Another method of expressing equations?

[MTannock]

I've written about this on another mathematics forum, but with no reply, so I was hoping I'd have better luck here.

Essentially I've found a way of constructing equations as diagrams, and I'd like to know what people think, because I unfortunately don't know enough mathematics to properly test it myself.

I start by drawing a line around some related variables and/or constants.
This allows multiple sets to overlap, in much the same way as a Venn diagram.
I then separate the contents with commutative elementary operations and horizontal lines of equivalence.
These lines allow the inverse operations to be represented at a glance as well, without the need for rearranging (not to be confused with division lines, though can function as such when the adjacent operation is multiplication).
It seems to work as well for functions like integration or differentiation, but with the choice of a primary function (both will be represented regardless, but the difference in notation makes one more apparent).

I've attached a key, along with two of my simplest examples of this system in practice.

-Michael.

 

[SteveL27]

I've written about this on another mathematics forum, but with no reply, so I was hoping I'd have better luck here.

By b + e/d did you mean what you wrote? Or did you mean (b+e)/d? I couldn't make either of those interpretations make sense relative to your first triangle diagram.

I think the math teachers of America are doing a terrible job teaching kids about parens and the rules of precedence. Must be the calculators.

Also, when you said a = b+c, that didn't make much sense either in terms of your triangle diagram; since the area of a and the area of b+c are so obviously NOT equal.

Can you just start with one simple example and describe what you're doing?

 

[MTannock]

I'm thankful for the reply even though it wasn't one I was expecting, since it explains why I've been having trouble finding out what people think of this.

I'll do my best to explain how the diagrams work.
Firstly, the shapes aren't describing areas, but creating distinct sets for variables. They could be any size and shape, but I chose triangles for their compactness. Where two of these shapes overlap, anything in the overlapping region is shared between them. You asked me for an example, and how the relationship between variables is expressed inside the triangles will require such an example. So for this, I'll use my second diagram, and focus on the bottom triangle. This triangle has a line running through it from left to right. Anything above the line (but inside the triangle) is equivalent to anything below it.
So because v is above this line and u+a*t is below it, this part of the diagram represents the equation v=u+at. Subtraction and division operations are generally unnecessary, because they can be deduced visually.

I've tested this method by converting a sample of equations, and then using the new diagrams to reproduce not only the original equations, but also the entire sets of equations they belong to. This system also seems to work with other functions, and division algebra, but I don't feel I know enough to test this thoroughly.

 

[Diffy]

Hi, what is the usefulness of this?

It seems awfully more confusing then just writing the relationships of the variables in either a list of system of equations.

 

[Whovian]

It is confusing, but it's also nice. Not quite the sort of thing I was looking for, but compact nonetheless.

EDIT: You might be wondering what on Earth I mean by "the sort of thing I was looking for." As I argued, with the username bdejean, on this thread, I expect that either the world operates through the math that we know and the math's, well, one of those concepts of the Universe that just is, or the same applies for some higher form of math. This was sort of like what I was looking for.

Oh, and now do calculus with this new representation. I'd have to argue that it's easier to do so the conventional way.

 

[MTannock]

I've actually found many equations to be far less confusing and much easier for me to remember when they're written in this form, so I'm guessing that either some people will find this more useful than others, or the usefulness could depend on how familiar someone is with the method.

If you're asking about the possible advantages of this over the usual linear method of writing equations; a single diagram can contain and display every permutation, so the sometimes long process of rearranging equations in many cases becomes irrelevant, and you can find the desired solutions almost immediately. It also displays the innate symmetry of relationships, so it could be easier for visual learners to memorize than a list of equations.

 

[AlephZero]

This is not really new. I can remember seeing a similar "triangle diagram" for remembering Ohm's Law V = IR from back at school, and there are plenty of examples on the web like this one:

 

[MTannock]

I'm aware of these, a popular set are the SOH CAH TOA triangles of trigonometry. They seem to be a subclass of the ones I've found.

 

[MTannock]

So do you think this has the same potential as the subclass of triangle diagrams AlephZero pointed out? Bearing in mind that it can express a wider range of equations.

