Euler's Equation and the Trinity/Godhead


Jamie123
 Share

Recommended Posts

I was talking to a mathematical colleague the other day about the curious "fact" that the sum of all natural numbers from 1 to infinity is -1/12. (If you've never heard of this Google it - it's quite a source of distraction!)

 

Anyway, he came up with an interesting factoid of his own - namely that i^i (where i as usual means sqrt(-1)) is real.

 

It took me a while to get my tiny mind around this, but eventually the penny dropped: Euler's equation gives us i=exp(i*pi/2) so i^i=exp(i*i*pi/2)=exp(-pi/2)=4.810477381 which is indeed real number!

 

But when I googled this online I found I had missed something: there is a more general version i=exp(i*pi/2*(1+4n)) where n is any integer, so i^i=exp(-pi/2*(1+4n)).

 

Each n, i.e. ...-2, -1, 0, 1, 2, 3... generates a different value. All values are real, none of them are equal to each other but all of them are equal to i^i.

 

1.67758E-05=i^i

0.008983291=i^i

4.810477381=i^i

2575.970497=i^i

1379410.706=i^i

 

but

 

1.67758E-05!=0.008983291!=4.810477381!=2575.970497!=1379410.706

 

What does this remind you of?

 

220px-Shield-Trinity-Scutum-Fidei-Englis

Edited by Jamie123
Link to comment
Share on other sites

.     Euler's equation:

e^(iπ) = -1

 

.     Take square root of both sides:

[e^(iπ)] = [-1]

[e^(iπ)]^½= i

 

.     Reversing and simplifying:

i = e^(iπ/2)

 

.    So,

 

i^i = [e^(iπ/2)]^i

 

.     which simplifies to:

i^i = e^(i²π/2)

i^i = e^(-π/2)
i^i = 1/[e^(π/2)]
 
.     This is easily calculated:
i^i  1/4.810477  0.20788
 

How about that? Though I am not seeing how including a factor of (1+4n) in the exponent is supposed to yield the same number. This would appear to be true for positions on the unit circle - you're essentially looking at the 90° point - but it is not obvious to me that there should be an expectation that this would generalize to all real numbers. My mathematical understanding is not deep enough to offer a firm opinion.

 

As to how this applies to concepts such as the Trinity. I think of Kurt Gödel, who published his famous and utterly revolutionary Incompleteness Theorem along with his own ideas about how this philosophically applied to the human condition. I am not convinced that the human conditon is well-represented by number systems, though. So I am not at all convinced that the consequences that Gödel himself suggested are even applicable.

Link to comment
Share on other sites

Thanks Vort! It's great to hear from you :) You are quite right - the value for n=0 is 0.20788, not 4.810... as I said. I did my calculations in MS Excel and I missed the minus out of the formula. The true values are the reciprocals of the values I listed. (My suspicions should have been aroused by the values increasing instead of decreasing with n - but oh well...)

 

Having said that though, what you quote at the start is not Euler's *equation* per se, but Euler's *identity*. The full version of Euler's equation is:

 

exp(i*theta)=cos(theta)+i*sin(theta)

 

which simplifies to exp(i*pi)=-1 (Euler's identity) for the special case of theta=pi. But if theta=pi/2*(1+4n), the right-hand side of the equation remains equal to i irrespective of n (so long as n is an integer) since all angles are modulo 2*pi. It therefore follows that the LHS, exp(i*pi/2*(4n+1)), is always equal to i, so i^i must be exp(-pi/2*(1+4n)). This takes a different *real* value for each n.

 

I'm not really suggesting that God or Man are really representable by number systems; it is just an analogy. I can imagine a simplistic Unitarian argument running something like this: Father, Son and Holy Spirit are all God, but Father is not Son, Son is not Holy Spirit and Holy Spirit is not Father (as represented in the Trinity shield). But if Heavenly Father is God, and God is the Son, then Heavenly Father is the Son. Reductio ad absurdum - Trinity disproven. 

 

One might of course say that "God" is more like an adjective than a noun - that more than one individual might "be God" (just as more than one person may old, fat, ugly etc.) but this is straying close to polytheism, which most Trinitarians reject. I have had this very argument with Jehovah's Witnesses who claim that all Trinitarians can say about the Trinity is that it is "a mystery" beyond human understanding - which explains nothing. But this may not be the case - this example shows shows something well within the field of human understanding which behaves exactly as the Unitarian claims the Trinity can't.

