Equations and Inequalities
I Like Abstract Stuff; Why Should I Care?
Mathematicians can find complications everywhere, even in the seemingly simple idea of putting things in order. If you date one, your sock drawer will never be the same again.
In algebra, the phrases "less than" and "greater than" have meanings based on the positions of numbers on the number line, because that is how mathematicians have agreed to order the real numbers. Trust us—it is an ordeal and a half getting all of those guys to agree on anything; appreciate it for what it is. This order makes a whole heap of sense. It stands to reason that two hundred should be greater than four because you have a lot more chocolate with two hundred brownies than with four brownies. If you don't, then we shun your brownies. Shun them.
We can order things besides the real numbers. Pairs of numbers can be ordered using the "lexicographic order." Say what? You know, Lexico. That country just south of Lamerica? Okay, it actually means that one pair is greater than another if its first number is greater than the other pair's first number.
(1, 4) < (5, 2) since 1 < 5.
If the first numbers in each pair are the same, we compare the second numbers:
(3, 1) < (3, 2) since the first numbers in each pair are equal and 1 < 2.
The lexicographic order is also called the "dictionary order" (the word "lexicon" means "dictionary"), because this order is similar to how we alphabetize words: we compare the first letters, then we compare the second letters, and so on. Look at that—your library shelving skills just improved.
In some orders, you can't necessarily compare every element to every other element. If Alice, Bianca, and Carol are climbing a mountain as shown below (careful, ladies—climb with your knees!), we could say Bianca is "greater" than Alice since Bianca is higher up the mountain. If we only use strict inequalities, we can't compare Bianca and Carol, because neither of them is higher than the other. Even though Bianca clearly has better hair.
This mountain example is an instance of a partial order, meaning an order with incomparable elements. Here, Bianca and Carol are both killing it equally, and so cannot be compared. Get it together, Alice. This is getting embarrassing.
An order in which we can compare any two elements is called a total order. Actually, a total order needs to have a couple of other things going for it. See, for instance...