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Inverse Problems in Mathematics

Inverse Problems

Let us illustrate an inverse problem using one of the fundamental theorems in mathematics - the Pythagorean Theorem.

Pythagoren Theorem (PT)
If a triangle is right with hypotenuse c and legs a and b then a2 + b2 = c2.

The inverse statement is this:
If a triangle is NOT right then a2 + b2 ¹ c2, where a,b, and c are the sides of the triangle.

This is a useful statement. That is, whereas all right triangles satisfy the equation a2 + b2 = c2, there exists no non-right triangle for which a2 + b2 = c2. However, books seldom present this inverse statement of PT. Here is the reason they do not.

To explain the reason, let us invoke the notion of the "counterpositive" or contrapositive.
The counterpositive of A imples B is NOT B implies NOT A. A statement and its counterpositive are logically equivalent statements. That is, when one holds, the other does too. Now consider the counterpositive of the Inverse of PT. It is this:
If a2 + b2 = c2 in a triangle, then the triangle is right.
But that is the converse of PT!! So that is the reason we do not see the "inverse" of PT stated in books... because it is logically equivalent to the converse of PT. And the converse is stated in books. for instance, the converse of PT is Proposition 48 in Book I of Euclid's Elements.

What is the use of the inverse statement?
The inverse statement explicitly says an object not satisfying the premise of the direct statement does not satisfy the conclusion of the direct statement. Thus, given a theorem about objects satisfying a given property, if the inverse is also a theorem, we get "the whole picture" concerning the property. The inverses of some theorems are stated in books. Here is an example.
Isosceles Triangle Theorem:
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Inverse of the Isosceles Triangle Theorem:
If two sides of a traingle are not congruent, then the angles opposite those sides are not congruent and the larger angle is opposite the larger side.

Notice the ending part of the inverse theorem: "and the larger angle is opposite the larger side." The inverse statement does not need this. However the ending part gives additional information.That is, inverse theorems are presented when there is additional useful information that can be provided. Without this additional information, what we would have is just a statement that is logically equivalent to the converse of the theorem. Clearly, not every theorem has a converse that is also a theorem. So the inverse statement does not hold for some statements.
Example: A person lives in New York implies the person lives in America. The converse is not true. Therefore the inverse statement is also not true: If a person does not live in New York the person does not live in America.

To summarize: Sometimes one sees inverse theorems stated. That is when additional information can be provided that can not be provided in the converse theorem.

Inverse Problem
Now that we know what an inverse statement is, what is an inverse problem? Here it is!
Direct Problem: Given the cause, find the effect.
Inverse Problem: Given the effect, find the cause.
In PT the "effect" is a2 + b2 = c2.
Here is another way of saying it:
Direct Problem: Given the question, find the answer.
Inverse Problem: Given the answer, find the question.
Inverse Problem: Given the conjecture, the question is "Is it true?"

Conjecture: Every positive even integer greater than 3 is the sum of two primes. Question: Is it true?

Mathematics advances by proving conjectures - great and small!
So Inverse Problems comprise the core activity of mathematical research.

Inverse problems are particularly relevant to the sciences as well as the applied sciences. An apple is said to have fallen near Newton. He pondered the cause why the apple travelled towards the earth. The Laws of Physics are discovered by the inductive inference on inverse problems. Consider the applied sciences. Computer-assisted Tomography (CAT) scans the human body and generates findings and then guesses the cause. Carbon Dating observes the fraction of C14 among the carbon atoms in a specimen and calculates the number of years needed to cause the ratio to have the value found. Crystal structure is inferred from X-ray diffraction patterns. RADAR interprets the time taken by radio waves to return from various points of an object to give the cause - the picture of the object. Doppler shift is used to deduce the speed at which an object is moving. And how do they figure out how many fish of a given type are in a body of water? Inverse problems not only advance mathematics and the sciences, they are also in the mundane service of mankind.



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Last updated - May 15, 2005