The Language of Math

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Sometimes I forget how much math I've learned over the years, including the different vocabulary!  In addition, I realize how funny it can be for people who have never learned some "Math Language" and how they choose to solve a problem.

I find this picture so funny! I feel like if I never learned what "finding x" really meant, this would be me! But, I don't think there's an answer like this on the SAT.

Though that was a little funny, there are really mathematical terms that I have learned that mean a lot.  For instance, when we have to find an answer in terms of z.  I think this is one of the most confusing concepts, because we use variables in order to find an answer without an EXACT answer.  Also, the fact that there is a way to verify and simplify using sin, cosine, and tangent.  Honestly, things get so confusing with all these math terms.  And we're only going to learn more.  Eventually, we will be able to find different real-life solutions using the language of math!

The Binomial Theorem and Pascal's Triangle

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The binomial theorem is the expansion of a binomial to the nth term.  It seems like you would have to multiply and multiply continuously, but there's always a shorter way! Well, at least there is one this time.  For the binomial theorem, in order to find a certain term, you can use combinations, which include factorials.  The equation for such combination is: nCr = n!/(n-r)!r! But, there's something much simpler in order to find a total equation.  Which is using Pascal's Triangle. Pascal's Triangle is a triangle that distributes the coefficients for a binomial.  But, it's not that simple, you must still multiply that term by others as well.

Here are some examples of Pascal's triangle.

Well-Ordering Principle and Mathematical Induction

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The well ordering principle states that every non-empty set of positive integers contains a least element. Meaning that there is almost a smallest term in a set of positive integers. This principle is also equivalent to the well-ordering theorem. This includes a set of numbers, with a well-ordered subset, in which every nonempty subset contains a least element.  The whole set will depend on the framework of the natural numbers as well as the second property of numbers.  These are known as an axiom or a provable theorem.
Here's a video on the process of the well-ordering principle.

Something equivalent to the well-ordering principle is mathematical induction.  Which is what we applied in our lesson today.  Mathematical induction is basically proving how a set of numbers work, and if true, proving how it will work for any variable put in.  There are two steps to mathematical induction, and it is a bit tricky.  But once you get it, it's quite easy.

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Personally, the hardest part for me is in step two. Mostly remembering that your previous term is equal to your given, and then substituting (n+1) for n.  In addition, simplifying and verifying that both equations are equal is a bit tricky, but once you remember your factoring, it gets a lot easier!

"Do I need this for my future?"

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Whenever we come across a lesson that seems utterly boring and not fun, I always say, "Am I ever going to use this?" or "Do I need this for my future?" And I think many students think the same thing.  But, apparently MANY students do, which is why a group of mathematicians have created a website called weusemath.org  This is where I found a video on why they believe math is really important.

The video is narrated by a few people who majored in math and ended up in various jobs.  Professors, scientists, analysts, and many other jobs use math within their works.  In addition, the video also talks about how majoring in math is a good basis for any job!  You have various amounts of options to choose from.  But, I realized that these people have a real hard-core passion for math.  One man even said that everyday he is excited to go to work and he has a big smile on his face.  I definitely can see why math can be a good use to many jobs out there, but I don't think I have the passion or love for math to have it as a job. #sorrynotsorry

Lesson 9.2

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Today we learned about arithmetic sequences, which is a sequence whose consecutive terms have a common difference.  An easy way to find out if a sequence is arithmetic is by subtracting the first term from the second term.  This will be the common difference, if the common difference is sustained throughout the whole sequence, then it is an arithmetic sequence.

In order to find the nth term of an arithmetic sequence, all you have to do is keep adding numbers until you get to that term! Just kidding, there's a formula (because there's one for everything!) and it makes things so much easier.
 The formula is An = A1 + (n-1)d
A1 is the first term of the sequence.  N is the number term you want to find. And d is the common difference.
Here's an example!

