The Relationship Between Coloring By Numbers And Fractions.

About once a week, I check out various sites for the latest news in Math.  Sometimes, I find something only a few days old, other times it is a bit older but the articles all sound interesting.  This time, I found an article that talks about how coloring by numbers reveals patterns in fractions.  I remember doing one of those color by number paintings when I was a child.  I don't remember what it was but I'm sure it wasn't a horse.

This particular discovery is found in the field of Ramsey Theory.  The field of Ramsey Theory focuses on the idea that mathematics structure exists within hostile circumstances. So what they've do is to break apart large groups of numbers such as integers, or slice up connections between points to show that certain structures cannot be avoided.

One way they do this is to select several colors and assign them to every number in a collection.  Even if you do it in a random or chaotic way, certain patterns will emerge as long as you use a finite number of colors and have a large group of numbers. Ramsey Theorists work on finding these patterns by looking for groups of numbers who have been assigned the same color.

The first result came back in 1916 when Issai Schur showed that no matter how you color the natural numbers, aka the positive integers, there will be at least one pair of x and y such that the x, y, and sum of x + y are all the same color. Then in 1974, another mathematician, Hindman, extended this result to an infinite subset of integers and proved that no matter how the natural numbers are colored (with a finite number of colors) the integers are all the same color and the sums are also this color.  These set resemble even numbers.

Hindman believes that it is possible to find an arbitrarily large set of numbers with the same color which will contain both its sums and products. This is based on how addition and multiplication are related and working with both sums and products at one time can be a bit difficult. However, as long as you limit this to rational numbers which are what fractions are, and voila, colors begin to appear frequently. This allows for the full {x, y, x + y, and xy}

So cool.  Let me know what you think, I'd love to hear.  Have a great day.