Isolating the equation, multiply both sides by a:

\[ a^2 = ab \]

Subtract \( b^2 \) from both sides:

\[ a^2 - b^2 = ab - b^2 \]

Now let us factor each side:

\[ (a+b)(a-b)=b(a-b) \]

We can now divide each side by \((a-b)\):

\[ \frac{(a+b)(a-b)}{(a-b)}=\frac{b(a-b)}{(a-b)} \]

This reduces to:

\[ a+b = b \]

Now since a and b are both equal (from the first statement), let us set them to 1 which produces:

\[ \begin{aligned}
1+1 & = 1 \\
2 & = 1 \; \; \mathbb{QED} \\
\end{aligned} \]

Can you spot the error? I’ll give you a second…

To most people, their immediate answer/reaction is that mathematics cannot be trusted and the above “proves” that Mathematics is sometimes wrong and untrustworthy (Mathematics is the problem, basically)… This could not be further from the truth! But why do they think this way? I believe it is a fundamental problem in the way that Mathematics is taught at a young age in the United States (I cannot vouch for other students around the world, this is just something I have noticed here in the states). The American education system is not a very good one when it comes to Mathematics (among other things). Case in point, more than half of the Mathematics I learned in college is learned by international students in their high school years! But I digress, this is not the main issue here. I do not necessarily care for this post that others learn more than we do (that is another issue), just that we learn things at a fundamental level wrong! It would be fine if the Mathematics that we are currently teaching is taught correctly, but it is not. This causes many problems down the road throughout people’s lives, such as balancing their checkbooks and keeping up with their finances. Most people are forced to learn Mathematics in the most boring fashion when they are young and it does nothing for them. They are not taught the beauty of Mathematics.

This is a tragic mistake in our education system. Mathematics is vital to being able to keep a structured and stable life, even if it just means that you can sit down and plan out your finances so your family can eat! These types of things are so important but the fundamentals of them are taught as if they are chores instead of activities. If Mathematics is mentioned to the layperson, they usually get a disgusted look on their face and say something to the effect of “Oh I never use math, who wants to use that? That’s for the smart people, it’s useless to me!” This cannot be further from how we should view Mathematics.

This viewpoint stems from how we teach Mathematics. Children/young adults are pushed to learn and memorize formulas without understanding how or why any of it works. Most students have a deep hatred for Mathematics starting at a young age. This is most unfortunate because Mathematics is a beautiful and rich subject that should be treated as such when being taught. Unfortunately, this is not so (I am not here to describe how we can reform the system, that shall be another post at some point). Even something as simple as the Pythagorean Theorem has a beautiful representation if presented right. It is not such some symbols you memorize for a test! However, Mathematics teachers do not teach this way… But, as I said, that is for another post! Let us look at that proof…

What was the flaw? Let us examine the proof and oust the fundamental problem that lies within!

First off, when dealing with Mathematics, you have to have a very good memory sometimes. You cannot skip over even the smallest detail. I believe this is why most people see Mathematics as laborious and do not want to put forth the added brain power to fully understand what they are seeing/learning. The whole problem with this “proof” would be mitigated if people really understood the very first line and the implications of it: \( a=b \). What does this imply? Many things actually. The most basic is that what ever a is, b is also. Or, some might see it as whatever b is, a is as well (it does not matter which way you look at it in this case). But when we say this, there are direct implications like if a is even, then b is even, and if we multiply a and b then the result is even. We can pull out many properties like this depending on what the properties of the values we give a (and thus b) are.

So now that we are thinking down the path of implications of knowing properties of numbers, what would you say about subtraction? What can we imply about subtracting a and b after we make them equal? Well, the first thing that you should think of is “0″! And that is correct!

But we are still not done. We know that this should now yield a 0 value if we subtract them. There should be a fundamental rule of Mathematics that should immediately be coming to mind at this moment and be screaming at you… NO DIVISION BY ZERO!

Ah ha! We have realized a fundamental law that could be violated if we simply use the statement \( a-b \) where a and b are equal. This logical progression should be almost immediate to anyone that has learned Mathematics at its most basic levels. And this all comes from just the first line, all of this should have been going through your mind on the very first line of this proof! If this did occur to you on the first line, or even throughout the proof, then you most likely would not catch the error in the proof. Even if this did occur to you at some point, it still might not have registered with you. But why is this?

