allofmymaths

[allofmymaths]

hihi! could use some help with understanding position vectors- just seeing it in questions confuses me :/

Hey there!  I must be a year late or something by now but hopefully this can still be helpful for someone out there. A position vector is what it sounds like: a vector describing position.  Now the word vector has many meanings in many different fields. They’re all very similar and some would argue they’re all just different ways of saying the same thing, but in this context what we mean when we say a vector is basically an ordered list of numbers: (1,2,3) or (5, -10, 12), or (1,5).  A vector can have as many entries as you want it to, but for our purposes let’s focus on just vectors of length two right now -- that is to say, vectors with two entries.  The idea of a position vector is to use every entry of the vector to describe some aspect of the position of something, relative to another point (usually the origin). Imagine you live on a 2D plane. You want to get to your friend’s house and you know it’s 4m east and 3m north of your house. Relative to your home then, you can describe the position of your friend’s house as (4, 3). Here the first entry describes how far east they are from you and the second entry is for how far north.  If there’s a store 5m west and 2m south of your house you could describe the position as (-5, -2) where again the first entry is how far ‘east’ they are and the second is how far ‘north’ (the numbers are negative since the store is in the opposite direction of east/north) Here you can see the first entry is really describing the horizontal position relative to your house and the second entry is vertical position. Putting these two coordinates together gives you the unique position of the location.  Notice, there’s no law saying the first entry HAS to be the horizontal position. You can flip them around if you want as long as you know what you mean and you’re consistent with it. Normally though, people use them the way described for convenience because then you can assume everyone is on the same page.  The real power of position vectors comes from adding or subtracting them. Let’s think about what that means.  If you imagine your house and your friend’s house on the plane, then you can think of the vector we talked about earlier (4, 3) as an arrow pointing from your house to theirs. This is great if you know the shortest path to their house, but if this is your first time visiting them you might not. Maybe you have a cryptic list of instructions from them which says:  1. Go 2m west and 4m north to the park  2. Then go 6m east and 2m south to the supermarket 3. Finally go 1m north to get to my house. You know then that relative to you, the park is at position (-2, 4).  Relative to the park the supermarket is at position (6, -2)  Finally, relative to the supermarket their house is at position (0, 1)  In the same way that we imagined the position vector of their house relative to ours as an arrow pointing from us to them, we can think of each of these position vectors in a similar way.  Intuitively, we know then that to get to their home we just have to follow the arrows from start to end. After all the steps we’ll arrive at their house.  The idea of ‘following the steps’ is simply the idea of adding the vectors, where you add them by adding the entries in the same positions together: (for example (1, 2) + (3, 4) = (1 + 3, 2 + 4) = (4, 6)) Therefore we know we can add the vectors for each step of the route to our friend’s house, and it should therefore just give us the position of their house relative to ours.  Doing so yields: (-2, 4) + (6, -2) + (0, 1) = (-2 + 6 + 0, 4 + -2 + 1) = (4, 3)  I appreciate this is a bit rushed and compacted so there may be parts that could benefit from further explanation, but regardless I hope that this is helpful for anyone who might read it. 

[allofmymaths]

I found it helpful to view dimensions very abstractly. I’ll highlight this with an example.  distance = speed * time 

Instead of thinking of the units as physical measurements, I just think of them as independent variables. Independent here meaning that they are completely separate, unrelated abstract quantities, in the same way that if I gave you two arbitrary variables x and y, they would be unrelated. With these abstract, independent variables we can then substitute them into formulas and dispose or any dimensionless constants or coefficients to help visualise the relationships between the various independent units. The “algebra” you then do will help you derive unknown units of quantities involved in the calculation. I use the world “algebra” loosely here because the rules are a little different (for example subtracting distance from distance still gives distance), but if we keep that in mind it can still be a helpful technique.  Using M for distance, T for time and X for the unknown unit of speed, we can just place them into the equation to get:  M = X * T  We can then solve this to get: X = M/T = MT^(-1)  Now we know speed is measure is distance (M) per unit time (T). With a more complicated example:  s = (1/2)at^2 + ut is a known suvat equation linking constant acceleration a, initial velocity u and time t to derive the distance traveled s. Again we can use M for distance, T for time and M/T for velocity, derived from our previous example, and X for the unknown units for acceleration. M = 0.5 * X * T^2 + M/T * T = 0.5XT^2 + M  Remember we can ignore the constants so we get: 