In addition to arithmetic sequences, there is the sum of arithmetic sequences! And of course, there is a formula!
The formula is Sn = n/2(A1 + An)
N would be the last term you want to find.  Then A1 is the first term and An would be the last term.  But, something that most people get confused with is putting the number of the term in An, when it is supposed to be the actual term.  So you must find the actual term first by using the explicit formula.  Here is an example:

Lesson 9.1

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Today we learned about sequences and summation notation.  This is a bit of a review from Algebra 2 last year, but with a few added steps.  To be able to solve a sequences and summation notation, you must start with the basics!

When given an equation, the variable n stands for the number of the term you are looking for.  All you have to do is plug in the wanted term, and you are able to solve for the term needed.
Here is an example:

The next thing you are able to do with sequences, is find the nth term.  Which is a term that can be found with any n given.  But, sometimes you are not given an equation.  When this occurs, you must first create an equation, and then plug the n in and you will get your term.  In order to find the equation, you must use the terms given to you, as well as the number in the sequence they correspond to.  After, you must use you brain (tough, I know) and find how the equations relate to n.  Once you find your equation, you just go back to your easy ways and plug and chug and there you have your nth term!
Here is an example:

Factorials, factorials, factorials! These were not my favorite in Algebra 2, especially because they are just dumb.  But, surprisingly factorials are sometimes used in sequences as well.  They may be given in an equation!  Just in case you forgot, a factorial (n!) means you must multiply the number given as well as the numbers preceding it. (So you might end up with a big number).  But sometimes, factorials cancel out, which is great!
Here's a simple explicit formula with a factorial.

Finally, summation notation!  It sounds really hard, but it is quite simple once you get it.  Basically a summation notation gives you a range of numbers, which will be your n's, and an explicit formula.  You must find every term needed and then add them up! And that's the answer to your summation notation! It seems a little bit confusing, but once you get it, it's a piece of cake.
Here's an example of a summation notation problem.

Chapter 8

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Chapter 8 was all about matrices and determinants.  We learned different ways to use matrices in order to solve a system of equations, in addition matrices are used to find determinants.  We also found out 4 applications of matrices and determinants.

Here's a link to our website all about Chapter 8!

Four Applications

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Today we learned about four different applications for determinants in matrices.

• Area of a Triangle
• Test for Collinear Points
• Equation of a Line
• Cramer's Rule

Here are explanations and examples of each type of problem.

Here is a voicethread on an example of an area of a triangle.

Determinants of Matrices

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Today we learned about finding determinants of matrices.  The process is simple, but it gets a lot more difficult with bigger matrices.  There are different tricks you can use in order to solve matrices quicker as well.  But basically, all you have to do is multiply diagonally to the right and then subtract the opposite diagonal.

Here are some examples.

Determinants eventually get a lot harder to solve once your matrices get bigger.  Here is a video on how to solve such determinants.

Cryptography

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Cryptography coding is an application in which you use matrices and their immerses to create a code. You start off with the inverse of an encryption matrix, which will be your decryption matrix, also known as your code. Next, you use the numbers given, and multiply by the decryption matrix to find the code corresponding letters of the alphabet to numbers. Here is my cryptography worksheet, which includes me solving Mama V's message, as well as a message I created.

Can you solve my message? Don't forget to correspond the code numbers to the size of your decryption matrix. For mine, you will separate groups of numbers by two!

Lesson 8.2

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8.2 is all about outing a system of equations into matrices and solving. It's quite easily, especially when you use the Gauss-Jordan formula. But sometimes you will encounter a system with no solution as well as a system with an infinite number of solutions.

You can find out a system has no solution when one of the rows ends up with all zeroes. This means that zero will equal a number, which would not be true.

Finding a system with infinite solutions is just like finding an equation in terms of z. When this occurs, you will have an infinite number of solutions.

Lesson 8.3

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Today we learned about inverse matrices and how to find them. Basically we make the given matrices equal to their corresponding diagonal ones, and then the answers are the inverse. It's a lot easier a simpler once you see an example.

But, there are times when there is no inverse. This occurs when the bottom row of the original matrice ends up as all zeroes. That is when it is called singular.