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