Notes on Representation – Equivalence

Let’s say that, to tell you how big the fish was, I spread out my two hands:

this big

this big

Spreading out my hands is a system of representation.  It gives me a way to talk about the fish even without the fish being there.  Even if the fish got away, or if the fish got barbequed and eaten last week.  That the fish isn’t there is kind of the whole point – if I had the fish to point to, I wouldn’t have to spread out my hands like that.

Within this system of representation – and it is a simple one, widely understood – there are many different ways I could show the size of the same fish.  I could hold my hands low, or high.  I could jump up and down wildly while I do this, or be very still.  I could take off my gloves or keep them on.  Those gloves could be orange, or they could be yellow.  None of that matters when it comes to communicating the size of the fish.  All of these ways are equivalent.

Other differences in how I spread out my hands aren’t so easily dismissed.  Some ways of spreading my hands wouldn’t be recognizable as a representation of how big the fish was.  I’m supposed to hold up both of my hands at the same time – if I raise my left hand and then half an hour later raise my right hand, nobody will recognize it as a gesture about the fish, let alone how big it was.
And then there are ways to spread my hands that aren’t equivalent because they indicate a bigger fish.

Almost all systems of representation have multiple ways of representing the same essential situation, ways that are equivalent.  In our normal decimal number system, there are many alternative ways to write the quantity 12345.6 – here are some of them:

12,345.6      12345.60      012345.6     012345.600      0000012345.60000000

Even if different ways of representing a value may be all equivalent, I may prefer some over others.  There may be one way of writing it that is considered standard.  You may never have much reason to write anything other than the standard form.  Yet alternative forms are common when you do computations, or when you show intermediary results.  In other situations, there may not be a single generally-agreed upon standard form, and multiple forms are in common use.  For a simple example of the latter, consider the number 345.6, written in standard form, which would never be written that way if it was a dollar amount: for dollar amounts, the universal standard is to give two decimal places or none at all – here $345.60.

Scientific notation, a form of which we introduced in a previous post, gives us the following equivalent ways of writing the same number 1,200,000:

1200000   120000E1     12000E2     1200E3     120E4     12E5     1.2E6      .12E7

and many others.  Depending on who you ask they’ll point at 1.2E6 or .12E7 as the standard way of writing this number.  Most calculators I’ve seen show 1.2E6 though all will accept the other forms as input.  In books  you usually won’t find 1.2E6 but instead something like 1.2 \times 10^6 , which makes it look like a piece of arithmetic rather than a way to write a number, but the intent is the same.

We’ll have other occasion to look at the issue of equivalent representations and their significance.

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