Another problem I would like to discuss is people’s fundamental block in going from symbols to numbers. They do not understand that a, in this situation, can be any number. So, when they see “a”, they do not see a number, they just see “a”. That is why many people cannot make the jump from \(a-b\) means \(1-1\) or \(2-2\) or any other number. That means they will miss that it could yield a 0 result which could be problematic with division further down the road in the proof!

Well, that is my two cents on the subject. I love these types of proof but I wish more lay people could spot the error(s) in them.

The first leg of my trip went off without a hitch! The flight from Roanoke to Detroit was uneventful. I got an aisle seat and the guy beside me respected the unspoken plane seat agreement so all was good.

Detroit’s airport was probably one of the nicest I had ever been in. They had lots of neat things around and the tunnel between terminals A and B/C was pretty awesome. They also had a really nice fountain in the main terminal.

The layover was 4 hours in Detroit so I had awhile to do… nothing pretty much. Unfortunately, the internet was not free so that blew. I (again) did not bring any movies or shows to watch. I did, however, bring a Differential Equation book so I read over that for a little bit. I just happen to look up at some moment and see that the gate I was at had changed its “next flight” status. So, I went to the screen and saw that my flight was delayed by 5 hours! Fortunately, I ran to a gate across the airport where another flight for Amsterdam was getting ready for take off and they let me on! This caused my bags to get delayed but I was okay with that (at the time) since I could still make my connecting flight in Amsterdam.

They provided a dinner half way through the flight to Amsterdam. It was so so. I was actually impressed by the salad, brownie, and roll but the main course (chicken or pasta, I chose chicken) was terrible! It had a side of green beans and potatoes and I don’t think I have ever eaten something so gross. I was hungry though that it didn’t matter. I was just glad to get food! I managed to watch 3 movies during the flight: Adjustment Bureau, Limitless, and Fast Five. Fast Five was the best of the 3 and I’m glad Delta provided these for free and not pay!

8 hours later I arrived in Amsterdam. I was expecting more from the airport but that doesn’t really mean much. My layover was uneventful and I then flew to Bremen, Germany from there. Only hitch was that I had to file a claim for my bags once I arrived but I received them later that night so it was all good!

I finally arrived 2 hours later at the Hermes Hotel in Oldenburg, Germany! And that pretty much concludes my journey TO Oldenburg.

• Charlie Bennett
• Aram Harrow
• Steve Flammia

These three people are brilliant physicists/computer scientists/mathematicians that have each made their impacts in Quantum Computation (and Science in general). Bennett is an IBM Fellow, Harrow is currently a lecturer at the University of Bristol (on leave at University of Washington where Dave just left), and Flammia who was a Postdoc at Perimeter until recently where he moved to IQI at CalTech. I think this selection of giants in the field was an excellent choice and I am looking forward to what they are going to pontificate on in the future!

Meagan and I got up this morning around 7 AM to prepare for our flights. She, unfortunately, will not be going with me to Germany so she is heading back to Clarksville, TN and Huntsville, AL to visit her family and friends (Hopefully next time she will have her passport so that she can go as well!) We decided to walk to the airport this morning since it was only 1.8 miles… mistake! It was too early to be walking to the airport. Next time we will definitely call a cab!

Meagan’s flight was a little behind (6 minutes or so) but she took off at around 10:20 AM from the Roanoke, VA airport. I am currently sitting in the terminal waiting on flight which takes off at 1 PM.

I am extremely excited about the summer school. I received a full grant to attend so it is essentially a free trip to do some awesome mathematics/computational science! I hope my trip to and from Oldenburg goes off without a hitch so that I can really enjoy my trip.

Well, that is it for now! I will post daily about my trip (and sometimes more) so check back for updates often!

