Tag Archives: math

Math Moments – Pointing the Direction

It might be good just to pause here for a minute or two and get our bearings.  If you were deep in the woods with a compass, you could get to any point you want by simply knowing the direction you need to travel and how far you need to go in that direction.  On a grid, we could call the point where you are “the origin”, you are starting at the point (0, 0).

You also have a map with a grid and want to be able to draw a line on the map from where you are to your destination.  We are going to draw this line using the general line parametric equations for a line we have already discussed:

x = x0 + Dx*S
y = y0 + Dy*S

With the origin as your starting point (x0, y0) = (0,0).  Notice that I used different simpler notation that says

x0 = 0 and
y0 = 0.

This is called vector notation, instead of stacking the parametric equations on top of each other, we put them side by side in parenthesis and separate them with a comma (x, y). Using this vector notation I can rewrite both general parametric equations for a line as:

      (x, y) = (x0, y0) + (Dx, Dy)*S

A “vector” is expressed as two numbers written in parenthesis, separated by commas. In this case, the first number is the “x term” and the second number is the “y term”. When we “add” two vectors we add both “x terms” together and also both “y terms”.  When we multiply a vector times a number, then we just multiply both the “x term” and “y term” by that number.  And so, the “vector” equation for a line can be expanded like this:

   (x, y) = (x0, y0) + (Dx, Dy)*S
   (x, y) = (x0, y0) + (Dx*S, Dy*S)  (multiply the Direction vector by S)
   (x, y) = (x0+Dx*S, y0+Dy*S)        (add this to the starting point)

You can see how this last equation looks like the original stacked parametric equations, but they are written side by side in the parenthesis instead of one on top of the other.

This vector notation has an advantage, it is easier to write, and it lets us express the direction of the line on as (Dx, Dy). 

This becomes very helpful.

If you are in the deep woods, then it would be very helpful to convert your compass heading direction into this Direction vector and find (Dx, Dy) for your desired direction heading so you can draw the line on your “grid” map.

Since you set the origin (x0, y0) = (0, 0) to be where you are, the vector equation for the line to your destination is:

   (x, y) = (Dx, Dy)*S

Now suppose that your destination is a distance “R” away from where you are. You can then draw a circle on your map with a radius “R” from the origin where you are.  You know that your destination is then somewhere on that circle.  You just need to know the direction (Dx, Dy) of your line to your destination point (see diagram above).

Let’s call the destination point (xR, yR).  Since we figure that the distance is “R”, then when S = R our vector equation is written:

(xR, yR) = (Dx, Dy)*R

And so using vector notation:

(Dx, Dy) = (xR, yR)/R = (xR/R, yR/R)

This tells us that if we know the point of our destination:

(xR, yR) and the distance R2 = xR2 + yR2, then we can find the direction vector and draw our line on the map.  And if we only know the Angle “A” off the ‘x-axis’ from our compass, We can use our calculator and the definitions of Cos(A) = xR/R and Sin(A) = xR/R, to find the direction vector:

    (Dx, Dy) = (Cos(A), Sin(A))

Let us look at an example:  Suppose we knew that if we walked north 3 miles and then east 4 miles, we would reach our destination.  Then our destination is at (3 miles north, 4 miles east) on our map:

   (xR, yR) = (3, 4)  and R2 = xR2 + yR2 = 32 + 42 = 25 or R = 5.

       So,

   (Dx, Dy) = (xR, yR)/R = (3/5, 4/5) North-east

       The line on the map could be drawn by using:

   (x, y) = (3/5, 4/5)*S    where S varies from 0 to R.

If we wanted to know the compass heading, then we could use the “inverse” Cos and Sin functions on our calculator to find the angle “A”:

   Cos(A) = 3/5   and Sin(A) = 4/5.

This might require a bit more knowledge of trigonometry.

Now we do something really interesting.  Suppose that we are still at the origin, facing directly at the point (xR, yR) but we want to find the point on the circle that is exactly 90 degrees to our left. If you look at the diagram, you will easily see that the point of destination would then be (-yR, xR).  The point directly behind us would be (-xR, -yR) and the point 90 degrees to the right would be (yR, -xR).

This is very useful if we want to draw lines that are perpendicular to the direction we are traveling along a line.  The direction (-Dy, Dx) is perpendicular to the left of the direction (Dx, Dy).