 

P.S. An additional thought; another possibility is Monarchianism which maintained that God is one being but has separate "roles" as Father, Son and Holy Spirit, just as I am a father, a husband and a son....though now I think about it maybe that's not such a good analogy because I am these three things to different people. OK - so maybe a teacher has his own son or daughter in his class - he is a father and a teacher to that kid, but the roles are compartmentalized. I have heard a Baptist minister I used to know teach this idea to his young people's class - and only years later read that it is considered a heresy (the "Monarchian Heresy"). Maybe someone learned in theology could explain why this is.

Edited by Jamie123
Link to comment
Share on other sites

Thanks for the correction of Euler's equation vs. Euler's identity, Jamie. I note that in addition to:

 

i = e^(iθ)

 

for all θ = ½π(1 + 4n), it's also true that:

 

i = -e^(iθ)

 

for all θ = ½π(3 + 4n). Not that this has any real philosophical relevance to anything we're discussing, of course. Just saying.

Link to comment
Share on other sites

  • 2 weeks later...

Thanks for the correction of Euler's equation vs. Euler's identity, Jamie. I note that in addition to:

 

i = e^(iθ)

 

for all θ = ½π(1 + 4n), it's also true that:

 

i = -e^(iθ)

 

for all θ = ½π(3 + 4n). Not that this has any real philosophical relevance to anything we're discussing, of course. Just saying.

 

That's very interesting! Imaginary numbers are a constant source of amusement. I remember reading this paradox a while back: -1=i^2=sqrt(-1)*sqrt(-1)=sqrt(-1*-1)=sqrt(1)=1, so -1=1 and 1+1=0. I once showed that to the head of our math school; she was stumped for a while, but later got back to me with the suggestion that you need to say sqrt(1)=-1 at the last step. (But why isn't the positive root just as valid?) Another solution I found on the web is that the rule sqrt(a)*sqrt(b)=sqrt(a*b) is not valid when a and b are both imaginary.

Link to comment
Share on other sites

That's very interesting! Imaginary numbers are a constant source of amusement. I remember reading this paradox a while back: -1=i^2=sqrt(-1)*sqrt(-1)=sqrt(-1*-1)=sqrt(1)=1, so -1=1 and 1+1=0. I once showed that to the head of our math school; she was stumped for a while, but later got back to me with the suggestion that you need to say sqrt(1)=-1 at the last step. (But why isn't the positive root just as valid?) Another solution I found on the web is that the rule sqrt(a)*sqrt(b)=sqrt(a*b) is not valid when a and b are both imaginary.

Sorry but you are mixing apples and oranges.  In this case mixing the "rules" of real and imaginary numbers.  This statement is not true "-1=i^2=sqrt(-1)*sqrt(-1)".  Your rhetorical error is applying the rules of multiplication (factoring) of real numbers to binary operations of imaginary numbers.  Factoring numbers from under the square root in the imaginary plain by definition requires that you multiply the result by i -- giving you sqrt(-1)*i*sqrt(-1)*i -- since you factored sqrt(-1) twice.

 

Btw the misplacing of rhetorical logic across non-congruent "sets" is a very common error in both politics and religion.  Usually when I point out such rhetorical errors with fellow scientists, mathematicians and engineers they apologize for their error and adjust their thinking - but when doing so with those of religious or political thinking - they generally get upset and accuse me of being arrogant or something else that is not nice.  There is a reason for a great divide between science and religion - that goes far beyond religious doctrine.

Link to comment
Share on other sites

Sorry but you are mixing apples and oranges.  In this case mixing the "rules" of real and imaginary numbers.  This statement is not true "-1=i^2=sqrt(-1)*sqrt(-1)".  Your rhetorical error is applying the rules of multiplication (factoring) of real numbers to binary operations of imaginary numbers.  Factoring numbers from under the square root in the imaginary plain by definition requires that you multiply the result by i -- giving you sqrt(-1)*i*sqrt(-1)*i -- since you factored sqrt(-1) twice.

 

Jamie did not factor √(-1). He just used the identity i = √(-1) in the expression i²:

 

i² = i × i

= √(-1) × √(-1)

= (-1)^½ × (-1)^½

= (-1 × -1)^½

= (1)^½

= √1 =±1

 

Not being mathematically sophisticated, I can't answer for sure why we must discard the +1 answer and retain only the (obviously correct, by definition) answer of -1. But when you take roots, you often get spurious answers, which is clearly going on here. (I openly admit that's a handwaving argument that asserts an answer without explaining it, but that's the best I can currently do.)

