Math-Journal:Understanding Math

Inversion

noreply@blogger.com (Unknown) — Fri, 11 Oct 2024 00:00:00 +0000

Inversion is a process of working backwards, or inverting operations to solve a problem. Example of inverse operations are addition and subtraction. They are each an inverse of the other. Multiplication and division are inverses. Just as you can work backwards to check a solution to a problem you can work through inversion to find many ways to solve problems from the mundane to the extremely complex.

We can use inversion to add two numbers without adding. In, fact we can use it to add through the use of subtraction. For example: we can add 3 + 5 by subtraction. First we will subtract 3 from 10 to get 7 then from 7 we well subtract the next number 5 to get 2. We can then subtract 2 from 10 again to get 8.

Now this seems an incredibly long way to add two numbers together. Yet, the importance is not in the specific inversion itself. It is in the way of thinking that it promotes. The ability to conceive of a new way of approaching a problem can often lead to a more profound discovery. It could lead you into finding a way to solve entire groups of problems.

It is a simple concept, work backwards. Look at the data and work backwards to determine the source.
An example can be found in inverse scattering problems. These problems look at the scattering to determine the source, or likely causal factors. Such as in sonar technologies. You look at the variations in the sound wave echos to locate objects.

The inverse scattering problem solving technique is also used in medicine. The Positron emission tomography (PET) scan uses the detection of photon bursts from the beta decay of radioisotopes.
Then working to generate an image of the targeted tissue from the scattered data.

It it is based on the category of problems called inverse problems. In these problems the likely source is the unknown, and only through inversion can you find trace back to the source. It is probably the most important class of problems, with this most important process of solving them.

Kindergarten Math worksheets

noreply@blogger.com (Unknown) — Thu, 10 Oct 2024 14:43:00 +0000

I know that I haven't posted in a while, I have been very busy getting my daughter ready to start kindergarten. In doing this, I found that worksheets have been extremely helpful. So I created a few math worksheets of my own to help out. I decided I would share them with you so you may download and print the the worksheets below. You may need to open them on a new page to get them zoomed to the right size for printing.

Forms and Algebra

noreply@blogger.com (Unknown) — Thu, 10 Oct 2024 14:32:00 +0000

When studying mathematics at any level, one might find themselves asking where did this stuff come from? , or why is this useful?. The answer to these questions can be summed up in one word , Algebra. Why algebra? Why is it so important? Well, let’s look at what algebra actually means. Algebra, throughout history, has developed many rules, relations, and symbols to answer some of the everyday problems we encounter. The history of algebra and how it was formed, in my opinion, has little to do with what it actually means. I will instead explain algebra by use of philosophy. We can call it the philosophical theory of algebraic reasoning.

The Theory of forms and Algebra

The theory of Forms is basically that real-world objects have forms. For example, when you look at your table, how do you know it’s a table? In fact, your table could be extremely different from all other tables, but you know it’s a table because it has the form of a table. The form of a table is that it has a flat surface at the top, and a support structure at the bottom. The Form of the table can be restated as the abstraction of a table. By having the abstract model or form of a table, we can then manipulate or customize our table to look how we want so long as it stays within the basic bounds of the form. We can also now take our base form and use it to define specific types of tables.

Algebra works in the same way as the theory of forms. In algebra we use letters to symbolize or initialize the form of a number. We call the letters variables because they can change their value. Let’s look at the statement 1+1=2. Even though we have numbers and not variables this is still an abstraction. This is a perfect abstraction for general purposes it gives us a specific form unity. The 1 is the form or abstraction of a real world object. It can be any object as long as it is a single object. So to translate this statement we would say that a single object and a single object together has the form of two objects. For most the meaning of this statement is obvious, however; the statement a + a = b is not as obvious. Also, this statement does not mean the same as 1+1=2. Table + Table = 2 tables this statement is equivalent to 1+1=2 only because we are assuming the Knowledge of what the symbols 1, 2, + , =, represent. So we build our philosophical theory of algebra off of the assumption that we don’t need to define the symbols for numbers and what they mean.

The symbols 1 and 2 have a very specific meaning, but their properties, and interactions have a more complicated philosophical meaning. So we have a gap in our ability to abstract numbers and their properties, which is why we use the letters or variables. The variables are in fact a form of a form. Sounds redundant, but it is necessary to provide abstraction from something that is already an abstraction. This concept is where it gets a little tricky. We can say a variable is the form of a number but not all numbers only numbers that have the specific properties we provide with the rest of the statement. Take our a + a = b statement. This statement gives us a relationship between any two numbers that can be represented in this form a + a = b which has many numerical equivalences. It can mean 1+1 =2, 2+2 = 4, 3+3 =6, 4+4 = 8…etc. In fact this statement provides an abstract definition of even numbers. Statements like this should then be called definitions of forms. We shall go back to our table example again. The basic form of a table is flat surface at the top and support structure at the bottom. What if we said table + wood? We are giving an abstract definition of tables that are made out of wood. So table + wood is the same as a + a =b they are both abstract definitions of forms. The only difference is abstract definitions of numerical forms take a little more interpretation. However both assume an understanding of something else. The table + wood example assumes an understanding of the form of wood and the form of table. The a+a=b example assumes an understanding that 2,4,6,8,10…etc are called even numbers. a+a=b just gives us a definition of even numbers without trying to list all of the even numbers up to infinity. So abstraction is actually necessary to define specific numbers and number relationships, because we cannot realistically list every even number to infinity.

Why is algebra the heart of mathematics?

The abstraction of algebra gives us strong foundation by which to build many more complex definitions with a wide range of uses in everyday life. A lot of these concepts we use every day without even realizing it. In fact nearly every branch of mathematics would not exist without the foundation laid down by algebra. For example statistics, uses many definitions that would not exist without the framework of algebra. We even use algebraic concepts in many other subjects. For example in composition classes we use abstraction as a tool to write papers we start with an outline, which is simply an abstraction. Our outline gives us a simple abstract definition of our paper which makes writing our paper much easier. Language itself is an abstract definition of our thoughts and feelings.

Multiplication with positional notation and the distributive property

noreply@blogger.com (Unknown) — Thu, 10 Oct 2024 14:32:00 +0000

In this post we will be learning how to use positional notation to perform multiplication.

If you need a review on positional notation please refer to my previous post on this subject.
Positional Notation

To refresh on what positional notation looks like we will write the number 235 with positional notation.

With a smaller example we can see if this allows us to learn something new about multiplication.

So to multiply 32 and 24 we will first write these two numbers into positional notation.

Does this look familiar? If we replace the 10 with x and since 10⁰= 1 we will

replace it to and then we have:.

To put into a more familiar form:

(3x + 2)(2x + 4)

With our number in this form we can now use basic algebra.

From distributive property.

a(b+c)

(ab)

(ac)

We can now multiply this out using what is sometimes referred to as the foil method *(First Outer Inner Last)

Now if we substitute the 10 back in for x

To see how this works scaled up we will multiply 3259 and 1564

And again we can put this into the familiar form:Now again to replace the 10 with x and since 1(x) is just x we remove the 1 in the second number:

Now we gather like terms:

If we put the 10 back in for x then multiply we will get

And here is our answer using positional notation and distributive property to multiply two numbers.

