Math, The Way It Should Be (Unit 1.2: Numbers, Part 2)

This the second part of the first unit on numbers.  If you want to start at the beginning of the unit, go here.  If you want to start at the very beginning, go here. This lesson doesn't contain any new symbols but does have a lot of vocab.

Recall that we started with the counting numbers (1, 2, 3, ...).  These are your very basic numbers, the first we encounter in our education, and this re-education is no different.  Next, we include the number 0 (which represents nothing) to the set.  This is a fairly big step that took human civilization thousands of years.  It is important because it is the "additive identity", in that zero added to any number results in that number.  After that, we include negative numbers, which are less than zero.  They mirror the counting numbers and are referred to as their opposites.  For example, the opposite of 4 is -4.  The interesting thing about two opposite numbers is that their sum is always zero.  This leads to the realization that subtraction is really just the addition of a number and the opposite of another number:

4 - 4 = 4 + (-4) = 0

The set that includes the counting numbers, their opposites, and zero is called "the integers".  You may convince yourself that the integers are an ordered set. Now everyone is happy.  We can effectively solve algebra problems that involve addition and subtraction.  For example:

x + 4 = 5

x + 4 + (-4) = 5 + (-4)

x + 0 = 1

x = 1

Ta dah!  An algebra problem is solved.  Don't be frightened by the unknown variable.  It is easy to find once you know how.  To eliminate the number that accompanies the variable, you add it's opposite.  Since the sum of a number and its opposite is zero and zero is the additive identity, you are left with only the unknown variable.  Of course, you must do the same thing on the other side, since you can't change one side and not the other and still have them be equal.

Side note: any variable can be used instead of x.  One of my math teachers was especially fond of "dead dog" and "house".  This flexibility will be useful later.

Now for multiplication and division.  I feel like these operations are poorly understood and difficult to explain. Multiplication can best be thought of in terms of rectangles.  Imagine a square with side length equal to 1.  The  following expression:

3 x 4

Can best be thought of as a rectangle containing 3 of these squares on a side and 4 on the other, like this:

Count the number of squares.  You'll find there are 12, a result that you should have expected.  This can be alternately be represented in the following ways:

3(4) = 12

3·4 = 12

3 x 4 = 12

3*4 = 12

Only the first two are commonly used, so forget about using the third one.  The fourth is used in computer applications because the asterisk is on the keyboard. Don't hand-write that one.  An important property to point out is that 1 is the multiplicative identity, that is, that one multiplied by any number results in that number.

Imagine now that you have 12 boxes in the above configuration.  If you want them put in three groups.  You would find that evenly split up, there are four in each group.  This is represented in the following ways:

12 ÷ 3 = 4

12/3 = 4

The twelve may also be put directly over the 3 in the second case (this is the most commonly seen).  Don't write the first one.  It makes you look like an amateur. One is also unique in division: divide any number by one and you will have the number you started with.  Also, never divide by zero.  Mathematicians don't agree on what happens when you do this, but it's always bad.

What if you want to put your 12 boxes in five groups?  There is no integer that can result from putting 12 objects evenly in 5 groups.  We will need a new kind of numbers to solve the following equation:

That is the subject of another lesson.  I hope by now that you see that I am presenting new groups of numbers as the solution to new kinds of problems. Counting numbers are required to count.  Integers are required to do addition, subtraction, multiplication, and some division.  But not all division can be done this way.  The next lesson on numbers will cover these numbers.

One last thing to take note of: adding and multiplication can be done in any order (for now).  Subtraction and division are best done in the order they're written in.

1 + 2 = 2 + 1

2 - 1 ≠ 1 - 2

Homework: 

• Learn your "times tables".  No joke.  They're important and will come in handy later.  
• Forget that you ever saw this: ÷
• Never, ever again say "times it by" in reference to multiplication.  To say something like "to do multiplication, one simply takes the first number and times it by the second" is wrong.  Not only is it incorrect, but it's against the morals and ethics of society.  Most importantly, it makes my blood boil.

-Lane

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.

Math, The Way It Should Be (Unit 0: The Beginning)

I have encountered many people who are not good at math. There are a number of reasons for this, many of them perfectly legitimate. Perhaps learning more math never seemed useful, perhaps a teacher wasn't able to explain fractions very well and things went downhill from there. My intention at this point is to start at the very beginning (waaaaay at the beginning) because I firmly believe that a lack of foundation is the root of many people's trouble with mathematics.

Why am I doing this? Reason number 1: math is part of the way I look at the world. At a sufficiently high level, the math becomes a beautiful expression of the way the world works. If the universe were an Italian opera, mathematics is Italian. You can enjoy it without the language, but there is something rewarding to me about getting as much of the full story as possible. Reason number 2: math is surprising useful. All it takes is a little creativity in application and all of a sudden you're making things a lot easier!

Right now, I estimate that there are going to be 12 units, each with multiple parts (this introductory post doesn't count as a unit). This is not intended to replace education, but if you've stopped where you left off, hopefully I can give you a little more understanding. Don't skip topics, they're very important because I'm going to go in the order that makes the most sense after looking back on my own mathematics education. That leads me to my next point: this series of posts is dedicated to the people who made me the mathematician I am today: Mrs. Peck, Mrs. Wells, Ms. Graves, Mr. Atherton, Mr. Alexander, Mr. McGowan, Mr. Anthony, Professor Volkov, Professor Humi, Professor Mosco, and Professor Mitrea.

One last thing: there will be homework, but it won't be collected for credit. This week's assignment: brush up on your times tables. If this is the only thing you get out of this series, it will have been completely worth it.

-Lane