Certain Things for my Math Blog

Note: This blog is heavily work in progress, so some of the sections might be incomplete. Please feel free to yell at me if I haven't worked on this for a long time. Thank you for your understanding.

Hello! If you are here, it's probably because I linked to this page on my Unsolved Mathematical Problems blog. If not, please feel free to use the table of contents, as this might get very long and messy. Here's a reference guide to most, if not all, of the notations presented in that blog post. Okay, I guess it's time to get to it!

Summations

Introduction

Anybody who has gone through primary education knows what addition is (if not, stop using this wiki and please go to school now). Here's an example, nonetheless:

$1+2+3+4+5+6+7+8+9=45$

Just some quick facts: the values before the equal sign (or the values being added, to be more precise) are called the addends or the summands, and the value after the equal sign (the result) is called the sum (this might already be obvious to some of you). Make sure you remember this, as I'm going to be using these terms throughout this section.

Many of you might be used to adding numbers together by writing out every addend one by one, and that's okay, but when adding larger amounts of summands together, this notation can get messy real quick. Let's say you want to add every integer from 1 to 100, and that you want to write it out so others know that you're making a sum. You could write out every term, but that would take up a lot of space, and would take a lot of time and effort in making. You could also write out the first few terms of the sum, put an ellipsis, and write out the last terms $1+2+3+\cdots+99+100$, and I'm sure most people would still understand the pattern, but there is a much more clear and and convenient way to notate this addition: sigma notation. Here is how the sum would look like written in sigma notation:

$\sum_{n=1}^{100}n$

What does this mean, exactly? Well, first, let's look at the definition.

Definition and Explanation

The Capital-sigma notation is defined as:

$\sum_{i \mathop =m}^n a_i = a_m + a_{m+1} + a_{m+2} +\cdots+ a_{n-1} + a_n$

where i represents the index of summation; a_i is an indexed variable representing each successive term in the series; m is the lower bound of summation, and n is the upper bound of summation. The "i = m" under the summation symbol means that the index i starts out equal to m. The index, i, is incremented by 1 for each successive term, stopping when i = n.

"Well, what does this mean?" you might ask. Well, let me try to explain it in terms that are hopefully easier to understand. First, let me present an example:

$\sum_{n=1}^{5}\frac 1 n$

That weird "E" looking sign is called a Sigma, it is a Greek letter (Σ is a capital letter); it is the operator of the function. "n=1", under the sigma, signifies the index being used, and the number the variable of the sum starts at (so in the first addend of the sum, n is equal to 1). The number on top of the sigma is the final number of the variable of the sum being added, basically where the sum ends (in the last addend of the sum, n is equal to 5). (Remember, n increases for each addend by a factor of exactly one.) Finally, the term on the side (${\frac{1}{n}}$) is the function of each summand of the sum. The following is an equation that compares the sum with the expanded traditional method of adding:

$\sum_{n=1}^{5}\frac 1 n = \frac 1 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \frac 1 5$

[remember to add more eventually]

Sets

Introduction and Explanation

Sets. Yes, sets! Sets are actually legitimate mathematical objects. Well, at least, that's the theory (more on this later).

A set is a (well-defined) collection of distinct objects, called its elements, and it is itself an object. These sets can comprise of literally anything, like numbers, functions, letters, people, planets, pictures, wikis, and even other sets.