4:43 pm

### Re: Where Algebra Fails

Math can assume an actual infinity for mathematical purposes, but don't endless sequences require special handling? What makes me suspicious is multiplying an endless sequence by a number.
~ Classical Arminian Christian Mystic ~

willowtree
Posts: 100
Joined: Sun Nov 02, 2014 1:56 pm

### Re: Where Algebra Fails

[quote="Paidion"
In the series you offered, there are an infinite number of elements which are larger than 1/2.[/quote]

My point exactly - your first series, with errors, includes 3/2, 9/4, 27/8 ? These are all greater than 1/2 and begin an infinite series which increase in value.

Graeme
If you find yourself between a rock and a hard place, always head for the rock. Ps 62..

Paidion
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Location: Back Woods of North-Western Ontario

### Re: Where Algebra Fails

Hi Dizerner,
Math can assume an actual infinity for mathematical purposes, but don't endless sequences require special handling? What makes me suspicious is multiplying an endless sequence by a number.
It is mathematically correct to multiply each term of and endless sequence by a number, as was done in the proof that 1/2 + 1/4 + 1/8 + ... =1
Paidion

Man judges a person by his past deeds, and administers penalties for his wrongdoing. God judges a person by his present character, and disciplines him that he may become righteous.

Avatar shows me at 75 years old. I am now 82.

Paidion
Posts: 5015
Joined: Mon Aug 18, 2008 10:22 pm
Location: Back Woods of North-Western Ontario

### Re: Where Algebra Fails

Hi Graeme,
You wrote:
Paidion wrote:In the series you offered, there are an infinite number of elements which are larger than 1/2.
My point exactly - your first series, with errors, includes 3/2, 9/4, 27/8 ? These are all greater than 1/2 and begin an infinite series which increase in value.
That is true that the original series which I gave increases in value. But you seemed to state that you were going to give the series 1/2 + 1/4 + 1/8 +... in reverse order, your having given 1+2+3+4 as equivalent to 4+3+2+1 as an example. But the series you gave, was 1/16 + 1/8 + 1/4 + 1/2 + ... and this is NOT the reverse of 1/2 + 1/4 + 1/8 +...
I gave reason WHY it is not the reverse, one being that it contains elements that are larger than 1/2. But my series 1/2 + 1/4 + 1/8 +... do NOT contain elements larger than 1/2.
The reverse of a series must contain the same elements as is the case with 4+3+2+1. It contains identical elements to the series 1+2+3+4.

I understood what you were trying to do by making the elements increase as in the original series. And you were on the right track!

So now I must do what I promised. The experts said that while algebra works with infinite series that converge, such as 1/2 + 1/4 + 1/8 +..., it does not work with series that diverge such as 3/2 + 9/4 + 27/8 +... They did not state WHY algebra doesn't work with divergent series. So I still do not know why. If you should find out, please post the answer. I would very much like to know!
Paidion

Man judges a person by his past deeds, and administers penalties for his wrongdoing. God judges a person by his present character, and disciplines him that he may become righteous.

Avatar shows me at 75 years old. I am now 82.