Intersection parallel fun math illustrations

That is, weвЂ™ll assume that every line can be written as an equation:

WeвЂ™ll imagine each line to be in вЂњslope-interceptвЂќ form, which you may remember from algebra class.

To investigate this claim we will create a model. (There are different ways to define parallelism, but weвЂ™ll go with this for simplicity). Imagine we want to investigate the following claim: Any two lines either intersect or are parallel.Д«y вЂњintersectвЂќ we mean the lines share a point in common, and by вЂњparallelвЂќ we mean they go off in the same direction but do not intersect. LetвЂ™s explore some of these issues using a simple model of a simple conjecture. So it can be hard to know that the evidence we collect from the model is truly evidence about the thing we want to know about. Nor can we be sure that our model is similar enough in the ways that really matter. We can never be certain that our model behaves enough like the thing we are actually trying to understand to draw conclusions about it. They are incredibly powerful tools that allow us to trade a problem we donвЂ™t fully understand for one we have a better handle on.Д«ut using models is inherently tricky. Mathematical models are used everywhere in science and can even be turned inward to study mathematics itself. It comes from the world of matrices, which the researchers used to model elliptic curves. The compelling evidence theyвЂ™ve collected doesnвЂ™t come from the world of elliptic curves. The work presents evidence suggesting that, just maybe, there are only finitely many elliptic curves whose rank is greater than 21. For many years the consensus view has been that the ranks of elliptic curves are unbounded, meaning there is no limit to how high a curveвЂ™s rank, or how complex its solutions, can be.Д«ut recent work has some mathematicians thinking that ranks may be bounded after all. In the study of elliptic curves, the вЂњrankвЂќ of a curve, roughly speaking, is a numeric measure of how complex that curveвЂ™s solutions can be. But there are other areas of math where evidence is being used to inform opinion in more controversial ways. We still havenвЂ™t proved that there are infinitely many pairs of primes that differ by 2 - the twin primes conjecture - but 2 is a lot closer to 246 than it is to infinity.ДЇor these reasons and more, believing that the twin primes conjecture is true, even though it hasnвЂ™t been proved, isnвЂ™t very controversial. Thanks to a subsequent public вЂњPolymathвЂќ project, we now know that there are infinitely many pairs of primes that differ by no more than 246. In 2013, Yitang Zhang shocked the mathematical world by proving that there are infinitely many prime number pairs that differ by 70 million or less. And results similar to the twin primes conjecture have been proved. The largest currently known pair of twin primes have nearly 400,000 digits each. Yet almost everyone believes it is true, because there is lots of evidence that supports it.ДЇor example, as we search for large primes, we continue to find extremely large twin prime pairs. The twin primes conjecture is not the Twin Primes Theorem, because, despite being one of the most famous problems in number theory, no one has been able to prove it. The twin primes conjecture hypothesizes that there is no largest pair of twin primes, that the pairs keep appearing as we make our way toward infinity on the number line. Twin primes are pairs of prime numbers that differ by 2 - for example, 3 and 5, 11 and 13, and 101 and 103 are all twin prime pairs. The twin primes conjecture is one example where evidence, as much as proof, guides our mathematical thinking.

These results shape our opinions, suggesting that we should try to prove some theorems but disprove others. We examine and weigh the evidence and decide what comes next. Often, before proving something, we play around, explore, consider examples and collect data.

Still, evidence is important and useful in mathematics. All available evidence might point to 32 as the next number in our sequence, but without a proof, we canвЂ™t be certain. ItвЂ™s proof that establishes mathematical truth. ThatвЂ™s one reason mathematicians strive for proof, not just evidence. Mathematics has a long history of defying expectations and forcing us to expand our imaginations.

Intersection parallel fun math illustrations