/r/mathematics

r/mathematics is a subreddit dedicated to focused questions and discussion concerning mathematics.

/r/mathematics is a subreddit dedicated to focused questions and discussion concerning mathematics. Submissions should state and outline problems or questions about a given field or link to an especially insightful article about a mathematical concept.

/r/mathematics is a moderated community. Please read the submission and comment rules before posting.

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/r/mathematics

5

The length of side AC of triangle ABC is 13.7 cm. From the vertex B, draw segments BD and BE so that triangles ADB and BEC are isosceles. Using the data in the graph, calculate the perimeter of triangle BDE.

9 Comments

2024/02/12

11:17 UTC

11:17 UTC

1

In a poker game, probability of choosing two pairs has been given as: (13×4C2×12×4C2×11×4C1)/2!÷(52C5)

My query is why can't I divide the upper term by 3! Since ,for example,JJQQA can be arranged among themselves as JJAQQ,QQJJA,QQAJJ,AJJQQ,QQAJJ?

What concept am I missing?

0 Comments

2024/02/12

10:52 UTC

10:52 UTC

1

Hello,

Can anyone tell mehether the zeta function can be represented recursively by the zeros - i.e. trivial and non-trivial together?

So can you use the non-trivial zeros Nr.1,Nr.2,Nr.3,.. etc. as z, z2, z3,...etc.

and the trivial ones, i.e. all even negative numbers -2, -4, -6-...etc.

to represent the function like this?:

Zeta=(x-z1)*(x-2)*(x-z2)*(x-4)*(x-z3)*(x-6)*(x-z4)*(x-z8)* .....

?

Kind regards

2 Comments

2024/02/12

10:06 UTC

10:06 UTC

6

0 Comments

2024/02/12

08:05 UTC

08:05 UTC

1

1 Comment

2024/02/12

07:00 UTC

07:00 UTC

10

I am currently a final year undergraduate of a university ranked in the 1000-1200 range. I have completely messed up my BSc which is a general three year program with 90 credits (about 30 credits each for pure math, applied math and physics) and I have a GPA of about 2.1. I did not get selected for a specialization program (a 4 year honours degree) because of my low gpa.

My goal is to be a theoretical physicist and I am currently extremely upset and have very bad mental health because of the state of my degree.

I want to study pure maths/mathematical physics at a graduate level. If I get into any masters program of a low rank and do well in it, would I then be able to possibly get into a PhD program in the top 50 or atleast 100 universities?

Alternatively, should I do a BSc from the beginning, get a pure maths specialization and assuming I do well, try to get into a PhD in a top 50? If I do this, my only option would be a university ranking in the 17000s.

Please give me some advice because I feel stuck and hopeless. I don't mind the amount of time that I'll have to spend on this. Please tell me which way gives me a higher chance of reaching my goal. This has been very important to me from a young age so even if it's a pipe dream I would like to atleast try.

24 Comments

2024/02/12

06:23 UTC

06:23 UTC

2

I am really perplexed right now. I tried solving a problem through two different methods and got two different answers. My math skills are still early in development, but I have no idea how this has happened.

The problem:

integral of dt/(200+2t)

When I start off with u substitution, I get an answer of 1/2 ln|200+2t| + C. But when I try factoring out 1/2 first, I get 1/2 ln |100+t| + C. At first I thought I made a mistake, but they differentiate to the same thing. I think it has something to do with the +C but that wouldn't account for the variable t having different coefficients. Why does this happen or did I make a mistake?

2 Comments

2024/02/12

05:52 UTC

05:52 UTC

0

I just want to study math in year levels

0 Comments

2024/02/12

05:19 UTC

05:19 UTC

1

Thanks!

5 Comments

2024/02/11

18:29 UTC

18:29 UTC

6

Should i be worried if i can make it as mathematician if i get around 5% of single variable calculu's exercises wrong or don't grasp them? Like my mind goes blank when facing certain exercises and i don't know what to do at all. Does this determine my success in context of math? Am i able to still come up with something special or if i wan't to contribute something math related, or am i supposed to be top notch student?

Math is not my major though, but it interest me alot. My major is economic science.

My dream is to come up with something unique -- even figuring out for company how to optimize their business in some unique way, it doesnt have to be that big, just something that i invented or reinvented and can feel grateful for creating.

