New Videos!

Inspired by a tutoring session, I have some new vidoes.

First is a two part series about how we can represent a function in different ways.

 https://www.youtube.com/watch?v=I4mNy9ktVWo
 https://www.youtube.com/watch?v=c7hb8UscaTM

One of the things I see when I tutor Algebra is that there is a disconnect between the graph of a function and the equation of a function. What I aim to do is show that we can use different methods to present the same function. In the second video, I go on to show how we can extract information about the function, like the slope and the intercepts from the different forms of the function.

The other video I put up is just running through a way to solve for x and y intercepts of a line, if all we are given is a chart of values.

https://www.youtube.com/watch?v=7ym7XhiHkGg

This will probably be all of the videos I post this week, as I am preparing for my in-person interview to try to be a member of Ada Developers Academy winter cohort!

Ask Aleph!

As I was working on a new math video today, I decided it was probably a good idea to have a separate account for my math videos. Thus Ask Aleph was born! Ask Aleph is my new YouTube channel where people can submit questions about difficulties they have with mathematics, and I try to come up with a simple way to explain the answer. I have additionally transferred this blog over to this account as well!

The first video "Differentiation is Linear " comes from one of my own personal questions I had when I took Calculus I. I never quite got the answer until I got to Linear Algebra, so I introduce the Linear Algebra definition of Linear to make sense of this statement for someone who has just taken Calculus I.

If you have a question that has confused you about mathematics, go ahead and email me, and I'll try to get a video up about it!

 Also, if you saw my earlier videos, you might have noticed that the quality of my videos has improved some. Instead of my incredibly janky setup where I stream to Twitch and export to YouTube, I use the record setting on Open Broadcast Software, and only use it to record video. I record and edit the audio through Audacity. To edit the sound and video together, I was going to try to Blender, but I was running into some troubles, so I stuck with just using Microsoft Movie Maker, until I can find some more time to tackle a more complex video editing program. Unfortunately, I lost a lot a video quality during some process. I'll have to figure out where that happened and improve it for next time!

What am I doing with this Math Degree?

This month I have started actively seeking out new students to tutor. With this, I have been trying a new tutoring style where I am trying to engage with my students on a more macro level. Instead of just helping them week to week, homework to homework, I am attempting to learn more about their math schedule and study habits when they aren't there. I am looking to find out when their tests and quizzes are to make sure they start preparing adequately ahead of time.

One of the other things I am doing is I'm making videos on topics I don't feel like I adequately covered, or just for questions that stumped me while I was tutoring. (I don't charge for the videos). What I hope to communicate through this method is, math isn't a knowledge base you just perfect memorize you go on. Even math majors can become temporarily stumped by a question, but with time and thinking, we can achieve understanding. (The videos are of questionable quality. I'm working on trying to get the static out of the background, and perhaps have a better system for recording videos than streaming on Twitch then exporting to YouTube). 

My videos right now, are very tailored to what we were discussing in the tutoring session. As in, I don't go into great detail about what the student already knew, but briefly cover it and then jump straight to the part which was more difficult to work through .But watching these videos now, I feel like I should perhaps do a more general video that describes the topic, and then do a second video talking about the specific (using a related problem, so I don't take away the experience of the student being able to engage with the material they were assigned). 

Aside from tutoring, I have also decided to apply to be a part of  Ada Developers Academy Winter cohort. This program is very appealing to me. It is a year long, tuition-free, intensive program for women interested interested transitioning to software development. I think this could be a great opportunity to transition my love for problem solving that I learned into mathematics into a career. When I was in junior high and high school, I loved spending my night looking through the source code on webpages, trying to teach myself html. When I reached college, for some reason I can't fathom, I never even considered how to transition this into a career. 

Another reason I am considering this program is, although I like tutoring and I have several years of experience doing it, I think it's going to be really difficult to make a salary in which I could support myself doing it. I think also, one of the things that bothers me about private tutoring is that I'm essentially only working with students whose parents make enough money to afford them. I could volunteer, but all of the hours that students generally can be tutored (4-8pm on weekdays) I need to use to private tutoring to make enough money. I think if I could get a solid career in web development, I could invest a little time each week to volunteering for a tutoring program for students who can't otherwise afford such a service. 

