Tag Archives: fraction

Fraction Arithmetic

Fraction Arithmetic

I hope everyone had a good holiday season. I certainly enjoyed mine. Over this season, I had a chance to speak with some youth and their parents. Funny that whenever we discuss that I have a PhD in applied mathematics, the topics of the children struggling in mathematics and the possibility of tutoring their children always seem to come up. I have no problem with tutoring and I actively participate in such sessions in my spare time. However I will say that it is sometimes a difficult task to do this job over such a short time period. Needless to say, I felt bad that I couldn’t have been of more assistance.

So, this being the holiday season and all, I decided to make somewhat of a new years resolution to focus this site more towards some of the things that the youth struggle with to hopefully be able to answer some of their questions.

With that being said, the first area that I decided to look at was fractions. This is one of the first areas where the youth begin to dislike mathematics. I feel like regardless of how much teachers and professors speak of the importance of understanding these processes, many students simply never grasp the procedures involved, partially because they never get used to the rules associated with these matters.

In this first script on fractions, I’ve focused on four types of problems corresponding to the four basic operations of arithmetic: Addition, Subtraction, Multiplication and Division.

To add two fractions of the form

num1
den1
+
num2
den2

We use the formula

num1
den1
+
num2
den2
=
num1
den1
+
num2
den2
=
num1*den2 + num2*den1
den1*den2

Lets take a moment to consider where this formula comes from. In order to be able to add fractions we first need to obtain a common denominator for the two fractions. One way that always works to obtain a common denominator is to multiply the denominators of the two fractions. So in the formula above, the denominator on the right hand side of the equals sign is the product of the two denominators on the left hand side. Once we have a common denominator, we need to rewrite each of the two fractions in terms of this common denominator.

num1
den1
+
num2
den2
=
num1*den2
den1*den2
+
num2*den1
den1*den2

The formula for subtracting fractions is similar, with the notable difference of a subtraction in the place of addition.

num1
den1
-
num2
den2
=
num1*den2 – num2*den1
den1*den2

To multiply two fractions (also known as taking the product of two fractions, the resulting numerator is the product of the two initial numerators, and likewise the resulting denominator is the product of the two initial denominators.

num1
den1
*
num2
den2
=
num1*num2
den1*den2

Finally, remembering that division is the inverse of multiplication, we can derive the formula to divide two fractions by multiplying by the inverse of the fractions:

num1
den1
÷
num2
den2
=
num1
den1
*
den2
num2
=
num1*den2
den1*num2

The next step in each of these operations is to reduce the fraction to lowest terms. One way of doing this is by considering Euclid’s GCD algorithm which is available here.

The script is available to practice your work on fractions at

http://www.learninglover.com/examples.php?id=31

How Could You Possibly Love/Hate Math?

Growing up, I never really liked math. I saw it as one of those necessary evils of school. People always told me that if I wanted to do well and get into college, I needed to do well in math. So I took the courses required of a high school student, but I remember feeling utter confusion from being in those classes. My key problem was my inquisitive nature. I really didn’t like being “told” that certain things were true in math (I felt this way in most classes). I hated just memorizing stuff, or memorizing it incorrectly, and getting poor grades because I couldn’t regurgitate information precise enough. If this stuff was in fact “true”, I wanted to understand why. It seemed like so much was told to us without any explanation, that its hard to expect anybody to just buy into it. But that’s what teachers expected. And I was sent to the principal’s office a number of times for what they called “disturbing class”, but I’d just call it asking questions.

At the same time, I was taking a debate class. This class was quite the opposite of my math classes, or really any other class I’d ever had. We were introduced to philosophers like Immanuel Kant, John Stuart Mill, Thomas Hobbes, John Rawls, etc. The list goes on and on. We discussed theories, and spoke of how these concepts could be used to support or reject various propositions. Although these philosophies were quite complex, what I loved was the inquiries we were allowed to make into understanding the various positions. Several classmates and I would sit and point out apparent paradoxes in the theories. We’d ask about them and sometimes find that others (more famous than us) had pointed out the same paradoxes and other things that seemed like paradoxes could be resolved with a deeper understanding of the philosophy.

Hate is a strong word, but I remember feeling that mathematicians were inferior to computer programmers because “all math could be programmed”. This was based on the number of formulas I had learned through high school and I remember having a similar feeling through my early years of college. But things changed when I took a course called Set Theory. Last year, I wrote a piece that somewhat describes this change:

They Do Exist!

Let me tell you a story about when I was a kid
See, I was confused and here's what I did.
I said "irrational number, what’s that supposed to mean?
Infinite decimal, no pattern? Nah, can't be what it seems."
So I dismissed them and called the teacher wrong.
Said they can't exist, so let’s move along.
The sad thing is that nobody seemed to mind.
Or maybe they thought showing me was a waste of time.

Then one teacher said "I can prove they exist to you.
Let me tell you about my friend, the square root of two."
I figured it'd be the same ol' same ol', so I said,
"Trying to show me infinity is like making gold from lead"
So he replies, "Suppose you're right, what would that imply?"
And immediately I thought of calling all my teachers lies.
"What if it can be written in lowest terms, say p over q.
Then if we square both sides we get a fraction for two."

He did a little math and showed that p must be even.
Then he asked, "if q is even, will you start believing?"
I stood, amazed by what he was about to do.
But I responded, "but we don't know anything about q"
He says, "but we do know that p squared is a factor of 4.
And that is equal to 2 q squared, like we said before."
Then he divided by two and suddenly we knew something about q.
He had just shown that q must be even too.

Knowing now that the fraction couldn't be in lowest terms
a rational expression for this number cannot be confirmed.
So I shook his hand and called him a good man.
Because for once I yould finally understand
a concept that I had denied all my life,
a concept that had caused me such strife.
And as I walked away from the teacher's midst,
Excited, I called him an alchemist and exhaled "THEY DO EXIST!"

Aside from its lack of poetic content, I think that many mathematicians can relate to this poem, particularly the ones who go into the field for its theoretic principles. For many of us, Set Theory is somewhat of a “back to the basics” course where we learn what math is really about. The focus is no longer on how well you can memorize a formula. Instead, its more of a philosophy course on mathematics – like an introduction to the theory of mathematics, hence the name Set Theory.

The poem above focuses on a particular frustration of mine, irrational numbers. Early on, we’re asked to believe that these numbers exist, but we’re not given any answers as to why they should exist. The same could be said for a number of similar concepts though – basically, whenever a new concept is introduced, there is a reasonable question of how do we know this is true. This is not just a matter of practicality, but a necessity of mathematics. I mean I could say “lets now consider the set of all numbers for which X + 1 = X + 2″, but if this is true for any X, then it means that 1 equals 2, which we know is not true. So the set I’d be referring to is the empty set. We can still talk about it, but that’s the set I’d be talking about.

So why is this concept of answering the why’s of mathematics ignored, sometimes until a student’s college years? This gives students a false impression of what math really is, which leads to people making statements like “I hate math”, not really knowing what math is about.