• Road To Reality by Sir Roger Penrose
  Awesome book so far and one of the most comprehensive texts I have ever read on anything. Roger Penrose is a master Mathematical Physicist and does a brilliant job at explaining a ton of ideas extremely well. The exercises are very interesting so far and I enjoy working them when I have time. I plan on spending a lot more time on the exercises in the next few months. If I had to recommend one book to anyone in the sciences, this would be it!
• Explorations In Quantum Computing by Colin P. Williams
  I am on Chapter 3 of this book and as of now, I think it is a good text for someone who already has a good understanding of both quantum computation and quantum mechanics. It seems that some of the information is misplaced and Colin goes into too much detail too early in the book, mainly in some of the examples. He introduces some concepts without explaining them (such as the Pauli gates) and doesn’t get around the telling you what they are till about 50 pages later. This is why I think that this book is more for the reader who knows a little about the subject. Other than that, I love the book. Colin gives a great introduction to the field and gives very interesting examples that I have not seen other places. I hope the rest is as good as what I have already read!
• Quantum Computation and Quantum Information by Michael Nielsen & Isaac Chuang
  An amazing book that I should have read cover to cover years ago. It is THE text on quantum computing and is extremely in depth and provides the best introduction to the subject around. I recommend reading this text and then picking up Colin’s book above for a more “Computer Science and Computer Engineering” viewpoint on the subject. Everyone who is anyone in quantum computation knows that this book is a great book. Peter Shor, Charles Bennet, and Michael Freedman all are recommenders for this book and they give it praise (if you don’t know who these people are, look them up and you will see my point). Read it!
• The Fabric of Reality by David Deutsch
  I am not very far into this book yet and I am sad to say it has taken me this long to start reading it. David is one of the founders of the field of quantum computing and I love many of his views on fundamental physics. I look forward to finishing this book quite soon.

Hopefully I will be finishing these books in the next few months and continuing onto other ones. I have a ton books on my bookshelf and I assume I will be adding to them just out of shear love for reading (and buying books!). We shall see how it goes, just thought I would gather my thoughts on which books I am currently reading.

Today I graduated with my Bachelor of Science degree in Computer Science with minors in Mathematics and Physics! Above is a picture of my family and I after the ceremony (Left to Right: Hannah [Sis], Dad, Mom, Me, Daniel [Bro], & Abraham [Bro]). My oldest sister Faith was the only sibling not present but she will be coming up next week to see my mother graduate with her Doctorate in Nursing (Go mom!) so it’ll be good to see her. It was great to have my whole family there and I really appreciate the support they give me and have given me throughout my life. Thanks everyone, I love you all very much!

Also at my graduation was my beautiful girlfriend Meagan South (pictured above). I’m not sure where I would be without her and I was so happy she came to see me finish a chapter in my life that will launch us both into the next chapter of our life together. I also want to thank her for putting up with all my science-y ramblings I’ve had over the past few years and I hope that she doesn’t mind a few more. I love her with all my heart and am glad we are together!

As some of you might know, I have recently been interviewing at multiple corporations and businesses for my next big move after I receive my Bachelors degree. Well, over the past few months, I have interviewed at a few places: Microsoft, IBM, Amazon, Google, Facebook, NASA, & a myriad of small businesses. I passed all of my initial interviews with everyone and was even given the pleasure of being flown out to some of the bigger names like Microsoft in Seattle, WA for onsite interviews. These experiences were quite amazing but I was left wanting much more from many of them. Some didn’t want to pay me what I thought I should be paid and some of the jobs were just plain boring sounding. And some I just plain didn’t make it pass the final interviews! But there was one exception to this. One place that fit like a glove on all accounts. And that was a small business research company in Roanoke, VA named Luna Innovations. My phone interview with them went amazingly well so they flew me to Roanoke, VA to have an onsite interview. I have to say, it was one of the most comfortable situations I had ever been in. Everyone was extremely nice and professional but they were also laid back. I could tell they cared about me as person and not just me as a worker. It was great! I was mostly asked about my diverse background in many areas of science and mathematics. Only a few technical questions were asked and I took this as a good sign. I was called the next morning and offered the job!

So…I am now officially a Research Engineer at Luna Innovations in Roanoke, VA! Meagan and I are moving up there the 22nd of May and we are looking forward to it very much! We already have a townhouse that we will be living in starting June 1st and we can’t wait to move in!

