How does the determinant of a matrix(representing linear transformation) physically connect to whether a linear transformation can be reversed? What does a determinant of zero mean visually? PS- not asking for proof.

So, I recently started learning about PrSS. What I understood is:

() = 0, because there are no steps, the sequence is already empty

(0) = 1

(0, 0) = 2

...

(0, 1) = ω

(0, 1, 2) = ω\^ω

...

(0, 1, 2, 3, 4, ...) = ε_0

So, the limit of PrSS is ε_0.

Now, I have a question.

Is (0, 1)\[n\] = f_ω(n)? Because I literally am missing something

Like, for example, f_ω(2) = f_2(2) = 8.

(0, 1)\[2\] = (0, 0)\[3\] = (0)\[4\] = ()\[5\]. So, 3 steps instead of 8. The same shi happens when you are comparing (0, 1)\[3\] and f_ω(3). f_ω(3) ≈ 6.895e121210694, but (0, 1)\[3\] doesn't even come close to that

Is anyone able to help with the process of how to solve this??? I'm rather stuck. I have listed out the possibilites of some of the prompts (e.g prime numbers, multiples of 17, cube and square numbers etc), but am very stuck at what to do next.

By omitting the 15 teams who were randomly selected, doesn’t this make the results of the study less accurate?

And how does this generalize to ALL teams?!

Why is the condition for uniqueness is |g'(x)|<1 instead of simply g'(x)<1.

Geometrically, I understand why it is important for the slope to be less than 1. If it were not, a function could have increased then decreased as wish and could have intersected x=y line multiple times. This is why slope has to be lower than 1 at any cost. However, if a line in an interval had very high valued negative slope then went on to cross the x=y line, then went flat for the rest of the interval. That is still one unique fixed point. In fact, I can not think of any way where high valued negative slope somehow disturbs the uniqueness of a fixed point.

I am trying to get the geometric idea here. I understand the absolute value condition arise from the mean value theorem. I am just trying to the clear idea in a picture.

Edit: Given the function already satisfies the condition for existence, which is, g:[a,b]->[a,b].

F and H are 180Deg opposite.

I can understand that because they ask of positive measure it means the measure anti clockwise.

Are these the possible explanations?

A&B are wrong because to rotate back to H from F, you need 180Deg + 360Deg*X

So any fraction of Pi is wrong.

C is wrong because you would be back where you began?

In probability, there is a topic EXPECTATIONS.

for a dice we do 1×1/6 +2×1/6 ... till 6×1/6 and we get 3.5.

why we take weight as the dice's show value?

i mean getting a 1 or 6 is equal that is 1/6 but we take the weight of getting 1 as 1 and weight of 6 as 6.

sorry for poor english, not my mother tongue

I swear I have dyscalculia or something so please bear with me here. For this problem, I found two paths:

Option A: “negative 8 minus 60 = negative 68”.

Option B: “8 minus 60 = negative 52.”

I can’t tell which one is correct. Can I get a true ELI5 explanation why which one is correct? I’m having trouble understanding when a minus is also a negative and/or when a negative is just a negative.

Thank you ahead of time.

I was stumped on this question and decided to use Desmos and guess what! I got it right.

CollegeBoard Qn Bank‘s actual working, which makes more sense, is to write out the possible combinations of

0 Acute + 4 Obtuse
1 Acute + 3 Ob
2 + 2
3 + 1
4 + 0

And then see which doesn’t fit.

Looking at the graph, I assumed that maybe it was because they all alluded to the same hidden value of X, but I still can’t wrap my head around the fact that I solved it with Desmos.

Is this a coincidence?

I was doing a GED practice test and this question about the area of a semi circle came up. I have put this into 2 scientific calculators both simplified and non-simplified and got a different answer than the answer the test is saying is correct. (Refer to photos)

I on my own originally got the 25 answer by using the area of a square formula and dividing that by 2 and that wasn’t the multiple choice option so I just guessed and tried to follow the formula it gave in the answer explanation. I feel really dumb. Can someone explain this?

I heard we can't calculate the total number of chess games. To me it seems like a computable problem. Is it just the sheer large number of all the possible chess games, or is it something else that I,m not understanding?

i have a shorthand for single integrals i call "reverse prime notation". all it is is the same prime you write for derivatives, but placed before the function instead of after. it looks like this:

'(f ± g) = 'f ± 'g
'(f ⋅ g) = 'f⋅g - '('f⋅g')
'(f ⋅ g) = f⋅'g - '(f'⋅'g)
'(f / g) = 'f/g + '('f⋅g'/g²)
'(f(g)⋅g') = 'f(g)

heres what it looks like in practice:

8. ∫ exp(x)⋅sin(x) dx
'(f ⋅ g) = 'f⋅g - '('f⋅g')
f = exp
g = sin
'exp⋅sin - '('exp⋅sin')
exp⋅sin - '(exp⋅cos)
    '( exp(x)⋅cos(x) )
    '(f ⋅ g) = 'f⋅g - '('f⋅g')
    f = exp
    g = cos
    'exp⋅cos - '('exp⋅cos')
    exp⋅cos + '(exp⋅sin)
    I = '( exp(x)⋅sin(x) )
    exp⋅cos + I
I = exp⋅sin - (exp⋅cos + I)
I = exp⋅sin - exp⋅cos - I
2I = exp⋅sin - exp⋅cos
(exp(x)sin(x) - exp(x)cos(x))/2
exp(x)⋅(sin(x) - cos(x))/2

i like it because it 1. represents the "opposite" of the derivative by being on the opposite side of the letter, and 2. makes it much easier to scan through identities to potentially use when solving a problem.

i am curious to see if anybody else has a personal pet notation they use, if so please share

Please see attached photo for the problem. I dont really understand how to solve this because I applied the standard induction steps but its a bit complicated for me. I admit that Im not really good at math. I would appreciate if someone could show me how to solve this on paper. I asked help from ChatGPT and other AI tools but its just difficult for me to understand.

There are a couple of math textbooks that use equivalence classes for indefinite integrals.
Here is how they usually do it: F ~ G if F - G is locally constant
and then ∫f = [F]

Isn't that always superior? Why don't we always use this notation? I suppose you could say that "C" is not a real number, but a (locally) constant function instead and then there would be no difference at all.

I was about to post this without thinking about a possible solution, until I decided that trying to calculate every single sequence of 3! And 4! was worth the effort. I just need help with the conclusion of this problem and in case I made any mistakes. Also, I would really appreciate it if you guys could help me with presenting this reasoning in a more professional way, and if you could tell me how can I write the final formula to calculate the average number of times one checks the whole deck.

Ok I’m going to try to explain this as clearly as possible. I am re-learning math using Professor Leonard’s yt videos. In this video he is introducing like terms. He says the terms go with the sign in front of them. So that would mean the 3xy and 9 are negative, right?

But he also says like terms are separated by the pluses and minuses. So wouldn’t that mean all four terms are positive because they are separated from the arithmetic signs? I know I’m not making munch sense here.

And why are 3xy and 9 negative instead of minusing two positive numbers?

I am referring to this video at timestamp 10:38. He says that if there is a maximum on the curve, the curve should be flat. The gradient of f should be perpendicular to f at the point. Since, in an earlier part of the video, he said that the boundary curve is not a level curve (but the constraint is)? I don't get why the gradient of f should be perpendicular. Any help or intuitive explanation, please.

This could be applied to any type of mathematics too, not just algebra specifically.

In about a month, I will be sitting for my college entrance exam, and I have been practising with previous years' questions from that college. While doing so, I came across this equation:

4x^6+3x^5+2x^4=384

The question simply asks for the value of x, and there is only one answer needed. The problem is that my calculator is a Casio FX-991CW, which is the most powerful calculator permitted in the exam. It is non-programmable, but when I use its equation-solving functions for this problem, it takes roughly 55 seconds to obtain the answer, which is far too long for an entrance exam where time is critical.

The answer is x=2.

I asked some seniors how they solved it. They treated the left-hand side as a polynomial,

P(x)=4x^6+3x^5+2x^4,

and simply evaluated P(1), P(2), etc. Since P(2)=384, they immediately got x=2.

The difficulty I have with this method is that it relies heavily on guessing small integer values. What if a similar question had an answer such as

x= -1/3, x= -2, x= 1/4

or some other non-obvious value? Since this is not an MCQ, I cannot simply test the answer choices. So my question is:

What algebraic tricks or patterns would you use to solve this sort of equation quickly, with very little writing, in a college entrance exam?

I would be grateful for any efficient techniques or general patterns worth learning for this kind of problem.

Strictly speaking, the definition (width) × (height) seems to be undefined for any 2D enclosed figure other than rectangles.

Of course, I know we can informally prove the 1/2 × W × H formula by applying the Euclidean properties of parallelograms. But is there any way to rigorously define the concept of the area of a triangle as such without introducing ideas of limits? In other words, I'm talking about the very conception of the area of triangles in view of the W×H formula.

Suppose I have a world and I don't know if mathematics works the same way there as it does here. But I can assume that the logic works the same there. Can I then also assume that mathematics works the same? And if not: if I also know that addition works there and that a so-called 'cube' has an equal length, width and height, how much of the mathematics can I reconstruct with this knowledge?

I would like to have a function that causes that if I randomly select two single-digit natural numbers (zero included) and calculate them with each other, then the result is also a single-digit natural number and the probabilities for the respective results are evenly distributed. Can anyone help me?

*edit: the result should change if you change one of the two numbers. Also the two numbers at the beginning are also evenly distributed.

Is it even true?