5,097,076

5th Iteration Minecraft Mighty Menger Sponge!!!!!!!!!!!!!!!

noreply@blogger.com (Unknown) — Thu, 10 Oct 2024 14:31:00 +0000

In an earlier post I created a 4^th iteration 160,00 block Menger Cube in minecraft. After I discovered the newly added structure block I was able to construct the 5^th iteration 3,200,000 block version.

After I cleared out a large enough area all the way down to bedrock I began construction.

Here is the bottom section with 5, 4th iteration menger cubes

The 32x32x32 limit of the structure block meant that I had to use 3rd iteration 8,000 block menger cubes Menger cubes to construct this because the 4th iteration was more than the 32 cubed or 32,768 limit.

An fun side note is that I constructed this with x = 0, z = 0 at the very center.

With the bottom section of 8, 4th iteration Menger cubes finished I it was time to do the second row of 4,

And the beginning of the top row of 8, 4th iterations menger cubes

From the above picture you can see the outline of the structure block for the 3rd iteration Menger cube. It takes 20 of the 3rd iteration to make a 4th iteration so for each of the 20, 4th iteration cubes I had place the structure block 20 times. That is still a huge drop from having to place each of the 3.2 million cubes.

If you can't make out the y coordinates below this structure goes from y = 0 to just under the minecraft build height limit. What this means is that the 6th generation is not gonna happen in minecraft unless that limit is changed.

Another side note a quick way to clear an area using the structure block is to save a 32x32x32 area of just air then use the structure blocks in the area you need to clear. You could also use the /fill command to fill an area with air but I like the structure block approach.

Here is a few shots of the finished product a megalith of 3.2 million blocks!!!!!!

And here is a few shots of me starting to decorate the inside.

Does poverty hurt academic success?

noreply@blogger.com (Unknown) — Tue, 08 Oct 2024 14:41:00 +0000

While reading through the various new articles about test scores, I have seen a common argument about poverty affecting test scores. Is this true or is it just an excuse?

I will start out with a personal note. I was a very impoverished teen in high school. I had to work two jobs while going to school. My parents were so poor I eventually had to live with my grandparents, who although were not much better off than my parents, they at least provided a steady home.

However they were unable to afford all of the things that I needed for school. My school required special clear backpacks which my family could not afford. I had to buy those and I was responsible for all of my clothing, car payment and insurance.
School supplies was another expense I had to work for.

While most students were sitting at home doing homework or out with friends, I was at work. I became an emancipated teen so I could work the late hours I needed just to survive financially. Unlike the students who would go out with friends I was unable to stay up late doing homework, or do it in the morning before class. I worked so late that I was already only getting a few hours of sleep. I did however pay attention in class, and read while on my lunch breaks at work.

Luckily most of my teachers, would focus on teaching an understanding of material rather that just rote memorization. My teachers also spoke to me on their level like I was a colleague in some situations, or they would at least speak to me like an adult rather than like a third grader. Not all of them were like this, I did have some bad teachers. However, most of it was due to the student to teacher ratio being so unbalanced that inexperienced teachers had to sometimes ":fill in" the gaps. No matter how old you are , most teachers treat their students like small children.

I believe it was the way some of my teachers would treat me that helped me not only understand the material, but also have the desire and ability to learn on my own.
Due to school policies homework was 2/3 of the grade and I did very poorly in my classes. However all of my final exams were A's and B's, which is the only reason I was allowed to graduate. ( thanks to an amazing teacher Mrs Janice Miller who rallied the rest of my teachers to get the school to allow me graduation and a diploma) I also was ale to achieve a 35 on the ACT test which shows that I was still able to learn despite my financial situation.

I have gotten a little off topic here, but I believe it is important to see that the way we treat students can go a long way in helping them learn. So if you have a student who is on a free lunch program or their parents can not afford all of the things they need for school, treat them like an adult who is at the same economic status that you are. They will respect you so much more for it , and will be more apt to get a lot more out of your lessons.

I will leave off with a question. Does how much money you have determine intelligence? Does a student whose parents can buy them a a brand new laptop, have an advantage over the student whose parents can barely afford pencils and paper? Should we change how we teach students of different economic backgrounds?

The answer to all of these is no. If a course in high school is designed to require a laptop the school should provide one, or not require it. If a school is in a lower income district it should still get the same amount of funding that a school in a high income district gets.

Minecraft 4th iteration Menger Sponge

noreply@blogger.com (Unknown) — Wed, 13 Apr 2016 12:59:00 +0000

Minecraft Menger sponge (Menger Universal Curve) :

The Menger sponge is a 3 dimensional cube that models the Sierpinski carpet. The Sierpinski carpet is a fractal generalization of the Cantor set.

The Sierpinski carpet is a 2 dimensional figure starting with a square and then subdividing into 9 sections. You then remove the center section.

The next step is to take the remaining 8 sections and repeat the process for each one.

After that you are left with an image where the middle section is missing from each of the 8 subsections of the original figure. You then take each of those subsections and repeat the process again. This will leave you with this figure:

You can then repeat the process again....

An amazing property of the Sierpinski carpet is that as the number of iterations of this process approaches infinity the area will approach 0

When this process is moved up into 3 dimensions you construct a Sierpinski sponge. The Menger cube is slight alteration of the Sierpinski sponge.

The Menger sponge has a similar amazing property that as the iterations approach infinity the Surface area approaches infinity and the Volume approaches 0..

I created a 4 iteration Menger sponge in Minecraft, starting with a 3X3X3cube.

Then I removed the center block from each of the 6 faces.

I Then cloned this cube using the Minecraft /clone command to get to 20 cubes arranged to show the next iteration of the Menger sponge.

I then cloned that cube to make 20 more copies to construct the next iteration.

At this point I will note that for the Menger sponge the number of cubes (or in this case blocks) is given by the formula (Number of blocks) N = 20ⁿ where n is the number of iterations and the above cube is iteration 3. which is N = 20³ = 8000 blocks.

For the final project I decided to move over into my realms server and construct a 4 iteration Menger cube with N = 20⁴ = 160,000 blocks...!!!!

The controversy with common core:

noreply@blogger.com (Unknown) — Sat, 09 Apr 2016 05:45:00 +0000

The controversy with common core:

Math has become a new phobia among youth and current parents. It has many wearing math illiteracy as a badge of honor. The problem began with adults experiencing negativity at a young age toward the subject. It was often taught in very rigid and mechanical tone. The rigidity is where the problem we face today finds its roots. Math was drilled into our heads with the idea that there is only one way to do things, and anything else leads to lower grades. Many teachers were strictly against alternative methods, and led to an ingrained aversion to the subject for their students.

Other issues include the teaching methods themselves, as well as the lack of guidance for students who may take longer to grasp the subject. In the interest of time, and testing, the teachers are encouraged to cover large amounts of material in a short amount of time. These factors in combination are what has resulted in the attitude toward math that parents possess today. This attitude is also being passed on to their children, by encouraging an avoidance of mathematics. The parents own aversion and lack of proper education in mathematics makes it difficult to understand the material their children are bringing home. The new common core standards and teaching tools are incredibly different from what many of these parents are familiar with.

