T_Scagel
June 18th, 2004, 2:46 AM
My theorum :
A continuous fraction can be defined as an infintisimal bound progressivly by a scaling speciffic infinty; and thus be represented as 2 equations instead of a division.
Proof :
-We are going to examine the equation (1/3)
-draw a number line from 0 to 1
- solve the equation for 1 decimal point (you should get 0.3). You now know the answer is between 0.3 and 0.4.
-draw 2 lines at 0.3 and 0.4 : Try to be as accurate as possible
-shade between the lines
-draw another number line from 0 to 1, identical to and directly below the first one
- solve the equation for 2 decimal points (you should get 0.33). You now know the answer is between 0.33 and 0.34.
-draw 2 lines at 0.33 and 0.34 : Try to be as accurate as possible
-shade between the lines
-repeat this series of steps for as many decimal places as you feel like, the important thing is that you notice the cone shape formed by the shaded areas. This cone shape is the infintisimal expression of the equations answer.
-Now for the tricy part : proving a scaling specific infinity
-draw the graphs again, but this time only graph the first number for each line (.3, .33, .333, .3333 etc.; dont graph the .4, .34, .334, .3334, etc.)
-you notice a parabolic shape to the position of these numbers on these graphs, make a guess out of context as to what the range should be (You would be correct to guess that it is [0.3 less/equal to THE ANSWER less/equal to ALL REAL NUMBERS])
-But this cannot being right as it demonstrates that the value of (1/3) is infinite [a number bigger than .3, 3, 300, or 3 million]
-This had me stumpted until I realized that I could define the bounds as parabolically changing values
-So i came up with this statement (I suggest you draw it on paper, it looks very bad on ASCII):
V = answer of the equation; </ is my less then/equal to symbol; D= amount of decimals you are solving to;
B1, B2= The min and max bounds of the answer
B1 </ V </ B2
B1= (3*10^(-D+0))+(3*10^(-D+1))+(3*10^(-D+2))+(3*10^(-D+3))... until one of these terms equals zero(D=The number added to it)
B2= B1 + (10^(-D))
-Therefore, we see that as a regular equation is ranged by a specific infinity, this is ranged by a specific infinity WHICH CHANGES UNIFORMALLY.
-So the 2 equations we would use to describe a continuous fraction would be :
A) A polynomial Function describing the shape of the infintisimal from the first part
B) A statement of the range at each decimal point, defined by polynomials describing the bounds of the answer
Well, thanks for reading, if you have any comments, questions, or criticisms of this, I would be like to hear them; as this is very first stages of developing this idea; and there are almost noone in my area who are interested in Mathematics.
S. P. Does anyone know how to find the asymptote of equation A?
A continuous fraction can be defined as an infintisimal bound progressivly by a scaling speciffic infinty; and thus be represented as 2 equations instead of a division.
Proof :
-We are going to examine the equation (1/3)
-draw a number line from 0 to 1
- solve the equation for 1 decimal point (you should get 0.3). You now know the answer is between 0.3 and 0.4.
-draw 2 lines at 0.3 and 0.4 : Try to be as accurate as possible
-shade between the lines
-draw another number line from 0 to 1, identical to and directly below the first one
- solve the equation for 2 decimal points (you should get 0.33). You now know the answer is between 0.33 and 0.34.
-draw 2 lines at 0.33 and 0.34 : Try to be as accurate as possible
-shade between the lines
-repeat this series of steps for as many decimal places as you feel like, the important thing is that you notice the cone shape formed by the shaded areas. This cone shape is the infintisimal expression of the equations answer.
-Now for the tricy part : proving a scaling specific infinity
-draw the graphs again, but this time only graph the first number for each line (.3, .33, .333, .3333 etc.; dont graph the .4, .34, .334, .3334, etc.)
-you notice a parabolic shape to the position of these numbers on these graphs, make a guess out of context as to what the range should be (You would be correct to guess that it is [0.3 less/equal to THE ANSWER less/equal to ALL REAL NUMBERS])
-But this cannot being right as it demonstrates that the value of (1/3) is infinite [a number bigger than .3, 3, 300, or 3 million]
-This had me stumpted until I realized that I could define the bounds as parabolically changing values
-So i came up with this statement (I suggest you draw it on paper, it looks very bad on ASCII):
V = answer of the equation; </ is my less then/equal to symbol; D= amount of decimals you are solving to;
B1, B2= The min and max bounds of the answer
B1 </ V </ B2
B1= (3*10^(-D+0))+(3*10^(-D+1))+(3*10^(-D+2))+(3*10^(-D+3))... until one of these terms equals zero(D=The number added to it)
B2= B1 + (10^(-D))
-Therefore, we see that as a regular equation is ranged by a specific infinity, this is ranged by a specific infinity WHICH CHANGES UNIFORMALLY.
-So the 2 equations we would use to describe a continuous fraction would be :
A) A polynomial Function describing the shape of the infintisimal from the first part
B) A statement of the range at each decimal point, defined by polynomials describing the bounds of the answer
Well, thanks for reading, if you have any comments, questions, or criticisms of this, I would be like to hear them; as this is very first stages of developing this idea; and there are almost noone in my area who are interested in Mathematics.
S. P. Does anyone know how to find the asymptote of equation A?