8 Comments

2024/02/11

15:57 UTC

15:57 UTC

5

Hello Mathematicians,

I have a set of non linear equations that I'm solving using ODE solver in Matlab. The equations represents a physical dynamic system with a known initial condition. My system goes through many periods before steady state, hence , I have to solve the ODE for the entire interval.

I tried to solve the ODE for the entire time interval - with my full transient signal, and I got the dynamic response but it was time consuming.

I then, wanted to solve for each period, like I start with initial condition, solve ODE for 1st period. Then update the initial condition with the new value obtained from ODE ( last value) and so on, and I got totally different results.

The 1st approach gave me a result signal that converge , but the second one keep diverging by higher end value each time.

Can someone explain the differences ?

( im using ode45 in matlab)

3 Comments

2024/02/11

13:15 UTC

13:15 UTC

1

**I have a question about integer partitions. I am familiar with 2 notations (example: (5, 5, 5, 4, 3, 3, 3, 3, 3, 1, 1, 1, 1) and (5^3, 4^1, 3^5, 1^4). I would like to know if there are other notations and if there are any good references to read on this topic.**

1 Comment

2024/02/11

13:10 UTC

13:10 UTC

5

how it is possible to calculate the trig. functions of numbers without calculator? Sample: Sin(30°=0.5 i just want know how (and who) discovered that sin(30°=0.5 without a calculator

7 Comments

2024/02/11

08:53 UTC

08:53 UTC

0

I found an ingenious way to create Hyper extensions of complex numbers with any even number of dimension, since complex numbers already have 2 dimensions, in a way that makes those extensionc commutative rings, so commutativity and associativity are preserved. The easiest way to create a n-dimensional extensional of complex numbers is when n is a power of 2, which can easily be done with recursion. To do this you create a representation of numbers in those extensions, and you can actually make a perfect match to C^n, for a 2n dimensional hyper extension of complex numbers. To do that just mae an array with exactly n complex numbers, which looks like this (a, b, c, d, e, f, g...) with n elements, where all elements are complex numbers. Multiplying 2 arrays just multilies the elemens with the same index, and it also happens with addition. Finally if you apply any function over the complex numbers in this array it just applies it to all elements.