Currently students are being introduced new concepts for learning math with the development of common core. Altering how things are done, can be beneficial to finding better teaching options. Sure many of these methods are breaking from tradition, but as with all things we are learning. Which forces us to integrate new information and approaches to teaching. Parents and educators all need to stay aware of this. As we learn and grow intellectually, knowledge must force us to change. However, this change conflicts with the fact that the parents of these children are unable to help them to better their skills in these areas. Here we see the roots of the problem. The parents were not taught by common core standards, and therefore cannot aid their children until first learning the methods themselves.

If parents or teachers see common core standards as a problem maybe schools could host a week long open seminar on what is actually going to be taught and how it is implemented. The schools could have qualified instructors in the common core standards come and talk with parents and teachers. They could even teach some of the new math strategies to the parents so they can be better informed on how to help their children adapt to the new material.

Given the chance to come to understand the common core math standards could help change the way some respond to them. I understand the problems many people have with common core math. Seeing a drop in test scores with new standards being put into place is a shock, but when you look into it you see it is not the problem with the tests, it is us. We are failing our children. The common core math standards are well informed and vetted by leaders in the stem fields. And these are the things our children need to learn to remain competitive with their peers around the globe.

The largest prime

noreply@blogger.com (Unknown) — Sat, 19 Mar 2016 00:51:00 +0000

The largest prime number ever seen by humankind was found on January 7, 2016. It weighs in at 22,338,618 digits in length. It is what is known as a Mersenne prime (2^74,207,281-1). Mersenne primes are a group of prime numbers of the form 2ⁿ -1 where n is is assumed to be prime. This is a fascinating find. Prime numbers are the "building blocks" of all other numbers. Their unique nature has led many number theorists and math enthusiasts to study them in great detail. Although they are used heavily in cryptography, the latest find is far too large to be used in any encryption schemes to date. But as computation theory evolves and computer architectures become more sophisticated these extremely large primes may find their place.

The "common core check guy"

noreply@blogger.com (Unknown) — Tue, 22 Sep 2015 19:15:00 +0000

In the news recently, we see the story of Doug Herrmann. Doug is a father from Ohio who out of frustration with his child's math homework, wrote a check using common core methods. He was frustrated because he was unable to understand what the school was teaching his son, and therefore was unable to help his son with his math homework.

The ten card system that his “viral” check picture was intended to highlight is not a terrible method of mental math. One exception is that for some it abstracts the idea so far that it seems to become separate from the actual “math”. Utilizing grouping techniques is in no way a “bad” technique to understand mental math. However, it becomes a problem if too much emphasis is placed on a single rigid method.

Using the ten card system is one of many useful techniques for understanding grouping. We only run into problems when the standardized testing systems force children to “believe” there is only one right way. The testing should only test for the ability to solve problems and not for the strict adherence to a single “correct” way of getting there. The practice tests I have looked at appear as though they are actually structured that way. The second grade tests do emphasize place value, but do not seem to indicate a single method. Now as far as the rigidity of the teachers, I have not seen how this comes into play. I do know that in many instances students fall behind due to teachers being too strict in the techniques that you use.

Please understand that I am well aware that there has been a rising trend in math illiteracy. I am not completely blasting common core. I , along with other parents, have some questions about the implementation and flexibility of these standards.

Some questions I have include:

How flexible are the methods used ?
- Are students being taught that the ten card system is the “only correct” method ?
- Will students be penalized for utilizing other techniques to arrive at their answers ?

How are conflicts between common core, and how the parents teach their children handled ?
- Do teachers penalize their students for the way their parents teach them to solve problems ?
Is there an awareness of the different ways people understand concepts ?
- Is there an accommodation in place for students who understand place value through other means than just grouping visualizations? ( e.g. positional notation instead of ten cards.)

These questions go beyond just common core. The rigid adherence to single techniques has long been the culprit behind the fall in mathematics education.

On a personal note, I have been a victim to the rigidity of teachers. When I was in high school(many years before common core) I failed a math class simply due to my teacher not approving of my mental math techniques. To be fair, I would like to emphasize the math teacher was not actually a math teacher, she was a soccer coach moonlighting as a math teacher due to a poor student teacher ratio.

The conflict arose when I was forced to show my work(and by show my work I mean that my teacher wanted to see the remedial addition, subtraction, multiplication, and division) This was an algebra class, where showing your work meant to show the steps taken to simplify the expressions. However, this instructor insisted we not just show that we had to multiply she wanted to see the actual steps we took to multiply. I thought this was odd in an algebra class, because learning to multiply was second grade. However, I did comply and attempted to be verbose with “showing my work”. However, my method of “long hand” multiplication was different from what she was taught. Most people are taught the “only” way to multiply is to start in the one column and carry and borrow and all these other concepts. I used a more “short hand” method which lends itself well to mental math. I would start with the highest power of ten column and work left to right instead of right to left. For example if I were to multiply two digit numbers I would start in the tens column and multiply no need for carrying and borrowing. You are just simply adding up zeros. This method isn't understood by all, but it was how I understood numbers and place value. Unfortunately, my math teacher was strict and rigid in how she wanted it done and failed me on all of my quizzes. I understood the algebra part and understood the basic calculations part, and arrived at the correct answer. I just applied a different method for the basic calculations.

Now my story is sadly similar to many other students. For most, these situations reinforce their frustrations with mathematics. It can quickly lead students to make snap judgments about the efficacy of learning and understanding math. They soon start to see math as a foolish endeavor with rigid methodologies not worth their time. Or they could simply just give up believing they will never understand and struggle to merely pass.

I don't exclusively place the blame on teachers for the problem of rigidity. The parents also hold a share of the blame. As with the “common core check guy” Doug Hermann and some of the comments on his posting, they also find themselves rigid in their understanding. Simply not understanding a technique of solving a problem does not condemn it. Rather, it simply means you should probably learn more about it before condemnation.

Propositional calculus: an introduction

noreply@blogger.com (Unknown) — Wed, 01 Jul 2015 22:48:00 +0000

Propositional calculus is the branch of mathematics that deals with the rules of logic and evaluation of statements or propositions. It is sometimes referred to as sentential calculus for its use of the sentential conjunctions. It is primarily concerned generating laws for evaluating conjugated statements. Propositional calculus is part of a broader science of logic and proof.

Some of the symbols used in propositional calculus:

Symbol	Meaning
¬	Not or negation
^	and
v	or
→	implies
↔	If and only if
P	Sentential Statement
Q	Sentential Statement
T	True
F	False

In propositional calculus the above symbols are used in constructing truth tables for evaluating compound or conjugated sentences. The truth tables provide a short hand tool for deriving new laws from the simple compound sentential statements. Truth tables like the one below shows some of the simple rules and the new rules derived from those.