M = XT^2 + M Here we must be careful and realise that we don’t want M - M = 0 and in fact M - M = M here to represent the fact that subtracting distance form a distance still gives a quantity that represents a distance. M = XT^2  X = M/T^2 = MT^(-2)

wild question but does anyone here understand maths

alright clearly i don’t understand dimensions and fundamental/derived SI units. can’t believe i’m posting about fucking maths but i was there for the whole fucking 3 hour lecture and i THOUGHT i understood but the more i think about it the more confused i become. i’m supposed to show how i got the dimensions for viscosity and they gave me some equations to help but i legit don’t get it. i’ve googled the process and i still don’t get it.

why the fuck am i in STEM i wanted to write stories lmfao clearly this maths is easy and i’m just not getting it hdgsjfagh

[allofmymaths]

There’s nothing here to solve. That’s just a complex number. 

hey i'm in desperate need of math homework help

how do you solve:

-1-√5i

please if you know how to solve this, help me

[allofmymaths]

It’s helpful to think of division as multiplication by a fraction and substraction as addition by a negative. In this way we can drop the division and subtraction parts and roll with PEMA or BOMA, which is a lot less ambiguous.

So my understanding is that the issue with pemdas/bodmas etc is that it instills a false understanding that multiplication/division has to be before the other when they're actually interchangeable. So I think it would make more sense for it to be B-O-DM-AS (because addition and subtraction are also interchangeable)

But I could be wrong...

Honestly I'm shit at maths but that sounds like it makes sense to me

Thank you sweetie!

[allofmymaths]

The Axiom of Regularity asserts that for any set, x, there must exist some element y of x such that y and x are disjoint sets.  So as an example, x = { a, b } where a = { b } and b = { a } is not allowed because it violates regularity.  The reason it exists is to assert that every set must be well-founded. This is why sometimes it is also called the Axiom of Foundation.  To understand the well-foundedness of sets, we have to define two things first:

1. Well-foundedness of Relations

If you have a relation ~, then the relation is said to be well-founded on a set/class X if every non empty subset of X has a “minimal” element with respect to the relation. That is to say: For all subsets S of X such that S is not empty, there exist m in S such that for all n in S, we have that n ~ m is false.  As an example, if X = { 1, 2, 3 } and the relation is the natural ordering of numbers (<), then < is well founded on X because all non empty subsets ( {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} ) all have minimal elements with respect to <.  2. Transitive Closures of Sets

Put simple the transitive closure of a set, X is the smallest (in terms on inclusion) transitive set that contains X. 

Here, a set Y is said to be transitive if given some element x of Y, and an element y of x, that y is then an element of Y (the element of an element is an element of the original).

Finally then, a set X is said to be well-founded if the set membership relation (a in b) is well-founded on the transitive closure of X. 

An immediate consequence of Regularity is that no set may contain itself. 

Hope this helped!

Is there anyone knowleadgeable about set theory here??? Can someone  *please* explain to me the Axiom of Regularity??? I’m going crazy…

[allofmymaths]

Something worth mentioning is that a trinomial is an algebraic expression involving exactly three different variables. Something like x + y + z or xy^2 + yz^2 + (xyz)^2  The term here: 9x^2 - 10x - 5 only has a single variable and would be called a monomial.  Here the highest power of the variable is 2 and so we call this is monomial (or more commonly polynomial) of degree 2 (or even more commonly, a quadratic) To aid in remembering this, the word polynomial comes from two Greek roots:  - Poly, meaning many - Nomen, meaning name Then “names” in this context are the different symbols used to represent each variable. 

your wish is my command. simplify this

(4x² + 3x + 3) + (5x² – 13x – 8)

:D thanks nick!

(4x^2 +3x +3) + (5x^2 -13x -8)

9x^2 -10x -5

(that trinomial can’t be factored, sadly! so that’s the final answer. try harder.)