That way of representing numbers is very useful, since the properties make it easy to do any transformation to it. The final thing is to find a way to represent numbers using sums, and not an array. To do that you start with some assumptions, the first being (a, a, a, a, a...) represents the complex number a, the second is that if any of the elements is 0, then the number is a 0-divisor, the third is that if a number only contains one or negative one as elements then it is a square root of 1, which will be used to extend the complex numbers. For a 4D extension of complex numbers you start with (1, -1) = j, a square root of 1, and using the formula ((a+b)/2+(a-b)/2*(1, -1)) = (a, b), which can be verified easily, you can find out that (a, b) = 1/2((a+b) + (a-b)j), and to convert back to the array form, a+bj = (a+b, a-b). This extension is just an extension of the Split complex numbers that allow a and b to be any complex number, rather than just reals. Now to make a 6D extension of complex numbers you start with (1, 1, -1) = j, (1, -1, -1) = k, and to convert a number from array form (a, b, c) to normal form d+ej+fk you have to solve the equation d+e*(1, 1,-1)+f*(1, -1, -1) = (a, b, c), which can be solved using a system of equations, which is d+e+f = a, d+e-f = b, d-e-f = c, which when solved yields d = (a+c)/2, e = (b-c)/2, f = (a-b)/2, so now you can convert from array form to normal form using the formula (a, b, c) = 1/2((a+c) + (b-c)j + (a-b)k), and to convert back you can just replace j and k by (1, 1, -1) and (1, -1, -1) respectively and add the terms, getting a+bj+ck = (a+b+c, a+b-c, a-b-c). If you try to use this formula to convert (1, -1, 1) to normal form you get that it equals to 1-j+k, so (1-j+k) is another square root of 1, equal to jk, so now you can make a formula to multiply any number in this extension with that result. To create an extension of complex numbers with an 2p dimension, if p is prime, create ceiling(log_2(p)) different square roots of 1 using numbers that contain only 1 and -1 in array form, and then solve a system of equations to convert to normal form. With that information, a 10D extension of complex numbers would need 3 different base square roots of 1 to work, which can be j0, j1 and j2, and a 14D extension of complex numbers would also need exactly 3, the only difference is that the products wouldn't be the same as the 10D extension, so it would work differetly. When 2n is not of the form 2p, you can use the prime factorization of n to create that extension, using recursion. To create a 12D hyper extension of complex numbers start by defining the square roots of 1 which will be used, which are (1, 1, 1, 1, -1, -1) = j0, (1, 1, -1, -1, -1, -1) = j1, (1, -1, 1, -1, 1, -1) = j2. Now you just need to solve a system of equations to get the for (a, b, c, d, e, f), using j0, j1, j2, j0j1, and j0j2, getting = 1/4 ( (2 b + c - d + e + f) + (2 a - 2 b - c + 3 d - e - f)j0 + 2(b - d)j1 + (c - d + e - f)j2 + 2(a - b - c + d)j0j1 + (c - d - e + f)j0j2), so now you can convert any number from array form to normal form in the 12D extension of complex numbers. This is much easier to do with powers of 2, like 8, since you can use recursion. In this case you start with 4 elements in the array, which can be simplified to 2, getting a number of the form (a, b), where a and b are elements from the 4D extension of complex numbers. Now define (1, -1) to be k, and the conversion gives you (a, b) = 1/2((a+b) + (a-b)k). Now replace a by (c, d) and b by (e, f), where c, d, e and f are complex numbers and convert them to normal form, getting a = 1/2((c+d) + (c-d)j) and b = 1/2((e+f) + (e-f)j). Now replacing those values into the formula you get (a, b, c, d) = 1/4( (a+b+c+d) + (a-b+c-d)j + (a+b-c-d)k + (a-b-c+d)jk). You can easily use recursion if n is a power of 2, by creating an array with 2 elements from the previous power of 2 extension and use the previous conversion formulas until all values are complex numbers. All of the extensions are commutative and associative. It is also possible to use the prime factorization of the number of dimensions to convert, but it is still hard for non powers of 2. Every equation with degree m has exactly m^n roots, for the extension with 2n dimensions, if no repeated roots exist. Finally there are 2 functions that can be extended from the complex numbers, being the absolutevalue and the conjugate. To take the absolute value of a number represent it in array form and take the absolute value of th geometric mean of all elements, works with any dimension. To take the conjugate you move all elements to the position n/2 to the right or left from their curret position, where n is the number of elements, so it only works when the number of dimensions is a multiple of 4. The conjugate of a number allows you to transform it to a number with half of the number of dimensions, if you add or multiply, just like the conjugate of complex numbers do. You could also create an extension with an odd number of dimensions by limiting the values of the array to be all real numbers, from the extension wth the double of dimensions, which lose the "algebraic closure" and equtions can have any number of roots, and adding square roots of 1 to the reals doesn't do much, so you can use them if you want.

If you have any suggestions on what I should improve in this post or you are having difficulties understanding just comment below.

2 Comments

2024/02/11

02:50 UTC

02:50 UTC

90

11 Comments

2024/02/11

01:34 UTC

01:34 UTC

12

I've read that an inner product on a vector space always induce a norm on the vector space by setting the norm to the square root of a vector dot producted with itself.

However, this clearly doesn't work if you consider a vector space over the rational field where the square root isn't necessarily defined. I guess one could argue that it is exists in R and Q is a subset of R and since norms have a range if R, we can still calculate a norm. However surely this doesn't make sense for a vector space over a more abstract field? The proofs I've seen always seem to assume the field is R^n or C^n, does this result really still hold true for ANY vector space?

If so, what would that norm be in the case of a vector space over an abstract field? Or does this result only hold when we are considering fields where the square root is meaningfully defined?

Edit: Thanks all, inner products are only defined for vector spaces over R^n or C^n, makes much more sense now

16 Comments

2024/02/10

17:27 UTC

17:27 UTC

1

Hello,

qualifications - BSc Mathematics Hons:

I would like to know the name of good colleges or universities in India for BSc Mathematics Hons who accept students based on only 10+2 result and don't require any test like JEE or CUET. Any information regarding that would be very helpful.

Thanks for any information.