P	Q	P→Q	P^Q	PvQ	(P^Q)→P	P ↔ Q
T	T	T	T	T	T	T
F	T	T	F	T	T	F
T	F	F	F	T	T	F
F	F	T	F	F	T	T

This is just a simple introduction to propositional calculus, it is part of a much broader branch of propositional logic . It is useful in the formulation of logic rules, and methods of proof.

The "Reality" of math and logic

noreply@blogger.com (Unknown) — Sun, 21 Jun 2015 01:55:00 +0000

Logic has been used as a methodology even before its formalization. In order to survive, our progenitors had to be able to deduce whether a particular place or event was dangerous. These more earlier versions of ourselves, needed to determine what was safe to eat and where food was located. In this sense they were using logic intuitively.

As societies evolved, we became aware of a need for a system of accounting. We developed symbols to serve as abstract models for real world objects. Empirically , we can demonstrate the need for a modeling system. We can also demonstrate that these models of real world objects work. The models functionality means they can be used for future discoveries, and aids in civilization building.

Enumeration permeates our entire civilization and history on multiple levels. It observably exists separate from language. From the simplest counting systems, patterns began to emerge. The laws of mathematics became more complex, and a method of proof was needed that was empirically verifiable. The earliest proofs were purely empirical as used by the Pythagoreans and Thales of Miletus. Over time, mathematical proofs slowly became less heuristic and a formalized system of logic was becoming more prominent. Logic was largely an emergent aspect of these methods of empirical verification. The methods of logical proof slowly grew over the next two and half centuries from Thales around 550 BCE to Euclid around 300 BCE. Euclid is credited with formulation of the axiomatic method of proof.

The Euclidean style of proof through axioms, appears assumptive, or only justifiable a priori.

The axioms served as descriptions or definitions of geometric objects. The definitions could be demonstrated empirically in general, but they did rely on assumptions. For example, take the statement 2 points determine a line. This is a postulate proposed as self-evident, and deduced from it is many other theorems. If you were to take this as meaning all statements derived rely on a non-empirical premise you would be incorrect. The statement can be demonstrated as empirically true. You would simply have to draw a line through only one point. The line drawn, would be a point and not a line. To construct a line you would have no choice but to have it pass through multiple points. So, we can see the statement is both self-evident and empirically demonstrable.

The evolution of mathematical systems of proof allowed for a more formalized system of logic. All of which finds its origins in our innate pattern recognition. The ability to recognize patterns and natural desire to seek them out led to the mathematical modeling of real world objects. The modeling techniques made a complete system of logic and mathematics more easily constructed and shared. We formulated logical and mathematically complete laws and theorems from some of the simplest models.

We see that these models are intuitive,empirical, and justified in their use.

Positional Notation

noreply@blogger.com (Unknown) — Wed, 03 Jun 2015 03:16:00 +0000

Positional Notation

Our number system is base ten, as many learned in elementary school. Which means that there is only ten symbols to represent numbers. 0,1,2,3,4,5,6,7,8,9 . To represent ten we use 1 and 0 (10). Which when we learned place value, we recognized this as 1 in the tens place and 0 in the ones place. What this means is that the position of the symbol will determine the value of the number. For example when we see the symbol 34 we recognize it as 3 tens and 4 ones. We would often see something like:

Tens	ones
3	4

We are mostly taught this and just accept it as is and move on. To dig a little deeper into to place value I will introduce a new way of looking at this (Positional Notation) . To rewrite our example ,34, into positional notation we get this:

3 x 10¹ + 4 x 10º

The reason I write it this way is that is represents a number as powers of the base it is in. In this case it is base 10. Positional notation gives us a way of looking at numbers relevant to their base. The reason this is useful to learn, is that it gives us a quick way to convert numbers from other bases to base ten. For example, if you need to convert the number 101 in binary to base 10 you can rewrite 101 in positional notation and convert it easily. Since 101 is in binary we will use powers of 2 instead of 10 and we get

1 x 2² + 0 x 2¹ + 1 x 2º

1 x 4 + 0 x 2 + 1 x 1

4 + 0 + 1

As you can see from the table above just by rewriting the number into positional notation relevant to the base it is in you add it all up and you get the number in base 10. So when we convert 101 from binary to base ten we get 5.

This is just a neat little trick I use to quickly convert numbers from other bases to base 10. With a little practice, you can convert numbers from most bases back to base ten almost effortlessly. There is other ways of converting number to other bases, I just really like this one.

Python and Prime number Lists

noreply@blogger.com (Unknown) — Wed, 31 Jul 2013 02:21:00 +0000

Prime numbers, although simple in definition, provide several complexities in the field of mathematics, as well as in computer science. For reference, prime numbers are natural numbers which are only divisible by one and themselves. On the surface, this definition seems straightforward and not an important area for research. However, there are a great many applications of prime numbers. Prime numbers are essentially the building blocks of all other numbers. For centuries, this unique set of numbers has mesmerized and even baffled some of the greatest mathematicians. The distribution of prime numbers is another big area of mathematics research. It would be great if we could find a formula for the nth prime. Yet, it seems as though it would be impossible. We can however use computers to do this for us up to the limits of the processing power available. No real efficient algorithms exist to find primes other than methods of iterating through every number up to the square root of a given number to determine if it is prime. Finding the patterns with some high level mathematics, is being used but no solid proofs exist yet to revolutionize this area of research. With this in mind I would like to continue with discussing python in mathematics.

I previously showed a fairly simple program for Slope-intercept using python. The program served to output a formula in slope-intercept form given two points. In another post on computer programming in mathematics, I included some Java code for testing if a number is prime. Below I have a Python code to ask for 2 numbers to serve as the starting point and ending point for listing primes within that range. I have not included any error checking or input checking, so it is important to ensure this code works you must enter an integer greater than 1. If you do not do this you will not get a meaningful output or you will get errors. This was written and tested in Idle 3.2 so it may not work as expected in other versions. This code also utilizes the standard sieving algorithm to perform its prime checking so it is by no means efficient for larger inputs and can be very processor intensive

. CAUTION!!! only use this code to list primes in small ranges CAUTION!!! The caution is here because I have locked up my computer trying to list primes from 2 – 10,000,000 I would recommend running it on ranges in the 100's It was originally written for educational and research purposes only. I generally use it to write primes to text files to print out as educational resource materials.

#primecheck(x) checks if x is prime returns boolean#

def primecheck(x):



    i=2

    for i in
range(2,x):

        if x%i == 0:

            return
False

    return True

#primelist(start,stop) lists all prime numbers between start
and stop#

def primelist():

    start =
int(input('Enter starting point must be an integer greater than 1:  '))    

    stop =
int(input('Enter ending point must be an integer greater than 1:   '))   

    i=2

    primeList = []

    for i in
range(start,stop):

        if
primecheck(i) == True:

           
primeList.append(i)

    return primeList

print (primelist())

Python Slope-Intercept

noreply@blogger.com (Unknown) — Sun, 28 Jul 2013 12:44:00 +0000

Python is a wonderful programming language for quickly coding solutions to many problems. You can download python by going to python.org/download and selecting the version that’s best for your operating system. Python can run from command line or though Idle that comes with most python downloads. I have been working with python for almost a year now and it has become my “go to “for

a lot of work in mathematics. It is not necessarily the “best” language for processor intensive mathematical algorithms, but it can offer efficient and quick solutions that can be easily applied to larger projects. Recently, I have been doing some review of linear algebra and slope-intercept formulas. This is a fairly simple subject in mathematics but I thought it might be interesting to use python to solve some of these problems. My solution might not be elegant but it is fairly simple. I designed the program to take in two points (x1, y1), (x2, y2) and output an equation of the line in slope-intercept form. For those who may not be familiar slope-intercept form it is y = mx + b where m is the slope and b is the y- intercept. The slope is calculated from the two given points by the formula (y1-y2)/(x1-x2) and the y -intercept can be found by

b = y – mx.

