Friday, November 2, 2012

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).


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.


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.)


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 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).


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.

Wednesday, October 31, 2012

Fundamental Theorem of Math Blogs

In the Internet age, there is a general demand that any question we have, there is an answer for. This convenience in some ways has corrupted our education. If there is a problem we don't know the answer to, it is extremely tempting to just Google the problem to find the solution. We have lost touch with the idea that discovery requires struggle, effort. This is a huge problem in mathematics. From arithmetic through things like the first calculus series, differential equations and linear algebra it is mostly possible to type verbatim a problem in a textbook and find an answer for it online. I myself have done this many times. My justification, and the justification I have heard for other people is.. I had no idea how to start and I just needed some direction. We feel like it is no different than going to office hours, or asking in class how it is done. The main difference here is, the teacher had no say about how much of the problem you saw, and if you did see the problem completed in completion, there is no possible way for the work to be your own anymore. I came to this realization during my Linear Algebra class, where half the class was accused of cheating by coping a written proof that used information that was not within the bounds of what we learned in this class. She did not say who in the class it was, but I think some people came to her office, possibly even ones that didn't cheat, with guilty consciences about to what extent they were using "outside resources."  My professor's point of view? She viewed using any answer that she did not give cheating. She truly saw homework as a line of communication between herself and the students. That circumventing her and consulting the Internet with your problems was not admitting that you were having difficulty with the subject. She also felt it cheated you out of genuinely struggling to find the answer on your own. She felt that this struggle was paramount to actually understanding what was going on.

This is the purpose of this blog. This blog is a journal of my struggle to understand mathematics in a truly meaningful way. This doesn't mean I need to understand the application of everything. I just want to be able to demonstrate true understanding of what I am learning about, and I want to start using the Internet in a new way. I want to share how I came to understand the things in math in my own language. By posting it on the Internet, I want to open up my work and my wording for the world to criticize. This blog may contain mathematically incorrect information for this reason. What this math blog does contain is mathematics to the best of my knowledge.

One of the roadblocks to starting this blog was actually quite silly. I wanted a clever name for it. Instead of struggling with this issue, I outsourced the problem to my overly clever friends, who came up with a fairly sizable list of good ideas, such as...

- Sum Blog
- Off on a Tangent
- Different(ial) Strokes
- Road Sines
- The Right (Angle) Stuff
- Integrated Education
- Linear Algeblog
- Fractions of a Thought
- Actue Math Blog
- You Can Count on Me
- Parabloggic Curve
- Complex Roots

I ended up choosing Complex Roots for its multiple meanings.  

First of all for it's mathematical meaning (which will be given in the next post).

Secondly, I like the imagery that the word "roots" brings. Roots in a more poetic sense can refer to a foundation. I think many people have complex roots in mathematics. For some people, at some point in their life, they got convinced that they were "bad" at mathematics. Or weren't a "math person." These people may not have been bad at math their whole life, but the just hit a wall at some topic and never broke through it. Some other people want to enter into a science or engineering career and see upper level mathematics as this huge roadblock to getting there. People's emotional foundations with math can be very complex.

Third, in a more academic sense, mathematics is an extremely cumulative subject. Rarely do we learn something in mathematics that is not a precursor for something to come. At some point, a class may have moved to fast for us, or we skipped a class, or we got sick and there ended up just being a hole in our mathematical education. There is an assumption as we progress through mathematics that we have perfect understanding of all the mathematics that came before that class. There is an assumption we all have the same mathematical foundation, the same mathematical roots. But this is not true. Some of our roots may be more developed than others. Even if we never missed a class, got a 4.0, over time we forget things that we don't use that much. 

As I've progressed through upper level math (I am in Linear Analysis right now, which is the last class in the Linear Algebra, Differential Equations series. This series usually comes after the first calculus series). I have found some of the questions that take the most time in class do with a gap in understanding from something much more basic. In my Linear Analysis class the teacher spent 20 minutes explaining how to factor a polynomial of degree 3, then how to divide polynomials, both topics covered in Algebra.

No matter what your mathematical roots are, I hope that you find this blog insightful in some way. I plan on most of my posts being about explaining a mathematical topic that I am currently working on. I also plan on returning to topics where my own roots are not well developed. Since one of my goals in life is to become a math teacher, I may also write my thoughts and research about mathematical education. I am also willing to write posts on roots you want to develop. In fact, I hope to be able to respond to any and all questions and criticism I receive. I want to live in a world where failure and mistakes in mathematics are completely fine. Where it is totally fine to not understand something at the same rate as someone else, or to understand math at the rate your school wants you. I will never insult *anybody's* question, no matter how basic they are. Nobody is ever stupid for not knowing a piece of mathematical knowledge. It is just simply where you are in mathematics.