definition: the value of a game state where if it's Left's turn he could win immediately, and if it's Right's turn she could also win immediately. this isn't equal to zero, greater than zero, or less than zero, but is instead "confused with" zero.
This description of star is correct except the conventional genders for Left and Right are swapped which doesn't matter for * but would matter for games that aren't nimbers.
If you saw a STEM outreach event was for "gender minorities" would you think that means "people who aren't men," "people who aren't binary men," "people who aren't cis binary men," or something else? I typically interpret it as "people who aren't binary men."
If you wanted to mean "people who aren't cis binary men," what language would you use?
“marginalized genders” seems like it might be more likely to include binary trans men, but I’d want to ask before telling a binary trans men “yup, you can attend”.
If you saw a STEM outreach event was for "gender minorities" would you think that means "people who aren't men," "people who aren't binary men," "people who aren't cis binary men," or something else? I typically interpret it as "people who aren't binary men."
If you wanted to mean "people who aren't cis binary men," what language would you use?
Like some others, I’d first think “people who aren’t (binary) men”, and that a trans binary man might not be welcome because of the potential privilege afforded by that gender.
Maybe something like “...event open to gender minorities and all trans people” if you want to avoid being explicit about the exclusion, like “Binary cisgender men have plenty of events available to them; STEM for Gender Minorities Event is for everyone else.”
Defining 1/0 as a number is a thing! It's a specific case of a process called localisation, which is basically picking random numbers and forcing them to have inverses. Sadly giving 0 an inverse automatically makes your ring have 0 as it's only element.
Huh yeah ok. That actually sounds really interesting.
Defining 1/0 is also a thing in a completely different type of way. Basically, work in a broader context and see what fits in with patterns that division satisfies other than the basic algebraic rule that we know must break down.
One common approach is the "real projective line" or the "complex projective line" which is better known as the "Riemann sphere". Basically, add one one extra "number", an unsigned ∞ which is near all the big numbers (in absolute value). Then since the reciprocal of a small number is big and the reciprocal of a big number is small, the reciprocal of 1/0 should be ∞ and 1/∞ should be 0.
The Riemann sphere and related ideas are actually really useful and help complex analysis and analytic number theory describe things in more tidy ways.
A different way to understand the analogy (and where the word "projective" comes in) comes from linear algebra. I'll stick with the reals for this. A real number r can be represented by a nonvertical line in the plane with slope r that passes through the origin. Note that arithmetic can be represented geometrically: add two numbers by adding the lines as functions, take a negative by reflecting the line over the x-axis, and take a *negative* reciprocal of a non-horizonal by taking a perpendicular line, etc.
From this perspective, there's no longer a good reason to leave out the vertical line, so if you add it back in, you a new "number" ∞. Except I lied because you can't add a vertical line like a function; you really have to say something like "the set of (x,y1+y2) when (x,y1) and (x,y2) are on the original lines" and then you get ∞+a=∞ always, like you might expect.
If you know linear algebra, you know there are a couple other subspaces of the plane ("graphs of linear relations", one might call them). We can throw those in, too and use them to define things like 0*∞ in a motivated way. But then multiplication is no longer commutative, so almost no one does that, and I don't know of even a theoretical application.
A platonic dittolid has its face one on face unless the faces are triangular and it's not a tetrahedritto. In the case of the icosahedritto, this follows from the dihedral angle being so large that its face slips onto an adjacent face. In the case of the octahedritto, this follows from the dihedral being more than a right angle (so you can see two faces well), but low enough to reasonably split its face into two pieces.
Exercise: Is there a uniform or Johnson dittolid (or similar) where the eyes and mouth would naturally lie on three different faces?
Question for the class: who are some successful amateur scientists/mathematicians?
They don’t need to be famous and it doesn’t need to be a massive contribution. I just want examples of people who 1. were amateurs 2. made contributions to technical fields 3. those contributions were accepted by the field
people like Yitang Zhang do not count, even though they weren’t fully employed in the field at the time they made their big discovery.
What are your definitions here? If amateur means "doesn't hold a PhD in math or a related field" and "accepted by the field" means "published in a real journal (not like a pay to publish thing no one looks at)" then tons of PhD students count as well as a decent number of PhD dropouts like me.
‘do the same thing on both sides’ is allowed, so i’ll differentiate both sides wrt x
2x = 3 -> x=1.5
I know you’re allowed to differentiate both sides sometimes, cause impilcit differentiation. is it because from the first equation, x CAN’T vary because it only has two specific values? or is there some other difference?
As you can see, it hits zero at x = 0 and x = 3 like you said.
But when you differentiate it, the equation you have no longer describes the value of the function but the slope. You can see that the slope is zero at x = 1.5
So, when you differentiate that function we got roots from, you get df/dx = 3 - 2x which you can see has a zero right at x = 1.5 where we would expect it.