Slope-intercept is useful in linear depreciation models. For example: if you bought a car at $81,292 and after 10 years it is only worth $33,121 to find out how much it will be worth after 13 years you can use slope-intercept to get a depreciation formula. The two points are at 0 years it is 81,292 (0, 81,292) and at 10 years it is 33,121 (10 , 33,121) substituting these numbers into our formulas you first find the slope which will represent the rate of depreciation. You use the slope formula (y1-y2)/(x1-x2)

(33,121 – 81,292) / (10-0) = -48,171 / 10 = -4817.1 and the y intercept is b = y – mx. 81292- 0(4817.1) = b= 81292 and putting it together you get y = (-4817.1) x + 81292 using this formula we can then determine the value of the car at 13 years by substituting 13 in for x

y = (-4817.1)13 + 81292 and we get 18,669.70 so after 13 years our car is worth $18,669.70

The Python code to find our linear depreciation formula is fairly straight forward

def slope():

result = ''



    x = float(input("enter x one"))

    y = float(input("enter y one"))

    a = float(input("enter x two"))

    b = float(input("enter y two"))

    slope = (y-b)/(x-a)

    yint = y - (slope*x)

    if yint < 0:

        sign='-'

    else:

        sign='+'

    result =
str("y="),str(slope),str("x"),sign,str(yint)

    return result

print(slope())







I use the input() method to take in the two
points (x,y), (a,b) 

then create a variable named “slope” and then immediately
assign it to the formula to find the slope given 2 points (y-b)/(x-a). I then
create a variable yint and assign it to the formula for the y-intercept. The
if-else statement I use to aid with formatting because generally the + (plus
sign) isn't displayed when you get a positive. The final line print(slope())
is there only for when you are running the code in console or idle if you were
to convert this to a python module it won't be needed.   
I am
sure there are more elegant and efficient ways of doing this as well as
built-in python methods to make it display much better but that’s the beauty of
python it’s easy to modify and play around with to get desired output style.

Standardized thinking.......

noreply@blogger.com (Unknown) — Sun, 30 Jun 2013 07:35:00 +0000

Despite efforts from Washington, math education is still in trouble in the U.S. We are now experiencing the so called “math wars”. Educators are competing with policy makers, and mathematicians are challenging both. The argument is that educators are focused on the mission and purity of education as a whole. Education is the basis for lifelong learning and free thought. When mathematics is concerned there is a wedge between mechanical learning and reasoning. University mathematicians believe that teaching reasoning before standard algorithms is crippling student‘s abilities to learn the more advanced concepts in college level mathematics. It is argued that before learning mathematical and numerical reasoning students must understand the standards of math.

The dangers of sticking to a mechanical standard algorithm approach are flexibility, intellectual growth, and individual ability. Flexibility refers to having a standard such as right to left based calculation. This concept alone can prove very confusing especially in younger students who are still honing their reading skills. In most languages, specifically English, students are learning to read from left to right. Every time they are exposed to any form of writing they are taught that it should be understood from left to right. When children begin learning math they are taught to read math in contradiction to all the other reading that they learn. In essence this can cause confusion in both subjects for most students. Not only is the left to right approach quicker and easier, it also flows logically with the way we are taught to read.

Another issue is that when teachers are grounded with this single algorithm method, students who understand it differently are often punished with bad grades. Some teachers are so set in the standard right to left and carry method, that if a student chooses to use a different method they are given a bad grade. The stigma put on outside the norm computation will stunt the students drive and understanding of math. It places in them the “why am I wrong when I got the same answer?” argument. I had a personal experience with this receiving a failing grade on a test because on the page where you “show your work” I did all multiplication from left to right. The teacher of the class I was in had no clue how that method worked and therefore failed me on my exam “for cheating”. The reason given was that the only way to multiply is the standard right to left so the work shown was an attempt to cover up cheating. Now this is a specific individual example that does not reflect all teachers, but there are still some that are so set in the standard that “out of box” thinking is “out the window”.

Aside from flexibility, we have the issue of intellectual growth. Intellectual growth is arguably the central idea of education. The ability to think and reason is devastatingly uncommon among many adults and children. Most educators wish to remedy this situation, by trying to step outside of the memorize and regurgitate on the test paradigm. Standardized testing is standardizing thinking and learning which is destroying innovation and lifelong learning. While most agree that some concepts should be tested and retained to be successful in college, the ways by which we test and use these concepts need to be modified.

The last issue is individual ability. Going back to the personal example of being labeled a cheater for utilizing a method that was easiest to me was a stab at my individual abilities. That failure caused me to have to take a much lower level math class then I should have been in twice. Luckily, the next time around I had a better informed teacher that removed me from that class put me into pre-calculus instead where I aced the exams. The point is that no matter how you try to standardize people, we are all individuals with individual ability to understand the world around us.

The Beal Prize

noreply@blogger.com (Unknown) — Sat, 08 Jun 2013 23:01:00 +0000

Andrew Beal a self-made billionaire has announced the prize of one million dollars to the person who can prove the Beal conjecture. The conjecture is similar to the famous last theorem of Pierre de Fermat. Fermat’s last theorem, originally written in 1636 remained unsolved until 1995. The theorem states that no three positive integers a, b, and c can satisfy the equation (aⁿ + bⁿ = cⁿ),for any integer value of n>2. The Beal conjecture begins with the equation A^X + B^Y = C^Z and states that if A, B, C, X, Y, and Z are all positive integers > 2 then A, B, and C must share a common factor. The novice to mathematics might ask the question, what does all of this mean or more importantly why is this conjecture important? This conjecture along with its predecessor Fermat’s last theorem is closely related to Diophantine equations. Diophantine equations are a special set of equations named for Greek mathematician Diophantus of Alexandria.

The Diophantine equations all dealt with whole numbers and involved multiple unknown quantities. The most famous of these equations is the Pythagorean Theorem, which is a²+b²= c ²where a and b are the two legs and c is the hypotenuse of a right triangle. The Pythagorean equation shows the relationship of the lengths of all of the sides. The real world applications of the Pythagorean Theorem are in engineering, architecture, cartography, and many other areas. Diophantine equations all find their way into many separate real world applications. The Beal conjecture is no exception the solution could lead to more applications, than just a simple intellectual curiosity. Hopefully, this one doesn’t take 3 centuries to solve as did its predecessor.

