## superposition of x^2 and euler’s identity

20 09 2014

I am meant to be thinking about economics for the construction of the ecosquared app. When I consider relationships of giving, I get sucked into math, and I end up gravitating towards Euler’s identity for some reason:

The identity is the one at the end. That’s the one I have in my head.

### starting with spencer-brown’s curiosity

The last time I saw Leon, he pointed me at this funny little morsel which happens to be at the start of Spencer-Brown’s Laws of Form:

If we begin with x^2 + 1 = 0

Transpose to x^2 = -1

Divide both sides by x, we get x = -1/x

This is self-referential of course, which we normally don’t like in maths. I certainly have been trained to avoid this kind of pattern.

Try substituting 1 in the equation and we get 1 = -1/1 which gives us -1. So 1 = -1

Try substitution -1 in, we get -1 = -1/-1 which gives us 1. So -1 = 1

Both are absurd. Or rather, this is an alternative way of interpreting root -1, or √-1, or i.

### my thought when i saw this

Instead of coming up with a geometric interpretation, where i is the y-axis, and hence all the subsequent interpretations of equations as rotations etc — in space — what if this is simultaneity in time? Or rather, we consider actual simultaneity in time of two states. In this case, then, i is now simultaneously 1 and -1 at the same time. It is in super-position, or dual, or whatever you want to call it. It is both 1 and -1.

Not that we are saying 1*-1, which is -1. We are saying both 1 and -1 as the same time. And I guess at some point, we might say neither 1 and -1, but that tends off to one of our interpretations of zero.

Oh, and remember we have the notion of negative as in ‘not’. So we have the rather interesting mathematical, or arithmetical, expression for 1 and ‘not 1’, or a thing and not a thing.

Extraordinary.

### now think about euler’s identity

That is, e^(iπ) +1 = 0

Becomes e^-π = -1 and e^+π = -1.

This can be turned into e^π = -1 and the reciprocal 1/e^π = -1, or -e^π = 1 and its reciprocal -1/e^π = 1

So, what is ‘1’ and ‘not 1’ becomes a value and… it’s reciprocal?… at the same time. I have no idea what this means.

The normal way to interpret π is in radians around a point, or 180˚, or half-turn around a circle. If we think of it as 3.14, we get

And the inverse is:

Which means what exactly?

Well, in terms of simultaneity, they are both… true… at the same time.

And here we are allowed the function of multiplying them together to get 1. Which seems to be a way of translating ‘1’ and ‘not 1’ into two absolute values (-23.140 and -0.043) that normalise to… 1.

The simultaneity of 1 and -1, can be made equivalent to multiplying these two numbers — negative numbers at that!.

And of course, if we are allowed any number, not just π, then we get multiplying any two reciprocals.

Clearly I have got my wires crosses somewhere along here. I just don’t have the brain right now to deal with this…

### so friggen what?

Well, I’ve had the intuition that the SEA is based on a summative basis. Which is fine for now. However, I do like the notion of 1^n, with n being the number of people. There’s something in that I would like to unpick. And it is related to the notion of numbers multiplying together to give 1. So when someone is at 0.5, another is at 2, one at 0.1, another is at 10. They are reciprocals. And I am thinking of reciprocals in a complex way. So, what are the reciprocals for three people, say? eg 0.5 and 0.5 and 4. Or 0.5 and 0.4 and 5. Interesting, no?

And consider Euler’s formula with sin and cos, something I never think about:

Since I am currently thinking about how values interact in the relative value algorithm, I can’t help but think there’s a relationship here which I haven’t brought out. Ratio of sides relative to the hypotenuse, which is 1; the ‘cos x’ and the ‘sin x multiplied by i’ gives us ‘e’ to the power of ‘i times x’. Relative to 1. Forget about the visual interpretation. Relative to the same value of 1.

It’s a relationship between addition and multiplication and powers. And what these mean temporally, I am guessing, not spatially.

Will take me years perhaps.

Advertisements