Math, The Way It Should Be (Unit 1.1: Numbers, Part 1)

This is the first lesson of the first real unit in the "Math, The Way It Should Be" series.  If you missed why this is happening, you should look here.  This post is going to cover some basics about numbers and the way we write them and some of the things that we do with them.  Again, don't skip this section, even if you think you have a pretty good handle on numbers (even though you probably do).

Each lesson will begin with an explanation of the symbols that you will need to know to get the lesson off the ground.  The symbols that we are going to need right now are as follows:

Now I know that you've seen these before and I'm not trying to be a wise-ass.  They are very important and I'm including them here for completeness and I want everyone to be on the same page as I am.  They are, in order: equal to, less than, greater than, less than or equal to, and greater than or equal to.  The first symbol means that the objects on both sides have same value (value to be discussed later).  The second symbol means that the object on the left has a smaller value than the object on the right.  The third symbol means that the object on the left has a greater value than the object on the left.  The fourth and fifth symbols indicate the same thing as the second and third, except that they also admit the possibility that the two objects have the same value.

Now that we know what the symbols are, let's talk about some numbers.  Numbers are essentially symbols that have certain values that we all agree on.  The first type of numbers are the first ones that you learn: 1, 2, 3, 4, etc.  They are called the counting numbers and refer to a set (or group) of numbers that one uses to count things.  If you have one duck and then get another and count your ducks, you have 2 ducks.

When you learn about new groups of numbers, it's nice to think about what properties the group has.  One nice thing about this group is that it's an "ordered set" meaning that if you take two of the objects, you can say one of three things:

1. The first object is greater than the second object (4 > 2)
2. The first object is less than the second object (4 < 6)
3. The first object has the same value as the second object (4 = 4)

This is an extremely useful property because it allows us to count, which is the whole point of these numbers.  From this set comes two very useful things to do with numbers: addition, subtraction.

Addition is the combination of two or more objects to create a collection of objects.  We use the + sign to indicate this type of combination.  Returning to the duck example, you had one duck and got another duck and ended up with two ducks.  This is written as 1 + 1 = 2.  Subtraction is the removal of one object from a collection of objects.  We use the - sign to indicate this process.  This symbol comes up again in a very interesting way later.  You eat one of your ducks and now you have one*.  This is written as 2 - 1 = 1.

As this is the first lesson and I hope that everyone has a pretty good handle on this material, if you're happy with the pace, there is no homework.  If you feel that I am insulting your intelligence, I encourage you to leave a rigorous, mathematical proof that 1 + 1 is always equal to 2 in the comments.

-Lane

*No ducks were harmed in the making of this post.