The proof of Fermat’s last Theorem was over 100 pages long and took 7 years of isolation for Sir Andrew John Wiles of Britain to complete. Will the Beal conjecture proof take as long? Now that the Fermat proof, which is closely related, is floating around, I propose that the solution to the Beal conjecture is within the Fermat proof or perhaps even an expansion or generalization of that proof.

Better learning through abstraction..…

noreply@blogger.com (Unknown) — Mon, 13 May 2013 13:52:00 +0000

There are many barriers to learning and teaching mathematics, some of which include: anxiety, comprehension, and focus. How do we get past these? First, let’s look at anxiety. Why does the mere mention of mathematics strike fear and confusion in students? It seems there are so many different causes of this to look at, that we are unable to pin point a single method to fix the problem. In fact to find the solution, we need use the solution. You will say “well that sounds impossible”. If we don’t know the solution, how can we use the solution to find it? There is a simple answer, abstraction.

I touched upon the idea of abstraction in Algebra: the true heart of mathematics. So rather than just focusing on a solution to all of the different barriers to learning mathematics, we instead use abstraction to find a general solution. We can then begin teaching abstraction rather than just mathematics. If we find ways to take the classic “math” out of the picture we avoid the instant fear that many students face. The statement “Today, class we are going to invent a new dictionary” is much less daunting than “Today, class we are going to learn algebra”. The first statement will immediately spark a curiosity. The students will begin asking questions, and wanting to know more.

What does inventing a new dictionary have to do with algebra? To answer this, we first need to think what is algebra? Algebra is basically a “dictionary” of formulas, theorems, and proofs which apply to a wide range of problems.

How do these relate to each other? Let’s look at the commutative property of addition. In algebra this is “defined” as (a + b = b + a) . To look at this in words rather than algebraic symbols, you get

Commutative property of addition: When adding two numbers together the order of the numbers does not change the result of the addition.

Where do we go from here? This seems no different than how algebra is introduced normally. What I am proposing is taking the math out of it in the beginning, and then slowly adding the math in after the concept of abstraction is understood. We ask the students to look at some common problems they may face in their daily lives. We will then list out some of the top suggestions. The students will then be asked to find the commonalities between the problems to group them together. Then the students are asked to make a name for each of the problem groups. They can then begin defining the general solution to those problem types. Keep it simple, only spend about a week introducing the concept of abstraction with simple problems. After the idea of abstraction seems clear then you could start adding the Math back into the process. You will begin with asking the students about math related problems. You could start with something simple like what is an even number. Many of the students by this point are familiar with the idea of what an even number is. However, this time you can approach the same way you approached your dictionary idea. You ask the students to create their own definition of an even number. Allowing them to create their own mathematical rules to use empowers them with the ability to not only recognize the steps of problem solving, but to understand it rather than just memorize it.

Teaching abstraction leaves a wide open doorway to many different possibilities. The hardest part of learning math for many students is the daunting amount of formulas and rules to remember. A student who knows how and why these rules work, and can look at a problem and devise their own algorithms for solving it, will be way ahead of the curve when they are faced with solving any problem they encounter. When you think about it you use this concept every day. As adults we face many different types of problems, and we solve them by first identifying the problem then abstracting a set of procedures for solving them. We sometimes have similar problems that we have solved before, and we encounter new problems that need new solutions. However, being successful in life is dependent on the solidarity and efficiency of our ability to solve problems. When I say problems I don’t mean just mathematical, it is the seemingly mundane parts of life for some and the more perplexing or dire situations we encounter that I call problems. Abstraction, logic, deductive and inductive reasoning are so important to a successful life that part of the curriculum of each grade level should be dedicated to them.

Analysis of the MegaMIllions lottery

noreply@blogger.com (Unknown) — Sat, 02 Feb 2013 22:59:00 +0000

I compiled some data in Excel using the winning numbers data from 5-17-2002 – 3-23-2012.A few of the charts created using the winning numbers for the Mega Millions lottery are found below.

This first chart describes the number of times each number is drawn for each pick. From the data we see that the number 2 is more likely to be the first number drawn because it shows up as the first pick more often. The number of times 2 has appeared first is 75.4% of the total number of times a 2 has been in the winning numbers.

How many times each numbers is drawn

first pick

second pick

third pick

fourth pick

fifth pick

mega ball

2's

4's

5's

6's

7's

8's

9's

10's

11's

12's

13's

14's

15's

16's

17's

18's

19's

20's

21's

22's

23's

24's

25's

26's

27's

28's

29's

30's

31's

32's

33's

34's

35's

36's

37's

38's

39's

40's

41's

42's

43's

44's

45's

46's

47's

48's

49's

50's

51's

52's

53's

54's

55's

56's

In this next chart I have laid out the average, mode, median, harmonic mean and other analysis of the number distributions across all of the different picks.

	first pick	second pick	third pick	fourth pick	fifth pick	mega ball
median all	7.5	18	28	38	49	24
Average all	9.278210117	18.6624514	28.3015564	37.42607	46.4542802	23.990272
mode all	2	17	27	39	52	36
standard deviation	7.137233978	9.10326996	10.2470703	9.59011029	7.6711884	13.842004
Geometric mean	6.565968069	16.139646	26.1471469	35.9559721	45.6978254	18.277583
Harmonic mean	4.207752217	13.1495089	23.5028125	34.0860285	44.7726832	10.698143
Mean - stddev	2.140976139	9.5591814	18.0544862	27.8359598	38.7830918	10.148268
Mean + stddev	14.63723398	27.10327	38.2470703	47.5901103	56.6711884	37.842004
avg mean+-stddev	8.389105058	18.3312257	28.1507782	37.713035	47.7271401	23.995136

This final chart is all of the data above rounded to the nearest whole number.

first pick	second pick	third pick	fourth pick	fifth pick	mega ball
8	18	28	38	49	24
9	19	28	37	46	24
2	17	27	39	52	36
7	9	10	10	8	14
7	16	26	36	46	18
4	13	24	34	45	11
2	10	18	28	39	10
15	27	38	48	57	38
8	18	28	38	48	24

It is important to note, that all these numbers represent the various distribution statistics of the numbers that were drawn at each of the televised drawings, and do not represent any regressive or other types of predictive models. These numbers are just simply a statistical analysis of the given data for educational purposes only. All data represents the real numbers published at http://www.megamillions.com/numbers/pastdrawings.asp The website offers a tool at the top of the page to select your data range and get all of the drawings

Brief Discussion on new Pell Grant law 2012

noreply@blogger.com (Unknown) — Tue, 30 Oct 2012 01:50:00 +0000

With the recent changes made to the pell grant eligibility rules, many low income students are left with basically two options. Dropout , or go deeper into debt by attempting to get more student loans. The changes to the pell grant were intended to save money in the budget by cutting education spending, and to eliminate the students who "abuse" the system. The main target being students who have spent six years attempting a bachelors or associates degree. However, with the way the law is setup is on percentage meaning that institutions or students that do not follow a "traditional" schedule will be cut off after just three years in some cases.

