Math, The Way it Should Be (Unit 1.5: Number Lines, Planes, and Vectors)

It’s been a bit of time since the last math post.  Hopefully you’ve used the time to review the previous lessons and done all your homework.

As always, the start of the series is here and the start of the unit is here.  This lesson doesn’t have very much new notation, either.  I think that later units will have more new notation.  This lesson is about looking at numbers in a more visual way and sets up the entirety of the next unit.

One artifact you’ve probably seen before is the number line.  It represents the continuum of numbers and looks something like this:

Here, you can see the integers between -5 and 5 and the arrow heads indicate that the numbers continue in either direction.  The line may be marked at any interval.  For example, if I had wanted, I could have put only even numbers on the line and the number line would have been equally legitimate.  This can be a very useful way of thinking about numbers and the things we do with them.  Addition can be thought of as movement to the right along the line, while subtraction is movement to the left.  To explain multiplication in a similar way, I need to introduce something called a vector.

A vector is commonly described as a magnitude and a direction.  In other words, it is an arrow with a known length and we know which way it points.  On the number line, the vector starts from zero and points to the number which corresponds to its length.  A vector representing the number two looks like this:

To multiply by two, we simply double the length of the arrow:

We get the expected result, which is four.  I feel like this is a much better way of imagining multiplication, rather than the rows/columns method presented earlier.  Division can be described in a similar way: instead of doubling the length of the arrow to multiply by two, you halve the length of the arrow to divide by two.

Once upon a time, a mathematician named Descartes put two number lines together like this (we call it a Cartesian Plane):

These two lines can represent anything, especially things that can’t be compared, like apples and oranges.  One line would represent apples, and the other oranges, and any combination of apples and oranges could be represented by a place on this “plane”.  The number lines are now called axes.  For example, if you have two apples and three oranges:

A vector could be drawn to this location (from the point representing a person having no apples and no oranges).  In either case, the point where you are with your fruit and the vector pointing to you are represented by (2,3).

This vector can be broken into two vectors, one for each axis:

The two red vectors are your apple vector (2,0) and your orange vector (0,3) and they add to your fruit vector (2,3).  BAM! You just learned vector addition.  When adding two vectors, you simply add their components.  If I have one orange and three apples (3,1) and give them to you (2,3), you will move to (5,4).  Vector subtraction is similar.  If you eat two apples and two oranges, you go down two on the orange axis and to the left two on the apple axis to (3,2).  Keep this in mind.

Homework:

• Review your times tables.  I'm not kidding, they really are that important.
• Draw a Cartesian plane and label the axes.  Label the vertical axis something made-up, like unicorns and the horizontal axis something real, like fire trucks.  Imagine yourself somewhere on this plane with at least one unicorn.  This is also important, as you will see later.

-Lane