I get THAT it doesn’t work, i was more asking why you couldn’t apply the general rule of ‘you’re allowed to do anything you want as long as you do it to both sides of the equation and it’ll still be true’ by differentiating both sides in this situation. There’s a reply talking about the ambiguity of the equals sign and the difference between equating functions (where stuff is true for all x,y etc.) where you can differentiate both sides, and equating stuff evaluated at a point or something along those lines (this is true for specific values of x,y etc.) where you can’t differentiate both sides of the equation.
Yep, that’s a good way to think of it too. Functions are equal Vs functions evaluated at a point are equal, and when you differentiate a function evaluated at a point you just get a value.
Right, exactly. If you were being formal and rigorous, you’d differentiate between x as in the placeholder for the value being mapped, and x as in the point at which the function is actually evaluated. Similar to how, when integrating time based systems from 0 to t, you differentiate between t, the time length over which the system evolves, and t*, the place holder time variable in the integral operator.
So there are things analogous to this sort of thing that are done often: implicit differentiation and related rates/multivariate chain rule. Like, the graph of x²+y² is a surface living in 3d. And the equation x²+y²=4 defines a bunch of points in the plane, and there are so many points that they make a curve (a circle) that's worth differentiating. For example, even if you didn't know a parametrization of the circle, you could use the chain rule to write "For any differentiable parametrized curve x(t),y(t) satisfying x²+y²=4, we have 2*x(t)*x'(t)+2*y(t)*y'(t)=0". This is true of good parametrized curves like (x(t),y(t))=(2cos(t),2sin(t)) and bad ones like (x(t),y(t))=(0,-2).
Now we can do the same thing with the original equation. Let's say I don't know whether or not x^2=3x is satisfied by an interval (like (x+1)²-x²=2x+1 would be). I can still say "If x(t) is a parametrized curve on the real line satisfying x(t)²=3x(t) for every t, then 2*x(t)*x'(t)=3x'(t) for every t, so that either x'(t)=0 or x(t)=3/2. This makes sense because no differentiable function x(t) can jump from x=0 to x=3, so it's got to stay in one place with x'(t)=0.
Method 1 - combine the givens and hope for the best:
We want to use that the sum of the x_i is 1. And we want to use C-S. The only bare sum appearing in C-S is the dot product sum, so make that match and see if it leads anywhere. That happens to solve half the problem. C-S only has one other type of term, so making that match the sum that's 1 is also worth trying. That happens to solve the other half of the problem.
I agree we should generalize “taking the stairs n at a time”, but I think I would personally prefer a different variant.
When you take the stairs “3 at a time”, you make 3*n stairs of progress after n steps. So if you step on each stair twice, you’ve made 3 stairs of progress after 6 steps, 2 stairs of progress after 4 steps, 1 stair of progress after 2 steps, so (1/2)*n stairs of progress after n steps. I would call that “taking the stairs (1/2) at a time”.
If you never leave your initial spot with your steps, then you’ve always made 0*n stairs of progress after n steps, so that would be taking the stairs 0 at a time. Then taking the stairs -1 at a time would be walking backwards normally. Taking the stairs -2 at a time would be like “2 at a time”, but backwards, etc. Similarly, (-1/2) at a time would be stepping on each stair twice but walking backwards.
Now if you vary your gait, we can get other fractions. For example, if you do something like “alternate between 1-stair and 2-stair steps” or “alternate between 3-stair steps and stepping on the same stair”, that’s like “taking the stairs (3/2) at a time (on average)”.
And if you vary your gait inconsistently, you can get other real numbers. For instance, if for each step you follow a rule like “flip a coin: if heads then step 1 stair and if tails then flip two coins and if they’re different then don’t step and if they’re both heads step 1 stair an if they’re both tails then repeat the ‘flip two coins’ step” then you’re taking the stairs (1/2)+(1/8)+(1/32)+...=2/3 at a time. And if you vary the rule based on the binary expansion of an irrational real number like pi, you could take the stairs pi at a time.
And if you stretch the definition of “stair” and walk on a checkerboard patterned floor, you could take the stairs/squares a complex/imaginary number at a time, because you have two directions you can make progress in at once.
And maybe there’s a complicated jungle gym with enough degrees of freedom to allow you to move between nodes a quaternion at a time.
But there’s no way to do octonions. That’d be ridiculous.
@notthedarklord42 While it's not the the main stuff that normal-group is studying, there are plenty of games with infinitely many states that can be played in finite time. For example, notable special cases of infinite Chess. For a boring example, "You name a natural number. And then we take turns counting down by ones to 0. The person who says 0 wins."
Given the "infinite games" tag I was thinking about the axiom of determinacy, but that's not compatible with choice so maybe it's just a large cardinal axiom? I don't actually know much about the infinite games they study in set theory so it could also be something crazy I've never heard of.
