Let’s explore the Pythagorean theorem as a measure of calculating distance. Today we’ll work through a simple example to see how it works. You’re spending a week camping and have set up your main camp at point A. After a day, you decide to ride a few miles and do a little fishing at point B (6 miles to the east or A). The next day you do a little hunting at point C (11 miles north of B), where you find that your vehicle has broken down. You’re going to have to hike back, but you need to make some decisions about how much gear to take with you on your trek back to camp site A. You can solve the distance using the Pythagorean theorem. First, let’s take a look at our triangle:

It is important to note that we are using capital letters to represent points and lower case letters to represent lines (distances). So, based on the info and the triangle above, we know that side a = 6 and side b =11, so, let’s set up the equation to solve for length of side c:

Now we need to perform a little algebra. Since c squared equals 157, our next step is to find the square root of 157 in order to get c.

So, our solution (fancy math word for answer) is c = 12.53 miles

Now you know you might want to bring a snack and a drink, because you’re going to be walking for a while!

So, now we know a little more about the Pythagorean theorem and Euclidean distance. More and unique uses coming soon! Also, please feel free to send me an email if you have any examples you’d like to see worked out!

Like the quadratic equation, today’s lesson will be about an infamous mathematical expression (it’s a polynomial!), except you can start using this one today. The Pythagorean theorem is widely used expression in geometry. Pythagoras (a Greek mathematician) is credited with discovery and proof, hence the name. The most common expression of the theorem is

Basically, you assign the letters in the formula to sides in a triangle (c should always be the longest side, though… it’s called the hypotenuse). So, now you can solve for a missing length on a triangle, if you have two of the lengths already, big deal, right? Wrong. Triangles are everywhere. For example, you are shopping for a TV. You wanna go big. I’m talking home theater, cover your wall big. In order to do that, it’s important to know how big, diagonally your wall space is so that you can find a projector that will give you the best picture that fills your wall space. You could get the tape measure and try to go corner to corner, but let me tell your wingspan doesn’t cover that span and your buddy will never get the tape measure straight.

Instead, measure the height and width of the wall. Take those measurements and plug them into the Pythagorean theorem. Now you know all your dimension! Warning, the next several days will be filled with uses and proofs of the Pythagorean theorem, prepare to have your life simplified.

The polynomial is one of the most important concepts to learn in algebra. In and of itself, it doesn’t have a lot of everyday uses (though I promise an example of one soon). However, polynomials are essential to statistics, calculus, engineering, etc. So, what is a polynomial? Well, it’s an expression that involves variables and constants bound together by operations. An expression is a fancy word for a function or an equation without an equals sign or f(x). So, the following would be a polynomial:

A polynomial by itself is usually pretty rough to deal with. So, we factor them out into their smaller components. Factoring a polynomial breaks it down to it’s more basic parts. For example, the above example can be factored into (x+3)(x+3). Likewise, these two factor expressions could be multiplied back together using the FOIL (First Outer Inner Last) method.

Factoring typically requires the dreaded quadratic formula. In order for the quadratic formula to make sense, though, we need to know the quadratic equation:

In the quadratic equation, x represents the variable and the other letters represent the constants. So, using the quadratic equation as a template, we solve for the quadratic formula. So, let’s factor a polynomial quadratically. First, the polynomial:

Now the quadratic formula (this is the quadratic equation in reverse):

So, let’s solve it:

There you go!

You are in the market for a new car and you are down to a two or three that you can’t decide between. Well, let’s build an algebraic model to help (technically this is algebra, but it’s also touching on statistics). Here is the equation we will use:

where i = factor importance

p = factor performance

n = number of factors

Σ = sigma, which means add everything up

Before you can work this problem out, an additional step must be taken. You must identify the factors that go into the decision. In the case of a car purchase, there are five main factors: price, performance, safety, image, and build quality. The next step is to assign an importance and performance rating to each factor. It is important that these ratings come from the same scale and that the scale range be equal to the number of factors. So, that means our scale in this example will be a five point scale, 5 being the highest point and 1 being the lowest point. Our first step will be to refine our equation, adding importance coefficients to each factor, since these values won’t change between cars; how important price is to you is going to change. So, let’s see how function looks now that I’ve assigned my importance values:

Notice that first, the sigma sign is gone and that there is no coefficient for the fourth factor. First, the sigma sign is short hand, so if we are writing out the whole equation, it is replaced by the plus (+) signs it represented. Now, the fourth factor doesn’t have a number next to it (the coefficient) because it’s coefficient is one. When a variable has a coefficient of one, you can omit the coefficient because it is assumed that means one times the variable.

With your importance values determined, the next thing to do is rank each vehicle in each of these categories like so:

When we plug these values in, we find that the total value of car 1 is 50, compared to the total value of 52 for car 2. Based on this, you should select car 2 as its performance more closely mirrors your values.

That $7 or $8 you spend a day at Subway seems like no big deal compared to your big ol’ paycheck. But how much is the cumulative effect on your income? Here’s the equation:

where:

• x = the dollar amount spent per meal
• n = the number of times you get lunch there per pay period
• y = how much you clear per pay period

Let’s solve this, assuming you make the average salary in the US and eat an $8 lunch everyday (this should cover your average sandwich, chips, drink meal).

That’s 6% of your take home pay… on lunch. And that’s assuming you make the average household salary of $48,201. Now, try working this out for any other small daily purchases that could have a big impact on your budget.

Middle schoolers, high schoolers, and even college freshmen have groaned about the necessity of learning the importance of isolating x. Yes, I’m talking about algebra. Most of algebra’s importance stems from it being a building block for many other disciplines of math and science. However, algebra has many practical uses in its own right.

First, let’s introduce some basic concepts of algebra: the variable and the function. Think of a variable as a place holder. In algebra, they are typically represented by single letters, but those letters could translate into anything. 4x could easily represent four times laps around a race track (aka four laps). A function, on the other hand is a collection of variables and possibly constants (those are the numbers) to which usually an arithmetic operation (addition, subtractions, multiplication, or division) is performed. For example, a function to calculate the cost of producing something might look like this: f(x)=3x+25. In this instance, the 25 would represent a fixed cost like let’s say a $25 blender. Meanwhile the 3 would represent variable costs for each unit produced, $3 worth of lemons and sugar. So, if you were going to produce 10 cups of lemonade, you’re total cost would be $55.

• f(10)=3(10)+25
• f(10)=30+25
• f(10)=55

Obviously, the usefulness of this is endless in economics. But there are practical uses for algebra as well. Tomorrow, we’ll work through an example.

Welcome to pamplemath.com, the math grapefruit. The name comes from the French word for grapefruit, pamplemousse (pronounced pomp-luh-moose) and mathematics. The obvious question here is what do math and grapefruits have in common. Well, growing up, no one ever really wants to deal with either math or grapefruit. Even into adult life, they still need a little sugar in order to be consumed. But, ultimately, both enrich our lives. My mission is to try to be the sugar that helps you learn some basic math skills that you can use in your everyday life, not just the abstract concepts you learned in school.

Now, I’m not a PhD in mathematics or a math teacher. However, I use math everyday for my day job as a statistical analyst and at home for DIY projects, finances, and web development. The basic flow will be that I will post an introduction into a math discipline (algebra, statistics, geometry, trigonometry, etc.) and then follow with a series of real life examples for the next several days. Any math professionals out there, I invite you to peer review my work and leave me plenty of comments. All others, please feel free to email me at thegrapefruit@pamplemath.com and let me know topics you’d like me to cover or specific examples you’d like to see. Be sure to keep an eye out for math news, tutorials on math software, math jokes, and possibly guest bloggers, too!