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What Are Funny Numbers And How Do You Find Them?

There are many types of numbers and in the typical starting point in math education begins with positive integers, which are usually called “whole numbers”. From here, we introduce students to the idea of negative whole numbers. This then leads to a debate about whether zero is a positive number or a negative number. Interestingly, within the field of numerical analysis, when one digs very deeply into how floating point arithmetic works, zero can actually be signed! Eventually, we introduce division and find that the ratio of integers is what makes a rational number (ratio — rational). From there, we can wonder about numbers between rational numbers and ask if there are numbers that cannot be written as a ratio of integers. This leads to the discovery of the irrational number. Much, much later in one’s math traing, we might learn about different levels of “infinite” and of “density”. The rationals are countably infinite. The irrationals are uncountably infinite. That’s usually when we get our first taste of head explosions. Somewhere in this tangled mess, the imaginary number \(i\) is introduced. It’s a number that solves \(x^{2} + 1 = 0\).

And on and on this goes. There are even numbers and odd numbers. As there are also prime numbers and composite numbers. The investigations can be endless! This is one of the joys of mathematics and math exploration. We don’t have to have context to think about mathematics. We can explore the properties of mathematical objects. Numbers are mathematical objects. What’s beautiful is that we can make up our own objects and explore them. Some of these creations can be boring. Some insanely deep. And some can be funny.

Intermission

Speaking of funny, let’s give a shout out to @icecolbeveridge for winning this challenge.

His entry

What does he win? Well apparently, he wins a Fiji jigsaw!

And now back to the show.

So What Are Funny Numbers and How Do You Find Them?

First, let me tell you where you can find funny numbers. Take any non-empty list of numbers so that the size of the list is an even number. For example, \(X = [4,3,3,3,59,23]\) is a non-empty list of numbers and its size is 6 (there are 6 numbers in the list). This list \(X\) has funny numbers in it. Specifically, they are \(3\) and \(4\). Any list that has funny numbers in it can only have two funny numbers and they must be distinct. For example \(Z = [2,3,3,8]\) doesn’t have funny numbers in it.

Let’s look at another example before I explain why \(3\) and \(4\) are funny.

Consider this list \(Y = [62,39,59,2,-14,94,17,0]\). In this list, the funny numbers are \(17\) and \(39\).

Let’s now see why. Let’s find the median of \(X\). To find the median of a list of numbers, we want to write the numbers sorted from least to greatest. For \(X\), this would be \([3,3,3,4,23,59]\). Next, we find the “middle” number. That number is the median. But for our list \(X\) there is no “middle” number. Thus, the rule is that we compute the average of the largest number in the bottom half of the list with the smallest number in the top half of the list. In this case, these numbers are \(3\) and \(4\) and the median is \(\frac{3 + 4}{2} = 3.5\).

The same holds true for \(Y\). The ordered list is \([-14,0,2,17,39,59,62,94]\). Once again since the list has an even number of numbers in it, the median is found by averaging \(17\) and \(39\).

So why are these numbers called funny numbers? Well, it took two numbers to construct the median. You could think of those numbers as the co-medians of the list.

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