Okay, so when I first began trying to figure out what to do with my last blog post I was clueless. I almost felt like I had too many options. So, I began working my way down Professor Golden's list of "Weekly Options" one by one, honestly skipping a few in between. None of the options were catching my eye at first. But, then I made my way back up to the top of the list (one by one this time) and I came across the Youtube video of Ramanujan's Magic Square and 27 seconds in the mind blowing began!

When we worked with magic squares the rows and columns of every 3x3 and 4x4 square had equal sums. True enough I struggled at first, but after a while I got the hang of it.This in itself was enough to keep me interested. But in comes the Ramanujan's magic square! We thought we were mastering something with the first squares and all along we were amateurs (just kidding)! There are so many interesting sums in this magic square, it's best I list them. Lets start with the "common" things (I'll refer to the lettered diagram pictured using the letters as labels only... the numbers would be that of the Ramanujan magic square on the right):

1. The sum of the numbers of any row is 139

2. The sum of the numbers of any column is 139

3. The sum of the numbers of any diagonal is 139

And then comes the good stuff!!!

4. The sum of the numbers of the corners is 139

*** i.e. a+d+m+p = 139

5. e+i+h+l = 139, b+c+n+o = 139, c+h+i+n = 139, b+e+i+o = 139

6. The sum of any central square is 139, in other words any 2x2 square inside of the 4x4 square has a sum of 139 (yeah, I was confused too)!

7. Srinivasa Ramanujan was born on the 22nd of December in 1887... wait for it, wait for it... (Go back and check out the top row of Ramanujan's magic square)!

BOOM!!!

1. The sum of the numbers of any row is 139

2. The sum of the numbers of any column is 139

3. The sum of the numbers of any diagonal is 139

And then comes the good stuff!!!

4. The sum of the numbers of the corners is 139

*** i.e. a+d+m+p = 139

5. e+i+h+l = 139, b+c+n+o = 139, c+h+i+n = 139, b+e+i+o = 139

6. The sum of any central square is 139, in other words any 2x2 square inside of the 4x4 square has a sum of 139 (yeah, I was confused too)!

7. Srinivasa Ramanujan was born on the 22nd of December in 1887... wait for it, wait for it... (Go back and check out the top row of Ramanujan's magic square)!

BOOM!!!

Infinity comes in TWO varieties: actual infinities and potential infinities.

Mathematician Karl Gauss once stated the "use of [infinity] is never permitted in mathematics".

Mathematician Rene Descartes saw the idea of infinity as an argument for the existence of God.

English mathematician John Wallis (1616–1703) suggested the use of ∞ as the symbol for infinity in 1655. Before that time,∞ had sometimes been used in place of M (1000) in Roman numerals.

On an encyclopedia website I also found a small tid bit about infinity and dividing by 0...

Although students are typically taught that "one cannot divide by 0," it can be argued that = 0 (read as "one divided by infinity"). How is this possible? Observe the following progression.

Note that as the denominator, or the divisor, becomes larger, the value of the fraction (or the "quotient") becomes smaller. What happens if the denominators become very large?

One can see that as the denominator becomes extremely large, the fraction values approach 0. Indeed, if one thinks of infinity as "ultimately large," one can see that the value of the fraction will likewise be "ultimately small," or 0. Hence, one informal (but useful) way to define infinity is "the number that 1 can be divided by to get 0." Actually, there is no need to use the number 1 as the numerator here; any number divided by infinity will produce 0.

Using algebra, one can come up with another definition of infinity. By transforming the following equation we see that infinity is what results if 1 is divided by 0.

If

Then 1 = ∞ × 0

And

Notice that this approach to informally defining infinity produces an equation (the middle equation of the three above) in which something times 0 does not give 0! Because of this difficulty, and because the rules of algebra used to write and transform the equations apply to numbers, some mathematicians claim that division by 0 should not be allowed because ∞ may not be a defined number. They argue that dividing by 0 does*not* give infinity, but rather that infinity is undefined.

Another method of attempting to define infinity is to examine sets and their elements. If in counting the elements of a set one-by-one the counting never ends, the set can be said to be infinite.

Although students are typically taught that "one cannot divide by 0," it can be argued that = 0 (read as "one divided by infinity"). How is this possible? Observe the following progression.

Note that as the denominator, or the divisor, becomes larger, the value of the fraction (or the "quotient") becomes smaller. What happens if the denominators become very large?