Link to comment
Share on other sites

Jamie did not factor √(-1). He just used the identity i = √(-1) in the expression i²:

 

i² = i × i

= √(-1) × √(-1)

= (-1)^½ × (-1)^½

= (-1 × -1)^½

= (1)^½

= √1 =±1

 

Not being mathematically sophisticated, I can't answer for sure why we must discard the +1 answer and retain only the (obviously correct, by definition) answer of -1. But when you take roots, you often get spurious answers, which is clearly going on here. (I openly admit that's a handwaving argument that asserts an answer without explaining it, but that's the best I can currently do.)

 

As I tried to say - the rules for real numbers do not apply to imaginary numbers.  What you are suggesting as a simple substitution is not transitory between the two sets or types of numbers.  The difference is by definition.  The binary operation of addition and multiplication of real numbers is defined differently for imaginary numbers - what we are all use to for real numbers  So that A+B=C for real numbers does not work out the same as A+B=C for imaginary numbers.  This is because by definitional real numbers are one dimensional and can be represented by a one dimensional line - the difference being the difference between them on the line.  But imaginary numbers are by definition are two dimensional numbers that exist in a pure two dimensional euclidean plane.

 

Just for fun - Einstein proved that our universe is not just 3 dimensional but that Euclidean geometry does not apply - allowing parallel lines to intersect based on definable rules.  It is my honest impression that many in the scientific world has difficulty warping their brains around non-euclidean 3 dimensional numbers because basic elements of mathematics they learned in grade school no longer apply and it so upsets them that they don't or won't actually make the transition.

 

Because many have learned that 2+2=4 from the real number system they get extremely upset when someone says 2+2=3 under certain conditions in 3 dimensional euclidean numbers and they all the more refuse, to even consider, the non euclidean number systems in 3 dimensions. 

 

Hopefully the reader can see the problem that being "right" creates when certain elements of definition are slightly altered.  If someone is thinking one dimensionally there is no way that they can honestly come to an agreement of what is right with someone operating with full understanding in a non euclidean multiple dimensional universe.  And then there is the other extreme of stupidity; that what is "right" is situational and then refused to accept the rigors of logic that applies to the multiple dimensions that they think they are considering.

Link to comment
Share on other sites

As I tried to say - the rules for real numbers do not apply to imaginary numbers.  What you are suggesting as a simple substitution is not transitory between the two sets or types of numbers.

 

Sure it is. If i = √(-1), then by definition i² = [√(-1)]². This is what the mathematical symbols mean.

 

Maybe I'm just misunderstanding you. If so, please feel free to clarify.

 

The difference is by definition.  The binary operation of addition and multiplication of real numbers is defined differently for imaginary numbers - what we are all use to for real numbers  So that A+B=C for real numbers does not work out the same as A+B=C for imaginary numbers.

 

I am no mathematician, but I'm quite sure this is incorrect. An imaginary number is simply a scaling of i -- so you could count i, 2i, 3i, etc. When e.g. an electrical engineer plots a time-varying voltage or current, he uses a real axis for x and an imaginary axis for y. All the math done is exactly the same in both cases. Even the multiplication of the imaginary part by itself is done with normal math, and i times i gives you a real number.

 

This is because by definitional real numbers are one dimensional and can be represented by a one dimensional line - the difference being the difference between them on the line.  But imaginary numbers are by definition are two dimensional numbers that exist in a pure two dimensional euclidean plane.

 

Interesting. I am aware of no such definition of imaginary numbers. On the contrary, the imaginary number line is exactly the same as the real number line, only each real number is multiplied by i. Thus, the imaginary numbers are equally well represented by a one-dimensional line. I wonder if you are thinking of real vs. imaginary plots, as used e.g. by electrical engineers?

 

Just for fun - Einstein proved that our universe is not just 3 dimensional but that Euclidean geometry does not apply - allowing parallel lines to intersect based on definable rules.

 

Not quite sure what you're saying here, Traveler. As far as I understand things, Einstein proved no such thing. Rather, he offered a non-Euclidian model of the universe. I do not believe Einstein ever proposed or seriously contemplated extra dimensions outside the three spatial and one temporal that we normally talk about. All the extra-dimensional talk came about in conjunction with string theory, I believe.

 

Obviously, the bare idea of extra spatial dimensions has been around at least since the nineteenth century's Flatland (and probably well before that), but I have never heard that such ideas took up any space in Einstein's theories or thinking.

 

Because many have learned that 2+2=4 from the real number system they get extremely upset when someone says 2+2=3 under certain conditions in 3 dimensional euclidean numbers and they all the more refuse, to even consider, the non euclidean number systems in 3 dimensions.