On a personal note, my wife is one of these affected students. She works full time and goes to school full time. When she first applied for the pell grant 3 years ago she was ensured to get through a 4 year bachelors program. Here we are 3 years later, and in the middle of a semester she is informed by the financial aid dept she needs to come up with $2000 to just finish the current classes she is in. This has put a great strain on our family. Now we are both going to have to work full time and hope that after paying for child care and bills there is enough left over to pay a fraction of the educational cost. If we were to save the money to cover the costs she would still have to take several months in between classes just to save enough for the next class. This will make her Aug 2013 graduation date be pushed far into 2014(if not beyond).

The situation for my wife is not an uncommon situation people are facing. There is a large number of students face similar circumstances. We have seen so much in the media about a growing need for qualified workers in the workforce. With these changes the media byline needs to be modified to "There is a growing need for middle to upper class qualified workers" The lower class has been effectively eliminated from the "qualified workers" category. So what message is our government sending us? Lower class shall remain lower class and barring a streak of extremely good luck, will never be able to crawl their way out. To most lower class families the middle and upper class is like an elite club that you can only be a member of if you have the right last name, win the lottery, or were lucky enough to get a "better than minimum wage" job.

Sorry for the rant, but I feel this was a very dangerous move on the part of the government. The number of dropouts and student loan defaults could potentially double if this is not amended. It represents a "huge" over site as to the problems this is going to cause and the damage it is going to do to the economy and the future workforce as a whole. A simple fix could be just adding a clause to the rule that allows students who are within a year from graduation to continue to receive funds until their reasonably anticipated graduation date, and if they do not complete their degree by that time then a cutoff of funds is reasonable. For students who already have a bachelors degree should be the ones with more strict cutoff schedules for funding a second bachelors or associates.

Computer Programming and Mathematics

noreply@blogger.com (Unknown) — Fri, 21 Sep 2012 22:23:00 +0000

Computers and Mathematics

As some of you may already know,there is a significant relationship between Mathematics and Computer Programming. Many mathematicians use programming to aid in their research. It can be an invaluable tool to achieving success in whatever proof you may be working on. My interest in how they connect is the algorithms you must create to apply to your program. In order to utilize a computer program to aid you in mathematics you must first write an algorithm,assuming you already know the programming language you will be using. The algorithm must then be translated to work in the programming language you chose.

Prime Number Checking

One of the most basic programming solutions is writing a program to find out if a given number is a prime number. The first step is to develop your algorithm. There are many different ways to check the if a number is prime or not. Most of them fairly complex, however there is one very simple technique. A prime number is only evenly divisible by one and itself. So the easiest way is to use modular division, which is simply dividing two numbers and keeping only the remainder. For example 32 modulo 5 = 2,

because 5 goes into 32, 6 times with a remainder of 2. Since prime numbers are only evenly divisible by one and themselves, prime numbers would always have a remainder greater than 0 when divided by all other numbers(other than 1 and themselves). With this knowledge we know that we have to test all numbers between one and the number we want to check if its prime.This can be limited even further by adding that the factors on any number can not exceed the square root of that number. Now our plain language algorithm is :

get number and call it n

calculate square root of n

test all numbers >=2 and <= square root of n

and if none of the numbers tested return a 0 remainder the number is prime

Java code to find if a number is prime

Our next step is to translate our algorithm into computer code. I will use java because that's my favorite so far. If you are not familiar with java you can find out more at http://docs.oracle.com/javase/tutorial/getStarted/index.html

The part of our algorithm where we calculate the square root and test all the numbers up to the square root is best achieved by a for() loop.the variable we will use to represent the number we want to test will be n and we will make it a "double" data type we will be using double for two reasons one the included Math.sqrt method we will be calling requires a double,.and the second reason is that we can check much higher numbers with double.The next variable we want will be used to store the numbers we are using in our modular division we will call it k and make it a double

lets code out our for() loop


double n;

double k;   

    for(double k = 2;i<=Math.sqrt(n);k++){

          if(n%k == 0){

              return false;

          }

  } return true;

That is our complete algorithm for prime checking translated into Java. I have tested this algorithm for numbers up to 500 digits long before it started to slow my pc. which is a few years old. I will gladly answer any question or comments on how to implement this into your java project as well as put more posts with how to make other java programs related to mathematics. Just leave a comment with you questions, requests, or suggestions . Thank you for stopping by.

Open book learning versus rote learning

noreply@blogger.com (Unknown) — Wed, 18 Apr 2012 11:35:00 +0000

What is rote learning?

Rote learning is the concept that you will memorize a procedure by endlessly repeating it until you “understand” it. Some believe this is the only way to truly understand a topic in mathematics. It is the “practice makes perfect” paradigm. The rote learning idea is that understanding is derived from memorization and recall over time. [edit:(9-24-2015) As we learn more about the human brain and the way it carves and hones pathways, we begin to see the the role of rote learning. We have come to learn that rote memorization of key concepts tends to in a way free up "ram" ,working memory, in our brains. With key concepts being stored in long term memory, our brains are able to process higher level logic faster. ] Understanding means to comprehend or to grasp the meaning of. Students with good memories will just simply memorize the formulas and recall them on demand at test time, and the teacher will believe they understand the concept and move on. Students who do not have great memories will struggle and get frustrated while trying to memorize the formulas. We can single them out in an attempt to give them extra help to understand, which in some cases actually embarrasses the students. [edit:(9-24-2015) A strategy to overcome singling out students with poor memory, is to implement a small section on memorization techniques. Proper application of memorization strategies can also teach concentration and focus techniques. (link 'www.psychologytoday.com')]

Embarrassment even in the smallest amount begins a psychological impairment on a child’s ability to learn. No matter how we approach the situation singling out one student or even multiple students can impair learning. Now I am sure there are some very exceptional teachers with the amazing ability to approach a student to find out why they are failing with minimal impact. Unfortunately, not all teachers can do this; it is a talent of charisma that we just simply cannot teach everyone. We are now left with the question: How do we give the children that need this extra attention help without singling them out? We first need to remove the extra pressure of rote learning exercises. We can do this by utilizing Open book learning approaches.

What is open book learning?

Open book learning takes away the pressure of having to cram for a test. Relieving a student’s stress will create a better atmosphere of learning. In introductory courses, having a system based on pure open book examinations can cause some problems of students not studying or reading the material before the test. To alleviate this problem you can implement both open book and closed book exams. The exams need to be structure for whether it is open book or not. For the open book tests do not put in direct information that can be simply copied. Present the questions where the students will have to apply the knowledge rather than recall or look them up and simply copy it from the book. Keep it random as well don’t let the students know what kind of test they are going to be having, so you don’t discourage study.

Don’t be one sided.

Stay flexible and implement all of these strategies together. The mechanical rote learning approaches should stay limited to introductory courses. Rote learning should never be the pass/ fail in math education. A final examination should be on concepts and understanding not recall of facts that anyone can look up.

The purpose of education is career preparation.[edit:(9-24-2015) The purpose of education goes far beyond career preparation. It is the cornerstone of human development and understanding our world. While career preparation is a tangible by-product of education, it is not the 'sole' purpose.] An engineer can still build a great bridge even if they have to look up a formula. However; if they don’t have a conceptual understanding of what to look up or how to look it up they won’t be an engineer for long.