0 Comments

2024/02/10

16:37 UTC

16:37 UTC

2

Hi, I’m trying to learn about Thompsons Groups F T and V, I have found loads of introductory material on F but haven’t stumbled across similar for T. Does anybody know of any good papers/articles/books that i could read? (I should say I have read Cannon-Parry-Floyd but i couldn’t get on with it very well)

0 Comments

2024/02/10

14:43 UTC

14:43 UTC

0

**Last digit rules**

**2**: The number ends in 0, 2, 4, 6 or 8

**5**: The number ends in 0 or 5

**10**: The number ends in 0

**Last 2 digit rules**

**4**: The last 2 digits are a multiple of 4

- The 10s digit is even and the last digit is 0, 4 or 8
- The 10s digit is odd and the last digit is 2 or 6

**20**: The number ends in 00, 20, 40, 60 or 80

**25**: The number ends in 00, 25, 50 or 75

**50**: The number ends in 00 or 50

**100**: The number ends in 00

**Last 3 digits rules**

**8**: The last 3 digits are a multiple of 8

- The 100s digit is even and the last 2 digits are 00 or a multiple of 8
- The 100s digit is odd and the last 2 digits are 4 times an odd number

**40**: The last 3 digits are a multiple of 40

- The 100s digit is even and the last 2 digits are 00, 40 or 80
- The 100s digit is odd and the last 2 digits are 20 or 60

**125**: The number ends in 000, 125, 250, 375, 500, 625, 750 or 875

**200**: The number ends in 000, 200, 400, 600 or 800

**250**: The number ends in 000, 250, 500 or 750

**500**: The number ends in 000 or 500

**1,000**: The number ends in 000

**Last 4 digits rules**

**16**: The last 4 digits are a multiple of 16

- The 1,000s digit is even and the last 3 digits are 000 or a multiple of 16
- The 1,000s digit is odd and the last 3 digits are 8 times an odd number

**80**: The last 4 digits are a multiple of 80

- The 1,000s digit is even and the last 3 digits are 000 or a multiple of 80
- The 1,000s digit is odd and the last 3 digits are 40 times an odd number

**400**: The last 4 digits are a multiple of 400

- The 1,000s digit is even and the last 3 digits are 000, 400 or 800
- The 1,000s digit is odd and the last 3 digits are 200 or 600

**625**: The last 4 digits are a multiple of 625

- The 1,000s digit is 0 or 5 and the last 3 digits are 000 or 625
- The 1,000s digit is 1 or 6 and the last 3 digits are 250 or 875
- The 1,000s digit is 2 or 7 and the last 3 digits are 500
- The 1,000s digit is 3 or 8 and the last 3 digits are 125 or 750
- The 1,000s digit is 4 or 9 and the last 3 digits are 375

**1,250**: The number ends in 0,000, 1,250, 2,500, 3,750, 5,000, 6,250, 7,500 or 8,750

**2,000**: The number ends in 0,000, 2,000, 4,000, 6,000 or 8,000

**2,500**: The number ends in 0,000, 2,500, 5,000 or 7,500

**5,000**: The number ends in 0,000 or 5,000

**10,000**: The number ends in 0,000

**Last 5 digits rules**

**32**: The last 5 digits are a multiple of 32

- The 10,000s digit is even and the last 4 digits are 0,000 or a multiple of 32
- The 10,000s digit is odd and the last 4 digits are 16 times an odd number

**160**: The last 5 digits are a multiple of 160

- The 10,000s digit is even and the last 4 digits are 0,000 or a multiple of 32
- The 10,000s digit is odd and the last 4 digits are 80 times an odd number

**800**: The last 5 digits are a multiple of 800

- The 10,000s digit is even and the last 4 digits are 0,000 or a multiple of 800
- The 10,000s digit is odd and the last 4 digits are 400 times an odd number

**3,125**: The last 5 digits are a multiple of 3,125

- The 10,000s digit is 0 or 5 and the last 4 digits are 0,000, 3,125, 6,250 or 9,375
- The 10,000s digit is 1 or 6 and the last 4 digits are 2,500, 5,625 or 8,750
- The 10,000s digit is 2 or 7 and the last 4 digits are 1,875, 5,000 or 8,125
- The 10,000s digit is 3 or 8 and the last 4 digits are 1,250, 4,375 or 7,500
- The 10,000s digit is 4 or 9 and the last 4 digits are 0,625, 3,750 or 6,875

