His head hurts because of the unnecessary brackets, right?
Please excuse my dumb ass self.
Notation without a definition of what notation you’re using is always going to be ambiguous.
If I wrote
6 9 * 6 9 + +
You wouldn’t know what that is, until I told you it was reverse polish notation, then you would know it resolves to 69 and does the same operations as the original equation.
You’re right, nobody defined the base of the numbers either. Come to think of it, what makes you think that those are numbers at all? That’s nothing but a random arrangement of pixels. I mean, who am I to tell you how to interpret the photons reaching your retina? You’re nothing but excitations in the electric fields in my neurons, anyways.
I adore structure of your comment!
Its not that there’s no definition for the notation, but if you fall back to commonly held definitions, there is divergence in common definitions without the parenthesis. Plenty of calculators, especially old ones, don’t respect PEMDAS, so the so by adding brackets your expression is going to fit the intended operations in more commonly used systems than had you left the brackets out.
I also do think its a bit more readable as your eyes are initially drawn to the first operation, you can start evaluating expressions without even parsing the rest of the equation, or you can just block out that entire chunk when you start looking at how many terms are in the equation. That’s subjective though, so to each their own.
Huh, that’s true of any number that ends in 9.
XY + X + Y = 10*X + Y
Y’s cancel,
XY = 9X => Y = 9 for any non-zero finite value of X.
so for 69? X = 6, Y=9
(6*9) + 6 + 9 = 10*6 + 9
54 + 15 = 69
69 = 69 (nice!)
429? X = 42 Y = 9
(42*9) + 42 + 9 =10*42 + 9
(378) + 51 = 429
429 = 429
Even if 10X+Y doesn’t equal something that ends in 9 it works
X=3.14 Y=9
(3.14*9) + 3.14 + 9 = 10*3.14 + 9
28.26 + 12.14 = 40.4
40.4 = 40.4
Doesn’t work if Y =\= 9:
68? X = 6 Y = 8
(6*8) + 6 + 8 ?= 10*6 + 8
(48) + 14 ?= 68
62 =\= 68
I wanted to try to properly prove that it didn’t work for y!=9, but I think you covered the edge cases - X=0 or unbounded. Well done!
I’m an engineering major, we learn all of the edge cases as “well technically this isn’t always true, but we’ll just pretend it is because the results are close enough”
Every value of Y works for X=0, the equation simplifies to Y=Y, so X=0 is just like Y=9.
In the limit as X->infinity, you get Y = 9 again.
X(1+Y) + Y = 10*X + Y lim X->inf Assuming Y is finite, you drop the non-X terms
X(1+Y) = 10*X lim x->inf
Here, because X is non-zero and equal to itself, you can cancel them (I assume, IANA Mathematician) 1 + Y = 10 Y = 9
Nice