[allofmymaths]

How it feels to be in the last 5 minutes of an exam with no idea how to solve the problem

you know what they say, let not your hand at time t know what your hand at time t + 1 is doing

[allofmymaths]

As the hint probably suggests, Player B has a winning strategy utilising the countability of the rationals:

1. Ennumerate the rationals in (0, 1)

2. On the nth guess, read the nth decimal of the nth rational in the ennumeration

3. Win

Q is a rational in (0, 1) and therefore exists somewhere in the ennumeration. Therefore you’ll always find it in finite time. 

A neat problem stolen from a math group:

At the start of the game, Player A chooses a rational number Q in (0,1). Player B is now allowed an infinite string of guesses: If - on their n’th guess - Player B correctly guesses the n’th digit in the decimal expansion of Q, then Player B wins. If their guess is wrong, they are told that it is wrong, and then must attempt to guess the n+1‘th decimal digit of Q. The question: Does Player B always have a winning strategy? That is, if Player A picks some rational number Q, is there some strategy such that - no matter what Q is-  Player B will always, after finitely many guesses, make a correct one? Notes: Rational numbers in (0,1) are either 1) Periodic, looking like 0 . (d_1 d_2 .. d_k) <- repeating 2) Finitely terminating- IE 0. (d_1 d_2 … d_k)(and then only zeroes) For a really slick approach, consider that the rationals in (0,1) can be enumerated.

[allofmymaths]

That is our break

Why do mathmagicians feel the need to just try and solve every puzzle they can come up with???? I see shit like “if two trains left their stations at the same time and one of the passengers was holding a multidimensional orb that you cannot shatter, how many kids does the conductor have?” Dude take a break watch the Simpson or something

[allofmymaths]

I think it’s talking about something like this

What does mathematics even mean as a Spotify genre. I'm so confused.

[allofmymaths]

The second derivative is like when someone has two middle names 

I’m sending my therapy bills to my maths teacher

What the fuck is a “second derivative” why the fuck do we have to multiply so many polynomials bitch my braincells are dying by the second

[allofmymaths]

Two things.

Firstly, there is an inherent assumption here that mathematics and the physical universe are intrinsically linked. Specifically, that any mathematics that doesn’t represent the physical world is inaccurate. Of course, this is nonsense since mathematics is just the consequences of made up rules. 

Secondly, there is in fact an axiom (an assumption so atomic that it’s not something that can be proven using pre-existing assumptions about how maths should work) that asserts something infinite exists. If you want, you can ignore this axiom and live in a mathematical universe that possibly doesn’t have infinity. 

The mathematics would be different to the mainstream mathematics understood and researched, but it’s still a valid universe of mathematics that would be interesting to explore. 

The physical universe can eat my ass though

Infinity doesn't exist.

Sure, there are unlimited numbers been 1 and 2, and 3 and 4, but they all come to an end once they hit "2" and "4."

There are an unlimited amount of numbers. The amount of 0's that you can add on to a 1 is endless, but it still ends, otherwise the resources of the Universe would be too much for it to handle.

Instead, there is a number out there, bigger than all of the other numbers. Perhaps the number of planets in space, or even the amount of atoms in the universe. That is the biggest number. Any higher numbers are instead metaphorical numbers which don't actually exist.

[allofmymaths]

This is a well known pattern and in some places (my home country at one point I believe) it’s even taught as a way to square numbers ending in 5. Congratulations on discovering it on your own!  In general, if a number ends in a 5, you can write it as 10n + 5, where n is the number that is made up of the other digits that aren’t the 5 at the end. Now typically when you square something like (a + b), multiply each pair of terms and add them together. As a worked example:  (a + b)^2 = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2  For our example then: (10n + 5)^2 = (10n)^2 + 10n * 5 + 5 * 10n + 5^2 = 100n^2 + 100n + 25 = 100(n^2 + n) + 25 = 100n(n + 1) + 25 If you look closely, this is exactly what you discovered! The term on the left: 100n(n + 1) is a multiple of 100 so the last two digits will be 00. Since you are adding 25 to it, this means that in the end, the last two digits of the square will be 25, like you’ve noticed!  Finally the remaining digits that aren’t the last two come from 100n(n + 1) and the digits will be the same as the digits of n(n + 1). Again, this is just like you’ve discovered! You multiply the number in front of the 5 by one more than itself and slap that in front of 25 to get the square! 

