Descartes and negative unity

[AntonioLao]

The invention of analytic geometry by Descartes circa 1637 is an indirect method of using negative unity as an alternative way of defining perpendicularity or orthogonality and the existence of right angles. These make sense if and only if 2 slopes exist and the relationship is given by

[math] mn=-1[/math]

where [math]m[/math] and [math]n[/math] are two different slopes of two given perpendicular lines.

[Guille]

Just something I noticed, and let me know if it's relevant?:

If, your equation [math] mn=-1[/math] is given that m=n, then, both m and n are the imaginari number (i).

[AntonioLao]

If, your equation [math] mn=-1[/math] is given that m=n, then, both m and n are the imaginari number (i).

m and n are real rational numbers in [math]R^2[/math] space representing the slopes of orthogonal lines. However, if m and n are pure imaginary numbers then

[math] i^2 = e^{i\pi} = -1 [/math]

[Guille]

m and n are real rational numbers in [math]R^2[/math] space representing the slopes of orthogonal lines. However, if m and n are pure imaginary numbers then

[math] i^2 = e^{i\\pi} = -1 [/math]

And this last equation is one of the most mathematically beautiful of all.

But, wait, if you simplify it, it's even more beautifull:

i=e^pi

And that includes the three magical numbers.

[AntonioLao]

How did you get that simplified term? Please explain.

[math]i=e^{\pi}[/math] ???

[Guille]

How did you get that simplified term? Please explain.

[math]i=e^{\\\\pi}[/math] ???

i^i=i*i

and as

i*i=e*(i*pi)

then

ei*epi=i*i

then

(i*i)/ei=e*pi

and

(i*i)/i=e^2*pi

so, simplified

i=e^2*pi

I just realised that when I wrote the other post I didn't include the squared, so this one is correct (but anyway, they equations continues to be incredibly beautifull, atleast for me):

i=e*(e*pi)

(using latex it wouyld be easier to understand).

[AntonioLao]

i still not sure if i understand. I'll check this and get back to you.