 

[MTannock]

Whovian I've only just noticed your edit. I'm not certain because as I said I don't feel that I know enough to fully test this. But it appears to be no more difficult to do calculus in this new representation than it is to do so in the conventional way: reason being that the laws and identities for functions are no more difficult to convert than the equations themselves. To illustrate this I've attached two very basic examples.

 

[Mark44]

Whovian I've only just noticed your edit. I'm not certain because as I said I don't feel that I know enough to fully test this. But it appears to be no more difficult to do calculus in this new representation than it is to do so in the conventional way: reason being that the laws and identities for functions are no more difficult to convert than the equations themselves. To illustrate this I've attached two very basic examples.

I don't see this as advantageous over just writing equations. Your example in the lower left corner has d/dx(xⁿ) on top and nx^n-1 below. Assuming that things above are equal to things below, why is this better than just writing d/dx(xⁿ) = nx^n-1?

Is there some significance to the two short lines that join the boxes in the top row? Is there some significance in the red line that separates the top row from the bottom row?

 

[MTannock]

This method is roughly equivalent to written equations. The advantage is in being a more visually explicit representation. An advantage that's clearly exploitable, otherwise there wouldn't be a limited version of this in use today. A version that I believe has only been limited, because until now a more generalised form hadn't existed.

The two short lines are there simply to show how the laws and identities for functions will naturally have an identical form to the laws and identities of their inverse functions. This means they can be deduced just as they can with written equations, but more easily because the steps for rearranging equations can be skipped.

The red line is just there to separate my two examples, and probably wasn't necessary.

 

[micromass]

This method is roughly equivalent to written equations. The advantage is in being a more visually explicit representation.

How is a box on top of another box a more visually explicit representation than just writing the = symbol?
The two are exactly the same, except that your representation takes up more space and ink.

 

[MTannock]

Ohm's triangle above takes up more space and ink than just writing the = symbol, but its form makes it easier to work with and remember, so it's used as a teaching aid for visual learners. I would argue that the limitation of the subclass it belongs to is one of the biggest reasons the diagram hasn't been adopted over linear expressions we're currently used to.

 

[micromass]

Ohm's triangle above takes up more space and ink than just writing the = symbol, but its form makes it easier to work with and remember, so it's used as a teaching aid for visual learners. I would argue that the limitation of the subclass it belongs to is one of the biggest reasons the diagram hasn't been adopted over linear expressions we're currently used to.

I never said Ohm's triangle is useful. By the time you learn Ohm's formula, you are already very acquainted with rearranging equations, so finding I in terms of R and V should be no problem. So really, I see no advantage in using Ohm's triangle or your method.

 

[dipole]

Yea I have to agree with Micro. These triangles are exactly equivalent and don't contain any new information. You're just replacing one simple set of symbols with a more complicated rule for how operations are written - it doesn't seem like it would be of any use, except maybe as decoration.

 

[MTannock]

Yea I have to agree with Micro. These triangles are exactly equivalent and don't contain any new information. You're just replacing one simple set of symbols with a more complicated rule for how operations are written - it doesn't seem like it would be of any use, except maybe as decoration.

One of the uses is the representation of full lists of equations, laws or identities and all of their possible rearrangements. This can sometimes mean using less space and ink.
This attached image below for example will give you most of the exponent laws and logarithmic identities.
I've also found these diagrams to actually be simpler, so I'm assuming by the reactions I've been getting that the appearance of replacing a simple set of symbols with a more complicated rule is because of the unfamiliarity of it. I of course don't know that this is the case, so that is one of the things I'm here to find out. I appreciate any reaction, positive or negative, so long as it's constructive.

 

[micromass]

OK, so all you did was erasing the equality sign and placing things in a triangle. I really don't see the use of this.

 

[Diffy]

I can see how this could help some students who are not mathematically inclined to try and remember exponent laws, and formulas.

I don't see any use beyond that.

 

[MTannock]

Looking at it, I think you're right Diffy.

I apologise for wasting people's time with this.