One can see that as the denominator becomes extremely large, the fraction values approach 0. Indeed, if one thinks of infinity as "ultimately large," one can see that the value of the fraction will likewise be "ultimately small," or 0. Hence, one informal (but useful) way to define infinity is "the number that 1 can be divided by to get 0." Actually, there is no need to use the number 1 as the numerator here; any number divided by infinity will produce 0.

Using algebra, one can come up with another definition of infinity. By transforming the following equation we see that infinity is what results if 1 is divided by 0.

If

Then 1 = ∞ × 0

And

Notice that this approach to informally defining infinity produces an equation (the middle equation of the three above) in which something times 0 does not give 0! Because of this difficulty, and because the rules of algebra used to write and transform the equations apply to numbers, some mathematicians claim that division by 0 should not be allowed because ∞ may not be a defined number. They argue that dividing by 0 does

Another method of attempting to define infinity is to examine sets and their elements. If in counting the elements of a set one-by-one the counting never ends, the set can be said to be infinite.

Although there are many definitions for infinity, none of them actually state infinity is the largest possible.

There are different sizes of infinity!

]]>When I first laid my eyes on Edward Frenkel's 292 page text written entirely about math, I was quite honestly frightened. I couldn't help but to think to myself how awful it was going to be to read a text that large about math... Yeah, even as a math major! Not only that, I thought he must be a brave man to not only put the words love and math into the same sentence and title, but to write an entire text. My initial thoughts were that I was going to be reading about how much the author loved math and it would be full of complicated algorithms, equations, graphs, math sub-groups, etc. Basically, I thought it would put me to sleep at every page turn. Yes, this is coming from a math major. But I could not have been more wrong in my assumptions.

When I began my journey as a math major surprisingly I was one of those people who thought I would only be working with numbers, you know all the "fun" algebra stuff. I couldn't have imagined being introduced to proof writing, Euclid, or derivatives, let alone actually learning to solve and/or prove parts of these maths. And then I hit the point in my college career where people gave "the face" when I told them I was a math major. At first I didn't understand why this sudden fear came over peoples' faces when I mentioned math. Then calculus showed up and I too was making AND feeling that face! But I decided to take on these upper level math classes and subjects as best I could. After 5+ years of spending 75% of my semesters studying a "new" math, I started to understand why math was seen as the horror movie of school subjects. When you're on the outside looking in, it can be scary trying to, for example, explain why everything you were taught was right because the teacher said so. And for some being able to little things like prove WHY even integers are even is a beautiful thing. And for others the question remain, "how can math be beautiful"?

With his text I think Frenkel set out to open up peoples' minds about mathematics and to show people that the fear they felt was not caused by math itself, but by the stereotypes we see everyday in the media and anywhere for that matter. When people think of mathematicians they think of withdrawn, nerdy, awkward, genius-like individuals. People who don't necessarily fit society's norm. And they think of math as number crunching and graph drawing. But this text does a great job of showing how math is beautiful. Not only that, it is a commonality among countries. It is an art form in itself. Instead of painting with paint and paint brushes, math uses proofs, graphs, tessellations, and many other things. It is more than numbers. It is the very foundation , in a way, of our world. When you read Frenkels book, you realize that it isn't money that makes the world go around... It is the math behind the money. It isn't the architects that create our objects, it is the math behind the architect. And when beautiful things are created, they are created using math. Think about it. What are shapes? What are lines? Numbers? Measurements?

Overall I think Frenkel was able to take some of the scary out of math. I found myself really connecting with his chapters on symmetry and love. It may seem a small matter to some, but for me it opened my eyes because I never really thought about why one object was more symmetrical than another. Like Frenkel, I provided answers without really taking into account why. The mentor goes on to ask Frenkel which was more symmetrical, a circle table or a square table. Like most Frenkel said the circle table, but when asked why he hesitated (like most). The mentor then goes on to explain that we know the circle table is more symmetrical than the square table because no matter how you move the table it remains the same. It has infinitely many symmetries. But, the square table only has four symmetries (90, 180, 270, and 360 degrees) because if you move it any other degrees' it would not be the same. For me, this was mind blowing. And it shows just how much math can open one's mind. just imagine if people took the way we question things (and prove) in math and used this in everyday life. We would have a world where a lot more people didn't just go along with things because someone said so. We'd investigate more!