Luna is a contract company for many government agencies: NSA, DoD, DoE, etc. I am not exactly sure what I will be working on yet but I will be working with the Secure Computing and Communications Group dealing with Anti-Tamper technologies in FPGA’s and other microprocessors. Lots of graph theory and applied mathematics are going to be involved so I am super excited about it! I will be able to use both my mathematical and software background in my work which is really what I wanted out of a job!

Last topic of the day… Graduate school for me and Meagan finishing her undergrad. As you might have noticed, we are planning on attending Virginia Tech. I am going to start in the fall as a Commonwealth Scholar (fancy word for people who work full time and want to take classes). I will be taking 1 class a semester for the first year to see how it goes. I am hoping to start with graduate Quantum Mechanics in the fall and finish the sequence in the Spring. I am very excited about taking graduate physics classes and can’t wait to start! I am still in debate if I want to obtain my Masters of Science in Physics or Mathematics as of yet. Right now it is Physics but it could change. I also still want to pursue my love of quantum computing because I ultimately want to pursue that as my PhD specialization but I am unsure yet of the interest at VaTech. Hopefully it will be strong! As for Meagan, she will have to wait a year before she gets in-state status for her transfer. Hopefully that process will work in her favor and she can finish up her Bachelor’s degree in Sociology as a Hokie! Until then, she will be taking online classes at Austin Peay, my recent Alma Mater.

Well, I think that is it for now. I am extremely excited about the next step in my life and I can’t wait for it to start! I have 2 weeks before my first day of work and I’m already itching to start working!

This is by far one of the best popular science books I have ever read. Lee Smolin is a brilliant physicist at the Perimeter Institute for Theoretical Physics who is able to create a fascinating history of how the main quantum theories of gravity were formulated and the part he played in them. I thoroughly enjoyed reading the book and would recommend it to anyone wanting to learn some interesting physics. Smolin is very modest in his own contributions to the field of quantum gravity which I appreciated. I do not like picking up a book and hearing about how awesome the author is and why everyone should give them standing ovations for what they have done (cough…A New Kind of Science… cough).

I had the pleasure of meeting Smolin very briefly while I was at Perimeter Institute. He had an office right down from mine and I only wish I could have spoken with him more. Unfortunately, I did not have much time to do so. He is just as interesting in person as he is in the book and I would (and am going to) read anything by him in the future! He is an excellent scientist and I highly recommend any of his works!

The author of this book, Seth Lloyd, is quite an interesting person. He is a theoretical physicist and professor at MIT in the Mechanical Engineering department, of all places. He considers himself the first quantum-mechanical engineer and has created quite a neat niche for himself in this realm. I have had the pleasure of meeting him on 2 occasions and his quirky personality and brilliance really do come out in this book!

Seth does a great job in defending one of his very controversial ideas that the universe itself is a quantum computer. Many people do not believe this is an accurate model for the universe based on varying reasons but I tend to like the idea. Seth starts off by gently leading the reader through some basic computer science and physics terminology and bringing the reader up to speed on concepts like Maxwell’s Demon and information-theoretic ideas. He then takes the reader on a winding path of why the universe is itself a quantum computer, ever computing with every collision of elementary particles (he rehashes himself quite a bit in the process, most likely because it is a popular science novel so for the scientific it can get slow at some points).

This idea of the universe being a huge quantum computer is the heart of this book. He portrays each fundamental particle as part of a symphony of bits computing till the end of time. He explains the circuit model of quantum computation and how a quantum computer is indistinguishable from the universe itself, he talks about how the different levels of complexity of system are direct effects from the Big Bang, and many more. All these are fascinating ideas but the real interesting twist for me came when he introduces the idea that this model can easily be used as a model for quantum gravity. This struck me as quite odd. I did not expect this from his book but it is nice to hear his ideas on some of the topics in quantum gravity as opposed to the leader in the field, Lee Smolin (See Three Roads to Quantum Gravity for more).

Overall this book was quite good. It gives a good introduction for the layperson on many things quantum and explains very straightforward how these quantum weird-ities can be used to compute. Even for someone who has a strong computer science or physics background, this book offers a good dose of fresh ideas about information and how complexity is viewed from a purely physical standpoint. I would recommend it to anyone interested in getting into quantum information processing!

