How can you prove the following equation false using real life physics and chemistry ideas, numbers, or equations?
Last week, one of the most intelligent math geniuses we've seen sent a letter to be published in which evaluating the following question which perplexed many math teachers. His letter is written below:
For half an hour, I tackled the inequality challenge you gave us today and came up with a solution. Considering x+y is positive and x-y is negative, the equation: -y<x<y is false in these following ways.
1. x and y can both equal infinity: "By this I studied online about the German engineer George Canner, who studied advanced physics and came up with this valid law in the 1800s: "In the 1995 Pixar film Toy Story, the gung ho space action figure Buzz Lightyear tirelessly incants his catchphrase: "To infinity … and beyond!" The joke, of course, is rooted in the perfectly reasonable assumption that infinity is the unsurpassable absolute—that there is no beyond.
That assumption, however, is not entirely sound. As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others.
Take, for instance, the so-called natural numbers: 1, 2, 3 and so on. These numbers are unbounded, and so the collection, or set, of all the natural numbers is infinite in size. But just how infinite is it? Cantor used an elegant argument to show that the naturals, although infinitely numerous, are actually less numerous than another common family of numbers, the "reals." (This set comprises all numbers that can be represented as a decimal, even if that decimal representation is infinite in length. Hence, 27 is a real number, as is π, or 3.14159….)
In fact, Cantor showed, there are more real numbers packed in between zero and one than there are numbers in the entire range of naturals. He did this by contradiction, logically: He assumes that these infinite sets are the same size, then follows a series of logical steps to find a flaw that undermines that assumption. He reasons that the naturals and this zero-to-one subset of the reals having equally many members implies that the two sets can be put into a one-to-one correspondence. That is, the two sets can be paired so that every element in each set has one—and only one—"partner" in the other set.
Think of it this way: even in the absence of numerical counting, one-to-one correspondences can be used to measure relative sizes. Imagine two crates of unknown sizes, one of apples and one of oranges. Withdrawing one apple and one orange at a time thus partners the two sets into apple-orange pairs. If the contents of the two crates are emptied simultaneously, they are equally numerous; if one crate is exhausted before the other, the one with remaining fruit is more plentiful.
Cantor thus assumes that the naturals and the reals from zero to one have been put into such a correspondence. Every natural number n thus has a real partner rn. The reals can then be listed in order of their corresponding naturals: r1, r2, r3, and so on.
Then Cantor's wily side begins to show. He creates a real number, called p, by the following rule: make the digit n places after the decimal point in p something other than the digit in that same decimal place in rn. A simple method would be: choose 3 when the digit in question is 4; otherwise, choose 4.
For demonstration's sake, say the real number pair for the natural number 1 (r1) is Ted Williams's famed .400 batting average from 1941 (0.40570…), the pair for 2 (r2) is George W. Bush's share of the popular vote in 2000 (0.47868…) and that of 3 (r3) is the decimal component of π (0.14159…).
Now create p following Cantor's construction: the digit in the first decimal place should not be equal to that in the first decimal place of r1, which is 4. Therefore, choose 3, andp begins 0.3…. Then choose the digit in the second decimal place of p so that it does not equal that of the second decimal place of r2, which is 7 (choose 4; p = 0.34…). Finally, choose the digit in the third decimal place of p so that it does not equal that of the corresponding decimal place of r3, which is 1 (choose 4 again; p = 0.344…)." Stated
By concluding x and y are infinity, they both represent vague values yet fit the factors, x+y is positive and x-y is negative, and prove this equation, -y<x<y false
Infinity-Infinity (considering there are different lengths of infinities)= negative
-infinity<infinity<infinity is false (considering the x variable is represented as a larger infinity in this equation yet smaller in x-y
Theorietically, this works, yet my representation of a mere and vague form of, "Infinty" may not deserve the extra credit. However, this challenge greatly excercised my mind. Thank you.
2. y represents 5 and x represents a system (thermodynamics) My variables represent math concepts for this one; not numbers.
when You have to prove -y<x<y wrong considering x+y=positive and x-y=negative; this means x has to represent different numbers and values which is impossible unless you work with unarguable math and physic concepts.
In thermodynamics and physics: a closed system's entropy is positive when it does work and negative when is it done work on.
Therefore x's representation of a heat system theorietically works because of the ability to change constantly from negative to positive therefore x can prove the equation false and still fit the other factors. The only concern, I share for this solution is that, the idea of a system may be too vague, and x as a physics concept may not be the "Number" you are looking for. Thank you for your consideration.