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Math Moments – A New Perspective

You can learn a lot about something when looking at it from a new perspective. In the diagram above we are looking at the exact same square as in the last post from a different vantage point. Again, I use my right to choose a reference point and direction. I keep the point O as my reference point, but I rotate the cartesian grid a bit clockwise.

Notice that I have not changed anything about the square, it has the same points. I have only changed the way I am looking at the square. The area of the square has not changed, it is still r2. I have just “tilted my head” a little bit. I chose an “x” direction that is not along the O O+ side of the square and the “y” direction to be perpendicular. This is certainly within my rights under the axiom of choice.

From this perspective, however, the way I measure the area of the square is a bit different, I cannot “tile” the square with the cartesian tiles in this direction without “cutting off” the corners of the tiles. From here grows the whole field of trigonometry. I need to rotate my ruler to measure distance from any point and in any direction in 2-D space.

In the diagram above, I am looking at the “bottom” line segment of the square, O O+, from this new perspective, I have also modified the grid spacing a bit for better clarity. In this diagram, I found the “coordinates” of the O+ point by finding the perpendicular distance from both the x and y axis, as is customary. The distance away from the y axis is labeled “x”, and the distance away from the x axis is labeled “y”.

It is obvious from the diagram that, by parallel sides, the respective distances from O+ form a rectangle with one side a distance “x” along the x axis and another side a distance “y” along the y axis. These distances are called the cartesian “coordinates” of the point O+. It is also obvious that the area of this rectangle is x times y, written as x*y ( we use “*” to mean “multiplied by” for clarity).

If I now choose the O+ point to be the point of reference and chose the “y” direction to be down and “x” direction to be to the left, as shown by the yellow script “x” and “y” in the diagram, I basically would have chosen to look at this same rectangle from an “upside down” perspective. By flipping the diagram upside down, you can see that the rectangle and O O+ line segment looks exactly the same from this perspective.

This shows an important symmetry of the rectangle. It also shows that the area of the triangle on the top half of the rectangle is exactly the same as the area of the triangle on the bottom half. Or to say, the segment O O+ cuts the rectangle in half. Thus, the area of both triangles is ½ * x * y. This also shows that the angles of both corresponding triangles are exactly the same. We will talk about this later.

The equivalence of areas (the real estate of 2-D), regardless of our choice of reference point and direction, gives us a way to “rotate” our ruler into any angle and direction. This relation was found by Pythagoras many millennia ago and is called the Pythagorean theorem.

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Math Moments – 2-Dimensional Real Estate

In the last post’s “cartesian grid” diagram, if you remember, the radial lines went through the point P. By reflection symmetry, I can reflect these radial lines onto any cartesian grid point I want. In the diagram above, I decided to reflect the radial lines so they now radiate from point O. I now choose point O to be the reference point.

I choose one of the “cartesian” directions to be from the reference point O to the point O+, I will call this direction the positive “x” direction, labeled by the red script “x”. I call this line the “x” axis, shown as a bold horizontal line in the diagram.

The second “cartesian” direction I choose to be from the reference point O to the point P. I call this direction the positive “y” direction (as customary), and label it with a red script “y”. This vertical line in the diagram is called the “y” axis and is bolded. There are always two perpendicular independent directions I can choose in a 2-D space. The axiom of choice allows me to choose whatever reference point I want and whatever two reference directions I want. As mentioned before, the right to choose is one of the fundamental concepts of math.

Alright, let’s talk money. In 1-D space, distance is everything, it is the most valuable asset, but in 2-D, distance alone is not worth anything. A piece of land that is 10 meters long has no value until you know how wide it is. The real estate of 2-dimensional space is called “area”. Area is measured by how many squares, or fractions of squares can fit within a given border. The number of squares determines the value.

In the diagram, we form a square by taking the segment O O+ (of length r) and the perpendicular segment O P (also of length r) and reflecting them through M+ to form the opposite sides P Q and O+ Q. Note that we found a point Q that is exactly the distance “r” from both the “x” and the “y” axis. This square (in green) now has value, it is the real estate of 2-D space.

