Proving Incommensurability in Number Theory with a Graph

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In summary, the problem involves finding two numbers, a and b, such that the area under the graph of the function y= x^2 is divided into two equal parts by two vertical lines. Point a is the intersection of the first vertical line with the x-axis, point b is the intersection of the second vertical line with the x-axis, and point c is the maximum number in the domain of the function. The goal is to prove that the line segment from 0 to a is always incommensurable with the line segment from b to c. However, it is unclear what is meant by "x sq." and "area under the graph", and there is no maximum number for y= x^2.
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Does anyone know a theorm in number theory or mathematics that could be used to prove the following problem: Given a function f(x) = x sq. Graph the function and with two vertical lines, divide the area under the graph such that the two areas are equal. Denote the point on the x-axis where the first vertical line intersects as point a. Denote the point on the x-axis where the second vertical line intersects as point b. Denote point c on the x-axis as the maximum number in the domain of the function. Thus, the area above the line segment from 0 to a is equal to the area above the line segment from b to c. Prove that the line segment from 0 to a is always incommensurable with the line segment from b to c.
 
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e2m2a said:
Does anyone know a theorm in number theory or mathematics that could be used to prove the following problem: Given a function f(x) = x sq.
What do you mean by 'x sq." ? x^2?

Graph the function and with two vertical lines, divide the area under the graph such that the two areas are equal.
What "area under the graph". If you meant y= x^2, that graph does not a have an upper bound. Do you you mean the area of the regions having y= x^2 as upper edge, for x> 0 and x< 0? I am going to assume you mean to find two numbers, a and b, such that the area of the region bounded by y= x^2, y= 0, and x= a is the same as the area of the region bounded by y= x^2, y= 0, x= a, and x= b.

Denote the point on the x-axis where the first vertical line intersects as point a. Denote the point on the x-axis where the second vertical line intersects as point b. Denote point c on the x-axis as the maximum number in the domain of the function.
There is no such maximum for y= x^2. And is c a point or a number?

Thus, the area above the line segment from 0 to a is equal to the area above the line segment from b to c.
??Didn't you just say that that was true for y= 0?

Prove that the line segment from 0 to a is always incommensurable with the line segment from b to c.
 

FAQ: Proving Incommensurability in Number Theory with a Graph

What is incommensurable proof?

Incommensurable proof is a concept in mathematics and philosophy that refers to the inability to compare or measure two quantities or ideas using a common standard. This means that the two quantities or ideas are fundamentally different and cannot be expressed in terms of one another.

What is an example of incommensurability?

A classic example of incommensurability is the comparison between the measurements of a straight line and a curve. Since a straight line has a constant length, it can be measured using a standard unit of length. However, a curve has varying lengths and cannot be measured using the same unit, making it incommensurable with a straight line.

What is the significance of incommensurability in science?

Incommensurability challenges the idea of universal and objective measurements in science. It suggests that different systems of measurement or theories cannot be compared or unified, and therefore may lead to different conclusions or interpretations of the same phenomena.

How is incommensurability related to the philosophy of science?

Incommensurability was first introduced by philosopher Thomas Kuhn in his book "The Structure of Scientific Revolutions" to explain how paradigms or dominant theories in science can change over time. According to Kuhn, incommensurability is one of the reasons why it is difficult for scientists to communicate and agree on theories, as different paradigms may have different standards of measurement and understanding of the world.

Can incommensurability ever be resolved?

Some philosophers argue that incommensurability can never be fully resolved, as it is a fundamental aspect of the human perception and understanding of the world. However, others suggest that incommensurability can be overcome through the development of new measurement systems or theories that can bridge the gap between two previously incommensurable ideas or quantities.

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