Back to Analytic Geometry
Analytic Geometry:
AG 001 Introduction:
By William E. Steinman:
With Analytic Geometry I will begin a new archive called Analytic Geometry. The original study Notes already contains enough material. If I add more, it will only add confusion. There is no need for that.
Let us go to Microsoft’s Bookshelf for a definition of our new subject:
Analytic Geometry:
A branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates. Analytic geometry was introduced by René Descartes in 1637 and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G.W. Leibniz in the late 17th cent. Its most common application-the representation of equations involving two or three variables as curves in two or three dimensions or surfaces in three dimensions-allows problems in algebra to be treated geometrically and geometric problems to be treated algebraically. The methods of analytic geometry have been generalized to four or more dimensions and have been combined with other branches of geometry.
That will do, thank you very much! You will notice, in the definition, that analytic geometry involves algebra and geometry. What it does not say is that trigonometry is also involved. Therefore, It might behoove us to review some of the concepts of algebra, geometry, and trigonometry before we go on. As to geometry and trigonometry that is part of what the previous Study Notes covered. If you followed that, you are okay with geometry and trigonometry. If not, you can still do it or you can get a Schaum’s Outline on plane and solid geometry and another on trigonometry. See the bibliography in the study notes.
What about algebra? There are a few things we must understand from algebra to make our journey through analytic geometry a smooth one. Let’s begin with the definition of an equation.
Again, from Microsoft’s Bookshelf:
Equation:
Mathematics. A statement asserting the equality of two expressions, usually written as a linear array of symbols that are separated into left and right sides and joined by an equal sign.
So, an equation is a statement asserting that two numbers are equal.
If X denotes a number, then the polynomials (X – 6) and (X + 4) are numbers.
Also, their product (X2 – 2X – 24) is a number.
Since the two numbers and their product are equal we can write it as an equation.
X2 – 2X – 24 = (X – 6)(X + 4)
How did we arrive at the product? Lets review.
Set it up as a straight vertical multiplication problem, like this:
X – 6
X + 4
Then multiply each term of one polynomial by each term of the other and add the results.
X – 6
X + 4
X2 + 4X
- 6X - 24
X2 – 2X – 24
To prove it we can substitute a number for X and do the arithmetic.
Try it!
Now we can call our equation a rational integral equation if we can write it in the form:
a0Xn + a1Xn-1 + a2Xn-2 + . . . + an-1 X + an = 0
Our example equation does not qualify because when we try to do that we come up with 0 = 0.
Try it!
The equation is true enough, but it is true for all possible values of X.
An equation that does qualify is 4X2 – 7X + 13 = (X + 2)(2X – 5)
Expanding that we get:
4X2 – 7X + 13 = 2X2 – X – 10
If we move the right hand terms to the left side we will get:
2X2 – 6X + 23 = 0
Since this is an equation of degree 2 ( n = 2) it has exactly 2 roots.
A first degree equation has one root, a second degree equation has two roots, etcetera.
A important equation of this type is the quadratic equation.
We usually write it this way:
aX2 + bX + c = 0
Now (a) cannot equal zero. That would make our first tern zero and we would not have a quadratic equation. We would have an equation of degree one, a linear equation.
Assuming a is not zero the roots of this equation are given by the quadratic equation
I’m sure you can do the manipulation to prove this.
That is enough for today. If necessary, we can review more about algebra as we go along. Next time I will begin with plane analytic geometry.