Lastly, I just have to hit on how Frenkel was able to effectively put love and math in the same text. I mean after reading the chapters before and taking in the information, just imagine if one could come up with a formula for love. I think he gives us something to really think about. I don't mean to dwell on being able to actually create a formula for love, but just think about what we can create with math. Not only that, bringing in love, another universal thing, and being able to tie to something a lot of people are fearful of, is genius. In a lot of ways math and love are similar. Both are something that can be so beautiful, but at times the road to get there is rough. We don't always get love right on the first try, but we keep trying. Just like with math.

***I highly recommend reading this text! It was worth every penny and then some! It goes to show one should love and learn often!***

]]>1. Hypothesis- Conjecture

2. Experiment- Proof

3. Observation- Proof

4. Analysis- Proof

5. Conclusion- Proposition (proven true)

Though this is a good argument, as I stated before, there are some who believe math and science are separate. Those who believe the studies are separate, for the most part, believe so because they view math as being inspired by nature whereas in science we go out into the world and observe.

Either way, my personal opinion is that the relationship between mathematics and science is a two way street. Mathematics draws from science when necessary and science draws from math when necessary. Often time when we are working in mathematics we are experimenting (specifically with proofs). And often time in science we need calculations and formulas, and that alone is large part of mathematics.

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Magic Math Squares originated in early 650 BC during a huge flood in China. From the flood emerged a turtle with a strange 3x3 grid pattern on its shell. What was found was that the sum of the numbers inside each individual column and row of the turtles 3x3 grid was 15! Magic squares also have origins in Persia, Arabia, India, and Europe. In 1510 mathematician Heinrich Agrippa expanded on the magical squares by creating magical squares of orders 3 to 9 whose sums add up to 15, 34, 65, 111, 175, 260, and 369 respectively

Then there is another square grid game, Soduko! The objective of Soduko is to fill the columns and rows of a 9x9 grid with the numbers.1-9 using each number once. The first modern Soduko puzzle is credited as being published by Howard Gerns in 1979. Since then it has become a world wide phenomenon! There are also variations of the puzzle/game that use Pentomino regions and Heptomino regions.]]>

So, this week instead of the usual full length math blog, I decided to play around with some tessellations that combined two different shapes. For my tessellation I combined a puzzle piece and a hand drawn flower. I began by taking a 11'8.5" piece of printer paper and splitting it into 6 sections. In the first square I sketched the original figure (the puzzle piece).

Step 1: The Puzzle Piece

Step 2: The Flower (Hand drawn in my tessellation, but can also be a pre made cut out figure)

Step 3: The Tessellations

I played around with how I was going to combine the two figures. I came to the conclusion that it was best to draw the more complex figure first (flower) and then draw the puzzle piece on top of it so that the puzzle pieces connected.

I played around with how I was going to combine the two figures. I came to the conclusion that it was best to draw the more complex figure first (flower) and then draw the puzzle piece on top of it so that the puzzle pieces connected.

Step 4: I completed my tessellation on half of the sheet and once I was finished I colored in the flowers first (making sure to keep a pattern), then I went back and colored in the empty space on the puzzle pieces.

]]>Overall I think the activity goes to show just how important it is to be mindful of wording in mathematics. Just like we are taught in science classes the importance of giving thorough instructions, it is equally important to word certain aspects of mathematics very carefully.]]>

While I know counting and being able to do algebraic operations in math is extremely important, I think that the logic and reasoning behind the math plays an equally important role. It is the underlying element in all things. Without even realizing it we use math in everything we do. When we cook, when we go to work, even when we're driving. Since math can be and is so complex, I think that is why it plays such an important role in peoples; everyday life.

What I am finding as I continue on into my academic career is the possibilities mathematics creates for people. Growing up I always thought the only careers in mathematics was teaching or banking, but I have learned there are just as many career opportunities as there are proofs in math. And I have also found that really studying math teaches one to question why things are and not just take them at face value. For the most part, up until I entered college, I was always told how to solve math problems and I was given the formulas and "steps", but I was not encouraged to explore further. And at times the fact that I went through 14 years of schooling (Preschool-12) hindered me from being successful in classes that were essentially centered around reasoning and logic.

I think after taking multiple math education classes and Euclidean Geometry is when I finally saw just how much can come out of using previous knowledge to both question and deduce mathematical reasoning. It really blew my mind how one man could stump so many people for so long. I mean he an entire subset of math was created behind this man! It always amazes me how much can be built from math just by questioning and drawing from previous knowledge. Math is essentially as complicated, complex, and mind blowing as it gets!

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