Some Mathematical FACTS about poverty in the U.S.

noreply@blogger.com (Unknown) — Sun, 01 Apr 2012 14:02:00 +0000

According to the Dept of health and human services, the poverty threshold in the U.S is as follows.

2012 Poverty Guidelines for the 48 Contiguous States and the District of Columbia
Persons in family/household	Poverty guideline
1	$11,170
2	15,130
3	19,090
4	23,050
5	27,010
6	30,970
7	34,930
8	38,890
For families/households with more than 8 persons, add $3,960 for each additional person.

Above Chart from http://aspe.hhs.gov/poverty/12fedreg.shtml

LET'S DO THE MATH

Let’s break this down a little bit. For the family of 3 the chart indicates that to be considered in poverty you have to make less than $19,090 annually. The National housing Conference gives the fair market average price of rent on a 2 bedroom as $960 monthly which is $11,520 annually. The U.S. dept. of Agriculture gives the cost of food for a family of 3 as $439 monthly which is $5,268 annually. The average cost of utilities (natural gas, water, electricity) varies widely but sits around $250 monthly for most families. This is $3,000 annually.

Now to add all of this data up we get a base cost of $19,788 that puts us close to the poverty threshold. This base cost does not include

Average fuel costs as reported by CNN are about $368 monthly which adds another $4,416 to the annual cost of living. Our total is now $24,204 annually

Let’s stop at this number for now and see how it compares to wages. The current minimum wage is $7.25/hour At 40 hours a week that is $290.00 weekly( $15,080 annually before taxes) Now if both parents were to be working 40 hours a week we double this to get $30,160 before taxes.(this puts you above the poverty level and makes public assistance unavailable) Now this is $5,956 more than the annual cost we have above, but with this situation we have a new problem(child care) Which makes a new expense.

USA TODAY reports that child day care ranges from $3,803 to $13,480 annually, for those lucky enough to live in an area on the low average that will still leave you with $2,153 a year. However we are talking about a family of 3 here with a child. Nowhere in our expenses did we include the other basics such as clothing and toys(yes toys they are essential for a child’s cognitive development)

If your child is in diapers and you shop wisely you will spend around $30 a week $1,560 annually on diapers (this number can vary slightly depending on special needs… sensitive skin allergies etc. ) Subtract this from the $2,153 we had left above we are now down to $593 However, with diapers we also need diaper wipes. A person could bargain shop and get away with around $12 month on those. That is only $144 annually, and when subtracted from $593 leaves us with $449 clothes, shoes, toys etc….

Another expense I left out is motor vehicle insurance which is now a legal requirement in most states. The average costs varies greatly but is around $1,500 annually leaving our money left over -$1051 There are more expenses that are “necessary expenses” That I have not included, but the fact remains It is impossible to live on minimum wage with a family of 3 and both parents working full time. This is why many people get caught up in major credit card debt, have poor credit and stay in a cycle of survival by destroying their financial future. Bad credit means higher interest and high payments on any loan they try to get, which keeps them always broke. .

For those that do manage to survive by whatever means they can are on a slippery slope. One emergency (car problems, health issues preventing work, cut in hours etc…) can completely devastate a family living off minimum wage.

What math means to me?

noreply@blogger.com (Unknown) — Sat, 10 Mar 2012 18:37:00 +0000

I love math, but I also realize that not everyone looks at math the same way. When I look at math, I see the beautiful mystery that can define everything in our world and beyond. I see a glimpse into unlocking the mysteries of the universe. Once you begin to delve deeper into mathematics, you begin to have an understanding of the world around you that only mathematicians get to enjoy.

After you have gotten into to higher level mathematics, some of the problems you solve will begin to fill up your page like a poetry of numbers. I know when I am solving problems that fill up a page I will take my finished product and just step back and look at the beauty of it. Even if you don't see the beauty in a page full of formulas, computers have allowed us to manifest formulas into beautiful works of art. Fractals offer us an in depth look at the artistic capabilities of even the most mechanical of ideas.
The connection math has to all other subjects is infinite. Even the art world uses mathematics. The golden ratio, related to Fibonacci numbers, is considered the formula for beauty. All of these various connections in mathematics further enhances the beauty it holds.
In addition to the art and poetry of mathematics, there is also the critical thinking aspect. Mathematics is logic conceptualized. It is about modeling our world into tangible coherent forms, and finding patterns and connections. Some refer to mathematics as the language of the universe, but it is beyond that. It is how we take abstract ideas and create a language to share and understand those ideas.

I know that most people do not look at math the way I do, and I am interested in getting some feedback on What math means to you? Everyone has their own story or way of looking at math, even if you are completely frustrated with math I would like to see the comments I get.

So please leave some comments on what math means to you so we can discuss and compare how everyone looks at it. I am very interested to see the variety of responses I get.

An interesting pattern with derivatives

noreply@blogger.com (Unknown) — Wed, 22 Feb 2012 13:32:00 +0000

I found an interesting pattern while finding higher order derivatives using the power rule. For readers who are not familiar with derivatives you might want to watch the video here ( MIT open course ware derivatives.)

I was curious what the 3rd derivative of x⁵ was.
So I used the power rule to find the first derivative and I got f¹(x)= 5x⁴ I applied the rule again to get the second derivative and got f²(x)= 20x³ I once again used the power rule for the third derivative and found the third derivative was f³(x) = 60x² Now by this time I started to get curious as to what the Nth derivative,fⁿ(x) would be for functions using the power rule. So I began to to work it out algebraically. I started with the general function f(x)= x^a and began taking the derivative by continuously using the power rule, and got the following results:

f ¹(x)= a x ^a-1
f ²(x)= a (a-1) x ^a-2
f ³(x)= a (a-1)(a-2) x ^a-3
f ⁴(x)= a (a-1)(a-2)(a-3) x ^a-4
f ⁵(x)= a (a-1)(a-2)(a-3)(a-4) x ^a-5

From these a pattern begins to emerge with, a , as I took the higher order derivatives. Here is the pattern of ,a, throughout each of the derivatives:

a
a (a-1)
a (a-1)(a-2)
a (a-1)(a-2)(a-3)
a (a-1)(a-2)(a-3)(a-4)

As we can see it looks like a pattern similar to a! or a factorial, but this alone is not the pattern we need to a be able to see how this relates to the nth derivative. Well lets look at the derivatives. The second derivative of f(x)= x^ais f²(x)= a (a-1) x ^a-2 ,and from this we see that it is not just a! it is a!/(a-2)! because a! keeps going until (a-some number) = 1 For the second derivative to make sure a! does not go past a (a-1) we use (a-2)! in the denominator because it cancels out all of the other factors past that point. That brings us to the general form of a!/(a-n)!

Now with the a in the exponent x^awe see that the a reduces by the the derivative we are on for example the second derivative , f²(x), the exponent is x ^a-2That makes the general form for the exponent x ^a-n

To put all this together we get the formula for the nth derivative using the power rule as

f ⁿ(x)= a!/(a-n)! x ^a-n

I am not sure if this pattern was already discovered by someone else or not. I have done some research and have not been able to find anything about it online.