We have seen that the whole 2-dimensional space can be “tiled” with the squares of a cartesian grid. The number of “tiles” within a boundary determines the area. For example, in the diagram above, the green r x r square has 4 cartesian tiles inside. Each side has 2 tiles and 2 x 2 = 4 tiles. Since we could shrink the size of the tiles, we could also fill the square with lots of tiny tiles. From this, it can be deducted that the area of the square is found by multiplying the length of the square by the width of the square (r x r = r2).

The area of the green square in the diagram is r2. In 2-dimensions, the “real estate” or area is always measured as a distance times a distance, like a square meter, or a square foot. In 2-D, a line segment alone has no width, and so it has no area, and thus line segments have no real value in 2-D space.

In concept, you can fit an infinite amount of parallel line segments into a square no matter how small it is. This becomes clear when we recall the same concept in 1-D space to show that an infinite number of points can fit between any two points on a line segment, no matter how small.

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Math Moments – Constructing Parallel Universes

The diagram above (which looks a little like Homer Simpson) shows you how to build a 2-dimensional universe from 1-dimensional universes. There is a lot in the diagram so we will go over it a piece at a time then you can use your imagination to build another universe.

Start with the original 1-dimensional line, the one that passes through the points labeled O, and O+ on the diagram. Remember that we need only 2 points on the line, any 2 of them will generate the whole line. We then need to find another point, labeled P, that is not on the original line. The lines radiating out of the point P represent all the lines connecting P with all the points on the original line “O”.

Each one of the line segments between point P with a point on the line spans a different distance. We find a line of symmetry with respect to P by finding the segment with the shortest distance. This segment goes to the point labeled O. Point O is an important point of reflection, a point of symmetry on the line with respect to P, any point on the positive side of O and its reflection on the negative side of O has the same distance to Point P.

We use the label “r” to refer to the distance between point P and point O. In math language, the symbol “r” reminds us of a “radial” distance from the point P. This r distance is referred to as the “distance” between P and the line. If we put together all the points (on the radial lines) that are a distance r from point P, we form a circle of radius r around P. Notice that since r is the shortest distance between P and the line, this circle only touches the line at point O and cannot reach the line anywhere else. Can you find this circle in the diagram?

The line passing through O and P is a special line of symmetry in this case. It is often called the “normal” of the line that passes through point P. It is also called the perpendicular to the line passing through P. Any set of points on the “positive” side of a line can be reflected onto the “negative” side by finding the perpendiculars passing through the points and then reflecting the points along their perpendicular lines the same distance on the “negative” side of the line. In this diagram, this symmetry is obvious, every point on the right (+) side of the perpendicular (O P) is a reflection of points the left (-) side and vice versa.

In the diagram, we show the point O+ on the positive side of the original line at a distance r from the point O. Its reflection on the negative side is labeled O-. We can easily see that the line (O O+) is also the perpendicular of the line (O P). If we wanted to, we could reflect this whole diagram through the original line and we would get the diagram flipped upside down on the bottom half. For clarity’s sake, I didn’t do this in the diagram.

You might be starting to see the amazing symmetries represented in this diagram. It gets even more obvious when you look at the midpoint between O+ and P. This is labeled M+. Its reflection is labeled M-. If we draw a line segment between M+ and M-, as shown in the diagram, notice that it is also perpendicular to the line of symmetry. To show this better I drew a circle around point M+ and M- that touches the perpendicular line at one point. It also becomes obvious that M+ and M- are the same distance from the original line O O+, by symmetry. All distances are preserved upon reflection.

Alright, now we use our imagination to see what would happen if we drew another perpendicular to the original line through point M+ and another through M-. We can use the small circles drawn to visualize where the circle touches the line to draw these perpendiculars. We can now reflect everything in the middle of these perpendiculars on either side. Like two mirrors facing each other, the reflections go on forever in both directions.
From these symmetries, the line M- M+ can be shown to always be the same distance away from the original line forever in both directions. These lines will never cross.

We have finally found two parallel 1-dimensional spaces that do not intersect each other. Do you think you could find an infinite number of parallel lines in the vertical direction by using reflections?

This is a concept utilized by the mathematician Rene Descartes. He built what is known as the Cartesian plane, composed of infinitesimally close parallel vertical and horizontal lines all normal to each other. These concepts give us many ways to look at 2-dimensional space. One of the most important qualities of this space is that it is assumed to be exactly the same in all directions. Very flat indeed.

