### Henry Reich

Shared publicly -This has nothing to do with physics, but I thought it was important so I made a video about it: uprisings in Venezuela. http://youtu.be/kmvAdRntWH8

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Henry Reich

Works at MinuteEarth

Attended Grinnell College

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This has nothing to do with physics, but I thought it was important so I made a video about it: uprisings in Venezuela. http://youtu.be/kmvAdRntWH8

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:)

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NEW DISCOVERY About the Big Bang! (and a new video explaining that discovery!)

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+Todd Scheving

That's because religion clearly doesn't work out. It only tells false stories about the universe. Science FTW!

That's because religion clearly doesn't work out. It only tells false stories about the universe. Science FTW!

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Are any Animals Truly Monogamous? NEW VIDEO

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Interesting video. I think monogamy is a word invented for the human species for practical purposes since human babies need to be cared for and nurtured. Also that we invented the concept of home, nation and country that has to be sustained by taxes. Most importantly, sexually transmitted diseases. The Old Testament has stories showing humans were not monogamous on the part of the male and was the accepted practice at the time. Without the order monogamy, there would be no cheating. Good or bad the order has been established.

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Evolution vs Natural Selection: NEW VIDEO

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That is what happened!

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What should you do when it's really really cold outside? Go swimming, for one...

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cool!!!!!!!!!!!!!!!!!!

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NEW VIDEO! Relativity Isn't Relative

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Calculation of mass of the body seems to be an illusion. Anyway thanks +VeryEvilPettingZoo

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How do Modern Light Bulbs Work? NEW VIDEO!

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great way to explain free electrons and holes of PN junction in LED.

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Excellent

There is a lot of discussion in various online mathematical forums currently about the interpretation, derivation, and significance of Ramanujan's famous (but extremely unintuitive) formula

1+2+3+4+... = -1/12 (1)

or similar divergent series formulae such as

1-1+1-1+... = 1/2 (2)

or

1+2+4+8+... = -1. (3)

One can view this topic from either a pre-rigorous, rigorous, or post-rigorous perspective (see this page of mine for a description of these three terms: http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ ). The pre-rigorous approach is not particularly satisfactory: here one is taught the basic rules for manipulating finite sums (e.g. how to add or subtract one finite sum from another), and one is permitted to blindly apply these rules to infinite sums. This approach can give derivations of identities such as (1), but can also lead to derivations of even more blatant absurdities such as 0=1, which of course makes any similar derivation of (1) look quite suspicious.

From a rigorous perspective, one learns in undergraduate analysis classes the notion of a convergent series and a divergent series, with the former having a well defined limit, which enjoys most of the same laws of series that finite series do (particularly if one restricts attention to absolutely convergent series). In more advanced courses, one can then learn of more exotic summation methods (e.g. Cesaro summation, p-adic summation or Ramanujan summation) which can sometimes (but not always) be applied to certain divergent series, and which obey some (but not all) of the rules that finite series or absolutely convergent series do. One can then carefully derive, manipulate, and use identities such as (1), so long as it is made precise at any given time what notion of summation is in force. For instance, (1) is not true if summation is interpreted in the classical sense of convergent series, but it is true for some other notions of summation, such as Ramanujan summation, or a real-variable analogue of that summation that I describe in this post: http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/#comment-265849

From a post-rigorous perspective, I believe that an equation such as (1) should more accurately be rendered as

1+2+3+4+... = -1/12 + ...

where the "..." on the right-hand side denotes terms which could be infinitely large (or divergent) when interpreted classically, but which one wishes to view as "negligible" for one's intended application (or at least "orthogonal" to that application). For instance, as a rough first approximation (and assuming implicitly that the summation index in these series starts from n=1 rather than n=0), (1), (2), (3) should actually be written as

1+2+3+4+... = -1/12 + 1/2 infinity^2 (1)'

1-1+1-1+... = 1/2 - (-1)^{infinity} /2 (2)'

or

1+2+4+8+... = -1 + 2^{infinity} (3)'

and more generally

1+x+x^2+x^3+... = 1/(1-x) + x^{infinity}/(x-1)

where the terms involving infinity do not make particularly rigorous sense, but would be considered orthogonal to the application at hand (a physicist would call these quantities unphysical) and so can often be neglected in one's manipulations. (If one wanted to be even more accurate here, the 1/2 infinity^2 term should really be the integral of x dx from 0 to infinity.) To rigorously formalise the notion of ignoring certain types of infinite expressions, one needs to use one of the summation methods mentioned above (with different summation methods corresponding to different classes of infinite terms that one is permitted to delete); but the above post-rigorous formulae can still provide clarifying intuition, once one has understood their rigorous counterparts. For instance, the formulae (1)' and (3)' are now consistent with the left-hand side being positive and diverging to infinity, and the formula (2)' is consistent with the left-hand side being indeterminate in limit, with both 0 and 1 as limit points. The fact that divergent series often do not behave well with respect to shifting the series can now be traced back to the fact that the infinite terms in the above identities produce some finite remainders when the infinity in those terms is shifted, say to infinity+1.