Also, I feel like once I learn more about web development, perhaps I can get inspired to link my expertise in both areas to create something that could really help out a lot of students. 

 With the week and half off I've had off between spring and summer quarter, I've been tweaking a setup that would allow me to stream and record math videos. To test my setup, I discuss why I feel it is a mistake to teach Algebra I (Pre-Algbera?) students FOIL when teaching them to multiply binomials.

Put your Mouth where your Money is

Today I learned that the 10 Deutsche Mark banknote depicts (prolific) German mathematician, Carl Friedrich Gauss.

Not only that, but the bank note also depicts one of his achievements: the formula and (detailed) graph of the Normal (or Gaussian) Distribution, one of the most important probability distributions.
(Click bank note to see the detail of the formula).

I wonder if a case can be made for understanding what is important to a country by what they put on their money. Canada has hockey players on their 5 dollar bill. Hmmm.....

Equal and Equivalent are not Equivalent

One of the things that I love most about upper level math is the precision of language. Words have very specific meanings and if you don't say something precisely right, then often it is wrong. This is not referring to the idea that "People like math because there is one right answer." This is literally referring to the words we use to write out mathematical statements.

Although this is always true in math, I did not realize the depth of it until my Linear Algebra class, where by the end of the semester we ended up getting points knocked off for incorrect language. In Linear Algebra you learn matrix mathematics as a way to solve systems of equations. In Linear Algebra you can make a statement like "the columns of matrix A are linearly independent" but you can't make a statement like "matrix A is linearly independent". Linear independence does not have a definition with regards to matrices.


Much of precision in mathematical writing has its roots in logic. For instance "A is true if B is true" is very different than "A is true if and only if B" is true. In this circumstance, the second statement is biconditional, meaning that you could swap A and B and the statement is still true. That is not the case for the first statement.

Today I learned that equal and equivalent are not words that you can use interchangeably in mathematics.  In general, numbers are equal and mathematical statements are equivalent.

To better understand this statement, let's look at what a mathematical statement is. 

A mathematical statement can be defined as any mathematical sentence that has a truth value (meaning, any mathematical sentence that is true or false).

Examples:

a. 5 is a prime number.

This is a mathematical statement because it has a truth value. This sentence can be either true or false. In this situation it is true.

b. 12 and 13 are solutions to the equation x + 1 = 0

Again, this is a mathematical statement because it has a truth value. In this situation, it is false. A statement does not have to be true for it to be a mathematical statement.

c. 20

The number 20 is not a mathematical statement because it has no truth value. 20 is neither true or false, it just is.

The Oxford-English dictionary defines equal as "being the same in quantity, size, degree or value" (as an adjective) or "to be the same in quantity or amount" (as a verb).  Meaning, equal refers to things that have a measurable size, like numbers. Mathematical statements do not have a measurable size, they merely have a truth value.

Equivalent is defined as "equal in value, amount, function or meaning." In the Oxford-English dictionary we do see some overlap in these definitions. First of all the definition of equivalent has the word equal in it, which I think for the purposes of mathematics, should probably be switched to "being the same in". The definition also reuses the term of value. Nonetheless, we can see the general, albeit subtle, difference between these words. Mathematical statements are statements that have function and meaning (true or false), not a measurable quantity, therefore it is correct to use equivalence rather than equal.

Complex Roots (Also quadratic factoring)

Although I am far past the initial idea of complex roots in my mathematical education, it seems correct to explain it here, since it is the name of my blog.

 The first time I consciously remember a teacher saying the phrase, "What are the roots of this equation?" I was in a Trigonometry class my first semester at Seattle Central Community College. I strained my brain, but I seriously could not remember coming across this term earlier in my high school (which included Algebra 1 & 2, Geometry and Calculus). That night I had go teach the concept to myself (since I was too embarrassed to ask in class, which is a great thing to learn to overcome).


Roots


The term root is used in reference to a function.
The roots of a function, f(x), are the values of x that will cause the function to be zero.

