A Square Deal

Geometry 4:

A Square Deal

When working with the right triangle, it is easy to build a "rectangle" by simply building a mirror image of the right triangle.  One might expect the term "right rectangle" to be used here, but that would be redundant.  This is because the prefix "rect" is a corruption of the original Germanic form of the Latin "rectus" which means "right."  The Old German was "reht" but the "h" was the hard "kh" sound that migrated up from the Greek "Chi" (looks like our "X").  The Old English form was riht, corrected (pun intended) by adding back in the hard sound.  This time, the hard "g" was used, rendering "ri-ght"  It did not take long before the hardness of the "chi" sound was abandoned.  From "rect" to "right" and back again!

Anyway, we have seen how a right angle can be constructed using the 3:4:5 ratio with the sides.  Using the largest side (the diagonal "5") a ratio of 5:4:3 becomes the mirror image forming a 3 by 4 rectangle.  The area of the rectangle is simple to determine in "square" units.  The area of a "right quadrangle" (four-sided with parallel sides) will be twice that of a right triangle.  Another way of seeing this is that the formula for triangle will be one half that of a rectangle.

The formulae for this are simple:

Area of a rectangle = base (width) x height (length).
Area of a triangle = 1/2 x base x height

The "square" is a special kind of rectangle.  Not only are all the angles inside a square equal, but so are the sides.  The ratio of the sides of the "half square" loses its whole number diagonal when "side a" and "side b" are forced to be the same size.  The diagonal becomes a multiple of the square root of 2! That works out to a little over 1.414.  I'd go out farther, but fourteen-fourteen works for me. But, when considering the area of the square, no irrational numbers need be involved.  I really prefer rational numbers.

When we move to other triangles and polygons, where the angles are not all right (no pun intended), then the "height" becomes a problem.  The height of a triangle is always figured in a right angle, which must be used to divide the shape in order to follow the rules of the square.  That is to say, everything must be reduced to right triangles and/or rectangles to be squared.  It's like playing with blocks!

I'll work out the bugs of those building blocks in another blog.  I know just enough geometry to be "dangerous."

So, what do I know?

Rectangle: Area = lw
Rt. Triangle:  Area = (1/2)lw = lw/2, where l and w form the right angle.

Square: area = l² (length squared) [length being equal to width)

All area within polygons are measured by the pythagorean theorem:  a² + b² = c²

Strange Facts about Exponents

Sometimes things just have to be proven.

We are taught to accept some things as fact, but they are not as evident as the facts I know intuitively.  These are the "math facts" that must be derived from the basic things.

This can be done using the "shorthand" known as the exponent.  It is like a "component" but set over to the side.  It is not part of the number but is a reminder that the number has been acted upon by multiplying it by itself.  The common exponents are the "square" [2], and the "cube" [3], drawn from the construction industry: flat panel and a "box."

When working with "whole" numbers, that is to say, zero and all the numbers we count with (aka "natural numbers), the exponent tell us at a glance that things just got a lot bigger.

Exponents also work with what are called "rational" numbers and even with 'irrational" numbers.  And lo, and behold, exponents work with "imaginary numbers" -- all these are things I know, but they need to be shown to be true.

Squares, Cubes and Beyond

First an illustration:

Consider that you are going to build a box, not quite as big as Noah's box (ark) but perhaps about the size of what that ancient Jewish guy, Bezaleel, did with some acacia wood (the box: the ark).  We'll skip the gold.  Anyway, the point is, measurements are made for the box that included its "footprint" (length and width) and its "body" (adding height).  The ends of the box were square, having the same height and width.

Given a width, call it W, then the board will be cut that wide in two (2) directions to get a square.  Or, in this case, W squared.

W x W = W^2

Well unlike Bezaleel or Noah, let us say the instructions to make the box the same measurements in all three dimensions -- put on your 3D glasses here.

Then, keep the measuring using the same mark on the stick, and you construct a cube!

W*W*W = W^3

No, not that "WWW" -- just a cube.  You know, like the description of the Holiest of Holies.  You don't know about that?  (leave a comment, I'll explain).

If you divide a cube by a square, you're back to a line the length of one of the edges.  So how does that help anybody?

Consider this algebraic express:

w^3/w^2 = W*W*W/W*W.

Going a step further:

W/W * W/W * W = 1 * 1 * W = W

OR

W^2/W^2 * W = W

Now, notice the relationship between the cube and the square:

W^3  = W^1.  W^1 = W^[3-2] or simply W.
W^2

The principle is, when dividing exponents, subtract those in the denominator from those in the numerator.

So, when when using the same base, the form N^D/N^D = N^[D-D] = N^0.  But wait, there is that zero again. N^0 = 1.  No matter what N represents!

All that to get to the first know fact of the day:

N^0 = 1.

Then, what happens if the exponent is larger on the bottom?

N^3   = N^[3-5] = N^[-2].
N^5

What in the world?

Simple, really. Look at it the "long" way:

      1*N*N*N     = 1/N*N.  The N^2 is in the denominator.
N*N*N*N*N*1

Think of it using the number 2.

2^3 = 8
2^2 = 4
2^1 = 2
2^0 = 1
2^[-1] = 1/2
2^[-2] = 1/4
2^[-3] = 1/8


So, what do I know?


N*D = multiplying N times D
But
N^D = multiplying by N, D times

N^0 = 1

N^[-D] =     1   
                 N^D

Coming Next: Making Arrangements

The One and the Only ONE

So if you can't use 0 to solve a multiplication problem, what is there that sort of "disappears" when you multiply or divide?  There is only one answer to that.  The answer is one, and one is the answer.

Here are the facts in algebraic expressions:

1*A = A

A/1 = A

A/A = 1

There you have it, the secret to "higher math".  Well, not really, but it helps a lot in problem solving.

Remember how you can rearrange multipliers?  Well that is where "reciprocals" come in.  Stated with variables that looks like this:

A/B x B/A = 1.

Got that? It looks better with pencil and paper, but my scanner is asleep right now.

Anyway, "A over B" times "B over A" equals "1".

It works out to something like this:

A*B/B*A
= A/A * B/B
= 1 x 1
= 1

Funny how that works out, huh?

Now we can use this unique upright integer to solve a problem.

Solve for A:

3A = 9
3A/3 = 9/3

Let's redo that second line:

(3/3)*A = 9/3
1*A = 3
A = 3

I really do need to get a scratch pad for Windows.  Or Something.

Anyway, here is what I know so far:

A+B = B+A
A-0 = A
A-A = 0
A/A = 1
A*1 = A
and
A*B = B*A
A/B x B/A = 1

Aren't I smart?


Well?

Perhaps I need to check to see if my hat is getting tight.

Tomorrow: Strange Facts about Exponents