Well, for the arithmetic mean, the geometric mean, and even some of the other means, we have some kind of ordered collection of objects, and we operate on that collection to some result.
A mean gives us the same end result, when we replace every item in the collection with it.
For example, with the arithmetic mean, we take the collection, add it up, and divide by the number of items in the collection.
{1, 2, 3} gives us 2 as the arithmetic mean.
So let’s replace every number in the collection with 2.
{2, 2, 2} is our new collection.
If we sum those numbers we get 6, which is exactly what we’d get if we summed our original collection.
So under summation, 2 is the mean of {1, 2, 3}.
Now we have a solid understanding of the fundamental working of an mean.
We take an ordered collection and operate on it.
If replacing all of those elements of the collection with another single value gives the same result, then that value is the mean.
But in this case we need to know what kind of operation we’re applying to the collection.
Existence and UniquenessBecause this definition is incredibly broad, and doesn’t even have to be an operation on numbers, but could be any kind of collection, we’re not always guaranteed to have a mean of a given collection.
The existence of a mean therefore becomes an important topic of study.
Likewise, a mean does not have to be unique.
There might be two values, or even an infinite number of values, that give the same result.
So uniqueness is also an important topic of study.
Relationship to Groups and nth RootsGroups are abstractions of numbers.
They function very similarly to our usual number system.
They consist of a collection of elements, along with a product operation, which assigns the product of every pair of elements in the group to another element in the group (the operation is closed).
Such an object also has to meet a few other conditions to be considered a group.
There must be an identity element, such that the product of the identity and an element in the group is the element, just like one is the identity under multiplication in our usual number system.
Every element must also have an inverse, so that the product of an element and its inverse is equal to the identity.
And the product operation must be associative.
By studying groups, in general, we can learn a lot about numbers and other types of systems that are similar to them.
One area of interest related to the discussion of means is nth roots.
An nth root of x is the value that if we multiply by itself n times, we get x.
For real numbers, which is the most common type of numbers we use, an nth root always exists, and it’s unique.
There’s only one value which will satisfy the requirement.
But with groups, there might be numerous values which fit, or no value at all.
Since the nth root fits the definition of a mean, under the operation of repeated product, the existence and uniqueness of nth roots is insightful for a general theory of means.
What It All MeansEssentially, while most of mathematics is well defined, for some reason, the concept of the mean has largely escaped rigorous definition.
But if we abstract the process enough, we can really get a feel of what’s going on: we want a value that we can use to replace all the values we’re operating on, while still getting the same result.
That’s a mean.
And who knows?.Such abstract ways of thinking about mathematical structures often lead to new discoveries.
That would be nice, right?.. More details