 

Again, I'm not exactly sure what you're driving at, Traveler. 2 + 2 does not equal 3 anywhere. It always equals 4. If you're talking about vector algebra on non-Euclidian surfaces, then that's fine -- but in that case, for example, "2i + 2j" is not the same as "2 + 2", even if you're talking about a curved surface.

Link to comment
Share on other sites

Sure it is. If i = √(-1), then by definition i² = [√(-1)]². This is what the mathematical symbols mean.

 

Maybe I'm just misunderstanding you. If so, please feel free to clarify.

 

 

I am no mathematician, but I'm quite sure this is incorrect. An imaginary number is simply a scaling of i -- so you could count i, 2i, 3i, etc. When e.g. an electrical engineer plots a time-varying voltage or current, he uses a real axis for x and an imaginary axis for y. All the math done is exactly the same in both cases. Even the multiplication of the imaginary part by itself is done with normal math, and i times i gives you a real number.

 

 

Interesting. I am aware of no such definition of imaginary numbers. On the contrary, the imaginary number line is exactly the same as the real number line, only each real number is multiplied by i. Thus, the imaginary numbers are equally well represented by a one-dimensional line. I wonder if you are thinking of real vs. imaginary plots, as used e.g. by electrical engineers?

 

 

Not quite sure what you're saying here, Traveler. As far as I understand things, Einstein proved no such thing. Rather, he offered a non-Euclidian model of the universe. I do not believe Einstein ever proposed or seriously contemplated extra dimensions outside the three spatial and one temporal that we normally talk about. All the extra-dimensional talk came about in conjunction with string theory, I believe.

 

Obviously, the bare idea of extra spatial dimensions has been around at least since the nineteenth century's Flatland (and probably well before that), but I have never heard that such ideas took up any space in Einstein's theories or thinking.

 

 

Again, I'm not exactly sure what you're driving at, Traveler. 2 + 2 does not equal 3 anywhere. It always equals 4. If you're talking about vector algebra on non-Euclidian surfaces, then that's fine -- but in that case, for example, "2i + 2j" is not the same as "2 + 2", even if you're talking about a curved surface.

 

Vort - I have thought long and hard about how to respond.  I have concluded that what I have already posted is about as best as I can expect of my self - lacking greatly communication skills.  Here is what I suggest.  You approach someone you trust in you circle of trusted friends - someone with a mathematical background - at least a minimum of a masters degree and versed in number theory.  Present the following two equations and ask them which is the most mathematically elegant (meaning in part complete) - or mathematically correct?

 

2+2=4

2>=2+2<=4

Link to comment
Share on other sites

Vort - I have thought long and hard about how to respond.  I have concluded that what I have already posted is about as best as I can expect of my self - lacking greatly communication skills.  Here is what I suggest.  You approach someone you trust in you circle of trusted friends - someone with a mathematical background - at least a minimum of a masters degree and versed in number theory.  Present the following two equations and ask them which is the most mathematically elegant (meaning in part complete) - or mathematically correct?

 

2+2=4

2>=2+2<=4

 

Honestly, I don't need to ask a math PhD to see that the second equation/inequality is incorrect.

Link to comment
Share on other sites

Honestly, I don't need to ask a math PhD to see that the second equation/inequality is incorrect.

 

I am suggesting that there is something about mathematics that you do no understand - It would appear to me that not only will you not believe there is something about mathematics that you do not understand but that you are unwilling to even consider expanding your understanding or consider any other witness.  Not at all what I expected.

 

Tell you what - tell your math expert friend that you have encountered a nut job on the internet and show them the two equations as proof.  Have a good laugh at my expense.

 

For the record - I may not agree with you on this point - but it has not changed my opinion that you are a good friend and ally to have and hope to continue discussions from time to time.

Link to comment
Share on other sites

2+2=4

2>=2+2<=4

 

 

Honestly, I don't need to ask a math PhD to see that the second equation/inequality is incorrect.

 

 

I am suggesting that there is something about mathematics that you do no understand - It would appear to me that not only will you not believe there is something about mathematics that you do not understand but that you are unwilling to even consider expanding your understanding or consider any other witness.  Not at all what I expected.

 

Traveler, look at the first part of the second equation (inequality):

 

2>=2+2

 

Two is not greater than or equal to four. This is not a matter of dispute.

Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.
Note: Your post will require moderator approval before it will be visible.

Guest
Reply to this topic...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...
 Share