For a more advanced example, I believe that the "field of one element" should really be called "the field of 1+... elements", where the ... denotes an expression which one believes to be orthogonal to one's application.

1+2+3+4+... = -1/12 (1)

or similar divergent series formulae such as

1-1+1-1+... = 1/2 (2)

or

1+2+4+8+... = -1. (3)

One can view this topic from either a pre-rigorous, rigorous, or post-rigorous perspective (see this page of mine for a description of these three terms: http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ ). The pre-rigorous approach is not particularly satisfactory: here one is taught the basic rules for manipulating finite sums (e.g. how to add or subtract one finite sum from another), and one is permitted to blindly apply these rules to infinite sums. This approach can give derivations of identities such as (1), but can also lead to derivations of even more blatant absurdities such as 0=1, which of course makes any similar derivation of (1) look quite suspicious.

From a rigorous perspective, one learns in undergraduate analysis classes the notion of a convergent series and a divergent series, with the former having a well defined limit, which enjoys most of the same laws of series that finite series do (particularly if one restricts attention to absolutely convergent series). In more advanced courses, one can then learn of more exotic summation methods (e.g. Cesaro summation, p-adic summation or Ramanujan summation) which can sometimes (but not always) be applied to certain divergent series, and which obey some (but not all) of the rules that finite series or absolutely convergent series do. One can then carefully derive, manipulate, and use identities such as (1), so long as it is made precise at any given time what notion of summation is in force. For instance, (1) is not true if summation is interpreted in the classical sense of convergent series, but it is true for some other notions of summation, such as Ramanujan summation, or a real-variable analogue of that summation that I describe in this post: http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/#comment-265849

From a post-rigorous perspective, I believe that an equation such as (1) should more accurately be rendered as

1+2+3+4+... = -1/12 + ...

where the "..." on the right-hand side denotes terms which could be infinitely large (or divergent) when interpreted classically, but which one wishes to view as "negligible" for one's intended application (or at least "orthogonal" to that application). For instance, as a rough first approximation (and assuming implicitly that the summation index in these series starts from n=1 rather than n=0), (1), (2), (3) should actually be written as

1+2+3+4+... = -1/12 + 1/2 infinity^2 (1)'

1-1+1-1+... = 1/2 - (-1)^{infinity} /2 (2)'

or

1+2+4+8+... = -1 + 2^{infinity} (3)'

and more generally

1+x+x^2+x^3+... = 1/(1-x) + x^{infinity}/(x-1)

where the terms involving infinity do not make particularly rigorous sense, but would be considered orthogonal to the application at hand (a physicist would call these quantities unphysical) and so can often be neglected in one's manipulations. (If one wanted to be even more accurate here, the 1/2 infinity^2 term should really be the integral of x dx from 0 to infinity.) To rigorously formalise the notion of ignoring certain types of infinite expressions, one needs to use one of the summation methods mentioned above (with different summation methods corresponding to different classes of infinite terms that one is permitted to delete); but the above post-rigorous formulae can still provide clarifying intuition, once one has understood their rigorous counterparts. For instance, the formulae (1)' and (3)' are now consistent with the left-hand side being positive and diverging to infinity, and the formula (2)' is consistent with the left-hand side being indeterminate in limit, with both 0 and 1 as limit points. The fact that divergent series often do not behave well with respect to shifting the series can now be traced back to the fact that the infinite terms in the above identities produce some finite remainders when the infinity in those terms is shifted, say to infinity+1.

For a more advanced example, I believe that the "field of one element" should really be called "the field of 1+... elements", where the ... denotes an expression which one believes to be orthogonal to one's application.

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Blah.....

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Why is the Solar System Flat? NEW VIDEO!

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Numan Salati

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+Henry Reich is +MinutePhysics g+ page abandoned?

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Why Winter has the Best Full Moons

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+Emily Elert ha! Yes, I am that one.

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- MinuteEarthCreator, 2013 - present
- MinutePhysicsYoutuber, 2011 - present
- Perimeter Institute for Theoretical PhysicsDigital Media Artist, 2012 - 2013
- Freddiew youtube channelvfx & other stuff, 2011 - 2011

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1st graduating class of PSI @perimeterinstitute!

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- Grinnell CollegePhysics and Mathematics, 2005 - 2009
- Perimeter Institute for Theoretical PhysicsPhysics, 2009 - 2010
- University of Southern CaliforniaFilm Production, 2011 - 2011

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