Proof: (Pigeon Hole Principle) from a Problem Solving Class

In summary, the task is to prove the existence of integers a,b,c, not all 0 and each with an absolute value less than 10^6, such that |a + sqrt(2)*b + cuberoot(3)*c| < 10^-11. The approach involves using the pigeon hole principle and rearranging the formula to solve for a variable in terms of another. This type of problem may be familiar to some as it requires creative thinking to find a solution.
  • #1
rad0786
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Homework Statement



Prove that THEIR EXISTS INTEGERS a,b,c NOT ALL 0 AND EACH OF ABSOLUTE VALUE <10^6 SUCH THAT
|a + sqrt(2)*b + cuberoot(3)*c| < 10^-11


Homework Equations




|a + sqrt(2)*b + cuberoot(3)*c| < 10^-11


The Attempt at a Solution



Well, I know that we have to use the pigeon hole principle, otherwise I am completely lost.

I did try to actually find values for a,b,c ... but no luck with that.

I am now attempting to rearrange the formula and solve for some variable in terms of another.

Has anybody seen a problem like this before? Does it have a name?
 
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  • #2
Never mind people, after a lot of hard thinking, i got it (well...close to it!)
 

What is the Pigeon Hole Principle?

The Pigeon Hole Principle is a mathematical concept that states that if n items are placed into m containers where n > m, then at least one container must contain more than one item.

How is the Pigeon Hole Principle used in problem solving?

The Pigeon Hole Principle can be used to prove the existence of a solution or to find the maximum or minimum number of elements that must be present in a given situation.

What are some examples of the Pigeon Hole Principle in action?

One example is the "Birthday Problem" where the Pigeon Hole Principle is used to determine the probability of two people sharing the same birthday in a group. Another example is in scheduling, where the principle can be used to prove that at least two people will have the same number of appointments in a given week.

How is the Pigeon Hole Principle related to the concept of cardinality?

Cardinality is the measure of the number of elements in a set. The Pigeon Hole Principle is related to cardinality because it can be used to prove that if there are more elements than containers, then there must be at least one container with more than one element.

Can the Pigeon Hole Principle be applied to non-mathematical problems?

Yes, the Pigeon Hole Principle can be applied to non-mathematical problems, such as scheduling conflicts or organizing data. It is a general principle that can be used in problem solving across various disciplines.

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