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Math Moments – How many Points are on a Line Segment?

If you have fun with the concept of being able to zoom in to a single Point forever, always getting closer and closer but never arriving, you are one step closer to understanding the concepts of advanced math and calculus.
You saw that with any two points you could always find a point that is exactly in the middle. You can then take this midpoint as a new end point and find another point that is in the middle of it. Since points have no size, you can keep on cutting these line segments in half forever.


No matter how close two points are together, you can still find a point in the middle. Thus, there are an infinite number of points in any line segment, no matter how short it is. This is an example of the concept of a countably infinite set of points. It is possible to write out list the location of these points as their distance from the reference point: 1/2, 1/4, 1/8, 1/16, … and so on forever.


You could just as well cut the segments into thirds, with the location list of points: 1/3, 1/9, 1/27, … and so on forever. Notice that this list of points does not have any of the same points as the one in the last paragraph. So, it appears like you could go on forever filling in all the gaps by dividing the segments up evenly and still never fill in all the gaps. So how many points can fit into the line segment between any two endpoints, no matter how short? Can we ever find a way to fill in all the gaps between points? To fill in all the points between any two points, we would need what is called an uncountably infinite number of points. This is a concept that mathematicians have not yet resolved.

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Math Moments – The Number Line

Let’s imagine that you have a universe with only two points in it. You now draw a line segment between the two points and choose one of them to be the reference point. The location of any point along the line segment is now determined by the distance it is from the reference point. Now make a third point on the line segment that is exactly half-way between the two points. This new point is located at exactly half of the distance between the two end points.


Can you see the symmetry that is formed by these three points? The distance from the center point to either of the two end points is exactly the same. If you now were to choose the center point to be the reference point, then both end points would be the same distance away, but in opposite directions.


You would now need more than just the distance measurement alone to determine the location of points along the line. With three points, now direction becomes important. Again, using the axiom of choice and can choose which direction from the center point is positive, and which direction is negative.


We can then, finally, determine the location of every point on the line by specifying both a distance and a direction (positive or negative) from the center point. This arrangement of determining the locations of points on a line is called a “number line”. We call the distance and direction of each point on the line the Point’s coordinates. A Point’s coordinates uniquely determine its location on the line. Finally, we have a way of determining the location of a Point.


The center point is a special point of symmetry between the two endpoints. We can “rotate” the direction (exchange negative for positive directions) and the location (distance and direction) of the two identical endpoints would be exactly the same. This is called “rotational symmetry”. It means you can switch reference directions and the list of all the point coordinates will be the same. Remember that since you can’t tell the difference between points, the order of the list of points is not important.


Please believe me, taking the time to understand these concepts makes math so much more fun and easier later on. Learning these concepts is like learning the basic vocabulary of the math language.

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Math Moments – 1-Dimensional Space

How can the exact location of the Point be described? This is a conceptual problem. It is also where some very advanced concepts of math (found in calculus and number theory) come to play. After some thought, it becomes obvious that if there was just one point in the universe, all by itself, it does not have an absolute location until you look at where it is relative to another point in the universe.


If there were only two points in the universe you could imagine a line segment drawn between the two points, and then there is some distance between the two points, you could pick one of the points as the reference point. Then you could describe the location of the second point as being a certain distance from the reference point. You only need one (1) distance measurement from the reference point to describe the location of the Point. This is why such a line is called 1-dimensional space.


You have now discovered quite a few interesting concepts, including the concept of the number “2”. Two identical objects in space that are distinguished by the distance between them. You have also discovered the concept of the number “1”, the concept of distance, and you are also starting to realize why Einstein’s theory is called the theory of relativity. Distances and times are only measurable relative to a reference point, and you can pick any of the points to be the reference point.


The idea that you can pick any of the points to be the reference point is called the axiom of choice. An axiom is something that is assumed to be true even though it has no definite proof. The axiom of choice forms the conceptual foundation for all mathematics.


Another concept that comes from using the distance from a reference point to describe location is the idea of symmetry. You can exchange the locations of two points, and you have not changed anything. It is impossible to tell the two points apart, and both points still have the same locations relative to every other point. This is called exchange symmetry. Symmetries makes math and physics much easier.

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