So, I am starting a new series called “A Book Review” (catchy title, I know) here at Viva la Science. I will showcase a newly read book, or possibly a book I read long ago, and just mention what I thought of the book and such. These books are mostly going to be scientific in nature but they could be works of fiction or even poetry, who knows! It will pretty much be a free for all at times and some posts might be rather short and others long. I might have other posts that stem off of these to talk about specific questions that arose while reading the book or they could be one line posts about how awful the book was and how I never want to open it again… Only time will tell what comes across in this series! I hope you enjoy the series!

]]> http://www.calebwherry.com/blog/2011/01/04/a-series-of-book-reviews/feed/ 0 http://www.calebwherry.com/blog/2010/11/20/infinite-sum-of-zeros/ http://www.calebwherry.com/blog/2010/11/20/infinite-sum-of-zeros/#comments Sat, 20 Nov 2010 10:26:05 +0000 caleb http://www.calebwherry.com/blog/?p=519 **Updated to make up for my ignorance in assuming things about infinite sums…** 12/01/2010

Given the following statement, what would the “logical” answer be:
\[ 0 + 0 + 0 + 0 + \ldots \; = \; ? \]

Someone without any mathematical knowledge (but knowledge of what the above mathematical symbols mean, of course) will almost without hesitation say 0. But is this correct? Even someone with some mathematical knowledge would jump to this conclusion (like me). I fail to see how this does not equal zero, which one of my professors, Dr. Ben Ntatin, thinks is so…

So here is his “proof” on the matter. Now I am no expert on analysis or algebra so maybe I am missing a fundamentally logical proof or something. So, here we go…

We start off with some statements that we accept as true:
\[\begin{aligned}
0 & = 0 \\
0 & = 0+0 \\
0 & = 1 - 1
\end{aligned} \]
And we assume:
\[ 0\;=\;0+0+0+0+\ldots \]

Okay so everything seems fine thus far. But lets start delving a little deeper:
\[\begin{aligned}
0 & = 0 + 0 + 0 + 0 + \ldots \\
0 & = (1-1) + (1-1) + (1-1) + (1-1) + \ldots
\end{aligned} \]

This is a legitimate statement given the rules we accepted as true above. All I did was replace each 0 with (1-1). My next step should be familiar to someone with a little mathematical background with the knowledge that real numbers are associative under the binary operator + (and also the fact that – is the same as + with the negative reals). So, we have associativity:
\[ (a+b) + (c+d) + (e+f) = a + (b+c) + (d+e) + f \]
All I did was rearrange the order in which I added the numbers together. This is a perfectly legitimate mathematical statement. But this is where my quarrel starts. Even if you move the parenthesis over one number as I did above, you will always have one number on the end that matches up with the beginning term. We now look at the above rule with Dr. Ntatin’s next step:
\[ \begin{aligned}
0 & = 0 + 0 + 0 + 0 + \ldots \\
0 & = (1-1) + (1-1) + (1-1) + (1-1) + \ldots \\
0 & = 1 + (-1+1) + (-1+1) + -1 \ldots \\
0 & = 1 + 0 + 0 + 0 \ldots \\
0 & = 1
\end{aligned} \]
Thus we have reached a contradiction so our original assumption of:
\[ 0 = 0 + 0 + 0 + 0 + \ldots \]
is wrong. But I don’t think that is true. I don’t think the algebra is correct above. I am still sticking to the fact that if you shift the associativity in the sum, you will still always have a term on the end that cancels out that first number in the sum. Because each zero is replaced by 2 numbers, 1 and -1, then you will always have an even number of numbers. So, you will always have a matching number to cancel out the first.

Now this is where I will admit that the concept of infinity is not at all intuitive. When studying mathematical analysis, there are things that make sense mathematically in the finite but do not make sense (and in some cases cannot be done) in the infinite. Maybe my reasoning above does not hold up because of this. Maybe my thinking of “grouping” infinitely many groups does not make sense. Maybe my thinking of there being an even number of 1′s does not make sense either because it is an infinite sum. I am at a loss…