Making observations like this and discovering patterns is the crux of pure maths so congrats again for doing this yourself! It’s an exciting feeling.

i just accidentally stumbled on a shortcut to squaring any number with two digits (may or may not incled a decimal at any position) ending with a 5.

multiply the first digit by the number after it and then chuck a 25 on the end. place your decimal according to usual rules.

thus 2.5×2.5=6.25, 45×45=2025, 3.5×3.5=12.25, etc.

[allofmymaths]

The product of a number n and 1/6 is (1/6)n  8 less than this product would be (1/6)n - 8 

You know this is no more than 94, or in other words it is less than or equal to 94, so you have (1/6)n - 8 <= 94 Now, if you were to add 8 to both sides, the inequality would still be true and you’d get: 

(1/6)n - 8 + 8 <= 94 + 8 The left hand side of the <= sign becomes (1/6)n and the right hand side becomes 102, so all together you have:  (1/6)n <= 102  Finally, if you multiply both sides by 6, since 6 is a positive number this means the inequality would still be true and you have: (1/6)n * 6 <= 102 * 6 The left hand side becomes n, and the right hand side becomes 612. All together then you have:  n <= 612 Hope that helps! 

Help please-

[allofmymaths]

Yeah the decimal system isn’t very good. The reason, as you’ve basically discovered, is because 10 doesn’t have a lot of factors. 

Multiples of 2 and 5 are easy to understand learn because 2 and 5 divide 10, so their multiples follow really nice patterns. But 3, 4, 6, 7, 8, 9 don’t divide 10 and so their multiples aren’t very neat or regular and are hard to learn.

In comes base 12. 12 is divisible by 2, 3, 4, 6 and is still a relatively small number to use as a base for a counting system. In base 12, multiples of 2, 3, 4 and 6 will all be very neat and regular. Let’s use ! for 10 and ? for 11 in base 12. Then: 

* Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...

* Multiples of 3 are 3, 6, 9, 10, 13, 16, 19, 20, ...

* Multiples of 4 are 4, 8, 10, 14, 18, 20, ...

* Multiples of 6 are 6, 10, 16, 20, ...

You can see these are now really neat and tidy and easy to memorise. 

It’s interesting how we choose to represent numbers in base 10. Not every civilization does this and there are plenty of examples in history that use different counting systems. You’re right, base 10 lends us the intuition of counting on our fingers, but something like base 12 would offer the comfort of easily recognisable patterns, which I’m sure our brains love. 

Me and my friend recently had a pretty long discussion about how the 10 number system whatever it's called isn't very good and only exists because we have 10 fingers probably

So we started thinking about what would work better, we found a system to changing numbers into other systems

We decided it should be an even number

6 would work with the thing that always annoyed us both - not being able to divide by 3 and while you can't divide by 5 and get a full number it's still a pretty nice one. 7 is when it becomes a problem but like there are no perfect numbers at some point I will find a number that fucks it all up

And we tried our changing system on it we had some fun with the 6

But it's not perfect

So maybe 8? It's 2x2x2 so that's nice that's a cool number.

Then I thought maybe 12 it's similar to 6 but a bit bigger so the numbers wouldn't be as long (6 in the 6 numbers system would already have 2 numbers in it) and I mean people use dozens sometimes

And idk I feel like that's some interesting stuff to thing about

[allofmymaths]

At a certain point maths (or at least pure maths) is about studying the consequences and implications of assumptions. You just see where these “rules” take you and come up with some convenient definitions that make it all easier to talk about and discuss.  Most of maths is useless and that’s okay. I’ve never liked thinking about maths as a way of modelling or improving the world, even though it definitely can. Pure maths feels more creative and abstract than I think most people expect. It’s rewarding in and of itself to discover proofs and results that slowly but surely give shape to the nebulous landscape of mathematics formed by our assumptions. 

Im sorry to all of you mathematics et cetera but at some point math starts not being real anymore and like i do it but it ain't real and has no meaning like there are some rules in fuckin equations or shit that make me think huh. You made this up. There's no way in the world this is not made up. But its ok im not saying this b/c i hate math, I don't hate math im bad at it but i can recognize that there are some fun and useful things about it but some of it is all made up just to make computers work and its ok and im glad but its made up

[allofmymaths]

Saying it to people just makes them think you’re crazy though

normalize using “iff” in everyday communication