Examples:

a. f(x) = 2x

This equation has one root. That root is x = 0. No other value of x will cause f(x) to equal zero.

b. g(x) = (x² - 16)

There are two ways you can think of solving this. The first thing that I see when I look at this is a difference of squares. Difference of squares can be factored every time using this method.

(x²- c²) = (x + c)(x - c)

Using that method in this equation we have

g(x) = (x² -16)

        = (x + 4) (x - 4)

At this point we can see that if either (x + 4) = 0 or (x - 4) = 0 then the whole function will equal zero due to the Zero Product Property (any number multiplied by zero will equal zero).


The roots of this equation are therefore x = -4 and x =  4.


The other way we could have done this problem was not by factoring but just considering the problem (x² - 16). What values of x would give us 16? What numbers squared equal 16?  4 and -4.



Quadratic Formula

Not all equations can be factored using simple algebraic methods, such as the difference of squares method. When it is not obvious what the factors are, or you cannot find a method to easy factor the equation, it becomes necessary to use the quadratic equation.

The term quadratic equation, refers to an equation with one variable (x, usually) with a degree of two (meaning, the largest exponent in the equation is 2).  The basic form for a quadratic equation is

ax²+ bx + c = 0

A quadratic equation will always have two roots. 

Where x is the variable and a, b, and c all represent constants. To use the quadratic formula, you must put the equation in this form. Once the equation is in this form, you plug the values from that equation into the quadratic equation.

(-b ± √ b² - 4ac ) / 2a .

(Full disclosure: For some reason, the quadratic equation was one of the gaps in my education. I still have hard time recalling it from memory.  I seriously had to double check it for this post. Don't be like me, just pound it into memory the first time you learn it.)

Example:

c. Factor f(x) = x² + 5x - 2 using the quadratic equation.

The value for a, b and c are  a = 1, b = 5 and c = -2. We simply just plug these into the quadratic formula.

[-5 ± √ 5² - 4(1)( -2)] / 2(1) 

[-5 ± √41] / 2

The roots for this equations are x =  [-5 + √41] / 2 and [-5 - √41] / 2.

Although these numbers look "complex" they are not actually complex numbers. They are two real numbers, real roots to the equation.

(Fun fact: There is also a cubic formula and a quartic formula, a general equation for solving for the roots of a third degree or fourth degree polynomial. The cubic formula is rather big, but still digestible, and the quartic formula is ridiculously huge.  There actually can be no quintic or higher general function, however. This was proven in the 1800s. The proof for it is called Abel's Impossibility Theorem).

Complex Roots

A complex number is a number that includes the imaginary unit. The imaginary unit is  (√-1). The imaginary unit is represented by a lower case i.  The imaginary unit is literally defined as  i² = - 1 All complex numbers can be written in the form a+bi where a and b are constants. a is referred to as the real part of the number and bi is referred to as the imaginary part of the number. The idea of an imaginary number, intuitively, is difficult to grasp. I will not attempt to explain it here (mostly because I have problems grasping it myself).

Some quadratic equations, when put into a quadratic formula, give complex roots instead of real roots. This happens when the 4ac term of (√ b² - 4ac) is larger than the b² term.

( b² - 4ac is also referred to as the discriminate of the quadratic equation. A quadratic equation can have either one real root, two real roots, or two complex roots. If the discriminate = 0, then the equation will have one real root, if the discriminate > 0 the equation will have two real roots, if the discriminate < 0 it will have two complex roots. This can be useful to get an idea of what you are working with before you star the problem).

Example:

d. Factor f(x) = x² + 2x + 10 using the quadratic equation.

The quadratic equation is (-b ± √ b² - 4ac ) / 2a.

a = 1, b = 2, c = 10

[-2 ± √((2)² - 4(1)(10)] /  2(1)

[-2 ± √4- 40] /  2

You can tell that we will have complex roots based on the fact that the discriminate will be a negative number. 

[-2 ± √-36] /  2
[-2 ± (√36)(√-1)] /  2
[-2 ± 6(√-1) / 2
[-2 ± 6i] / 2 

-1 ± 3i 

The roots of this equation are x = -1 + 3i and x = -1 - 3i.