Triangle of Power

Alternate notation for powers, logs, and roots.
An equally valuable form of support is to simply share some of the videos.

(This is the corrected version of the one I put out a month or so ago, in which my animation for all the inverse operations was incorrect)

[blackpenredpen]

Thank you for the excellent math video!

[AndyGoth]

"the notation for each varies radically"

I see what you did there.

[Phoenix_Vizvai]

2◿3=8 power
3◹8=2 root
8◺2=3 log

(2◿3)◺2=3 log inverts power
3◹(2◿3)=2 root inverts power
(3◹8)◿3=8 power inverts root
8◺(3◹8)=3 log inverts root
(8◺2)◹8=2 root inverts log
2◿(8◺2)=8 power inverts log

It’s beautifully symmetrical and the symbols don’t conflict with existing ones. Going left-to-right, exponents are slanting upwards as the height gets larger, implying growth. It also matches the current superscript and ^ notation. Roots are also similar to the current notation, slanting downwards with the height coming down from the horizontal side, implying reduction. The log doesn’t look much like the existing symbol but it shows the most shrinkage, implying its end behavior.

And on paper you can save time by not drawing the horizontal sides! Just 2 lines.

The fourth right triangle isn’t used. But that’s cool, I don’t like how equilateral the triangle of power is anyway, when the corners are clearly not meant to be interchangeable. The right triangle notation still shows that each flip is an inversion.

Parentheses still matter because you’re turning a triangle into a line. Not my fault, it’s inline.

Hope you guys enjoy! Leave a reply with thoughts and suggestions. :)

[CaryDominic]

As a student in 11th grade...I won't really use the triangle of power often...but it really helped me understand the relationship of the three ideas...

[SojournerDidimus]

In electronics - as you hinted - calculating parallel resistance is usually denoted as two forward slashes (two parallel lines). This is an easier symbol to type than the o-plus symbol you proposed. a // b = 1/(1/a + 1/b)

[Mr.Nichan]

What I like about this notation is that it kind of matches the positions of the numbers in our current notation, so, to some extent at least, you can just draw or imagine it over the current notation when necessary and get the same mnemonic effects without having to actually change the notation you use and create confusion, nor figure out how to type or write all these triangles all the time.

[reubenfrench]

The only notation I think really needs to be addressed is that there are at least 3 uses for superscripts: exponents, function iterations, and derivatives. sin^2(x)=sin(x)*sin(x); sin^-1(x)=arcsin(x); sin^[4](x)=d^4/dx^4(sin(x)). It gets really annoying tutoring someone in calculus when one equation uses sin^2(x), the next uses sin^-1(x), and the next uses sin^[4](x).

[eduardocortez]

1:10 "What the h*ll"

I've never heard him be that angry; I'm scared! Anthropomorphic pi creature, hold me!

[mistycremo]

I got permission from my math teacher to use this for all of my work, and it is just so much better to use.

[MustSeto]

I have an idea to make this work inline and on a standard keyboard. We already have a separate notation for x to the power of y is z using a the carrot symbol:
x^y = z
I noticed that the carrot looks like a triangle with the bottom part "cut off", with only the top corner remaining. So what symbols do we get if we keep the other corners instead?
You get symbols that look a bit like "<" and ">". But you could use those symbols to represent the adjacent number's place in the triangle. You'd basically be trying to make a triangle out of the symbols "<^>". As close as possible using a standard keyboard, anyways.

For example (hopefully I've kept everything straight):
logx(z) = y would become x<>z = y
y root of z = x would become y^>z = x
x to the power of y = z would become x<^y = z
... though in the last case you can say the "<" is implied, since the operation is so common, and it isn't ambiguous whether you're using the "<" and ">" symbols as inequalities or as parts of a triangle.
So you could keep writing x^y = z.

[LuisCarlos-kpjq]

This is genius! But in order to give more generallity to the name, please take in consideration this one: "operational triangle" (or maybe "op. triangle") instead of "triangle of power". Please keep going with your work, and let me say again: THIS IS GENIUS!!

[Tsskyx]

Rather than using this as a substitute for what we currently have, it would be more useful as a visualization tool. I for example understand logs perfectly, but working with them was always a bit hard for me. I also had no idea why writing a log of n of a base b is the same thing as writing it as a fraction of two logs with an arbitrary base z. This triangle really helped me understand logs more.

[franzluggin]

One problem I just realized when watching the video is that the triangle of power is probably confusing to students because it is not a function, but an equation, that, when a value is left out, _implies_ a function. That I think is not an easy leap of logic for someone who is probably not all that familiar with implicit functions, since polynomials, starting with y=ax^2+bx+c, are needed a lot sooner than logarithms and roots, meaning they were taught in my school in what probably amounts to the second year of high school. Of course, there is no harm in introducing them strictly as the lower left being the variable, the upper one being constant and the right one being the result you seek at first, and only afterwards telling them more.

It may give them the wrong impression, though, that implicit functions can usually be rewritten to be one explicit function on the whole domain (if you make the domain the positive reals, at least). Which is by far not the case.

[mokopa]

This was the first 3B1B I ever watched. Wow...here I am now, on my way to a degree in Mathematics thanks to this (and a few other) inspiring videos...

[soyoltoi]

An observation:
(a+b)(a⊕b)=ab
This follows from how we add fractions: 1/(a⊕b)=1/a+1/b=(a+b)/ab.
Thus we have a sequence of equalities:
ab = (a+b)(a⊕b)
= (a+b+a⊕b)((a+b)⊕a⊕b)
=
= ...
Probably not of any use but an interesting decomposition.

[krotenschemel]

Several years of math tought in 8 minutes... this is by far the most useful youtube video.

[corvidophilm]

Reminds me of Ohm's triangle:
V
I R

cover up the one you want, read the other two. So V=IR, I=V/R, R=V/I.

[Hivlik]

log() notation makes more sense knowing exp() notation. Since nobody uses exp() notation until long after learning exponents, log() seems out of place

[JM-usfr]

I'd actually love it if you did more notation rants. This was so beautiful!

[deleteaman]

I always come back for this video. It helps me a lot even years after finding this video. Just an OP method of exponent stuff.