**4,000**: The last 5 digits are a multiple of 4,000

- The 10,000s digit is even and the last 4 digits are 0,000, 4,000 or 8,000
- The 10,000s digit is odd and the last 4 digits are 2,000 or 6,000

**6,250**: The last 5 digits are a multiple of 6,250

- The 10,000s digit is 0 or 5 and the last 4 digits are 0,000 or 6,250
- The 10,000s digit is 1 or 6 and the last 4 digits are 2,500 or 8,750
- The 10,000s digit is 2 or 7 and the last 4 digits are 5,000
- The 10,000s digit is 3 or 8 and the last 4 digits are 1,250 or 7,500
- The 10,000s digit is 4 or 9 and the last 4 digits are 3,750

**12,500**: The number ends in 00,000, 12,500, 25,000, 37,500, 50,000, 62,500, 75,000 or 87,500

**20,000**: The number ends in 00,000, 20,000, 40,000, 60,000 or 80,000

**25,000**: The number ends in 00,000, 25,000, 50,000 or 75,000

**50,000**: The number ends in 00,000 or 50,000

**100,000**: The number ends in 00,000

**Last 6 digits rules**

**64**: The last 6 digits are a multiple of 64

- The 100,000s digit is even and the last 5 digits are 00,000 or a multiple of 64
- The 100,000s digit is odd and the last 5 digits are 32 times an odd number

**320**: The last 6 digits are a multiple of 320

- The 100,000s digit is even and the last 5 digits are 00,000 or a multiple of 320
- The 100,000s digit is odd and the last 5 digits are 160 times an odd number

**1,600**: The last 6 digits are a multiple of 1,600

- The 100,000s digit is even and the last 5 digits are 00,000 or a multiple of 1,600
- The 100,000s digit is odd and the last 5 digits are 800 times an odd number

**8,000**: The last 6 digits are a multiple of 8,000

- The 100,000s digit is even and the last 5 digits are 00,000 or a multiple of 8,000
- The 100,000s digit is odd and the last 5 digits are 4,000 times an odd number

**15,625**: The last 6 digits are a multiple of 15,625

- The 100,000s digit is 0 or 5 and the last 5 digits are 00,000, 15,625, 31,250, 46,875, 62,500, 78,125 or 93,750
- The 100,000s digit is 1 or 6 and the last 5 digits are 09,375, 25,000, 40,625, 56,250, 71,875 or 87,500
- The 100,000s digit is 2 or 7 and the last 5 digits are 03,125, 18,750, 34,375, 50,000, 65,625, 81,250 or 96,875
- The 100,000s digit is 3 or 8 and the last 5 digits are 12,500, 28,125, 43,750, 59,375, 75,000 or 90,625
- The 100,000s digit is 4 or 9 and the last 5 digits are 06,250, 21,875, 37,500, 53,125, 68,750 or 84,375

**31,250**: The last 6 digits are a multiple of 31,250

- The 100,000s digit is 0 or 5 and the last 5 digits are 00,000, 31,250, 62,500 or 93,750
- The 100,000s digit is 1 or 6 and the last 5 digits are 25,000, 56,250 or 87,500
- The 100,000s digit is 2 or 7 and the last 5 digits are 18,750, 50,000 or 81,250
- The 100,000s digit is 3 or 8 and the last 5 digits are 12,500, 43,750 or 75,000
- The 100,000s digit is 4 or 9 and the last 5 digits are 06,250, 37,500 or 68,750

**40,000**: The last 6 digits are a multiple of 40,000

- The 100,000s digit is even and the last 5 digits are 00,000, 40,000 or 80,000
- The 100,000s digit is odd and the last 5 digits are 20,000 or 60,000

**62,500**: The last 6 digits are a multiple of 62,500

- The 100,000s digit is 0 or 5 and the last 5 digits are 00,000 or 62,500
- The 100,000s digit is 1 or 6 and the last 5 digits are 25,000 or 87,500
- The 100,000s digit is 2 or 7 and the last 5 digits are 50,000
- The 100,000s digit is 3 or 8 and the last 5 digits are 12,500 or 75,000
- The 100,000s digit is 4 or 9 and the last 5 digits are 37,500

**125,000**: The number ends in 000,000, 125,000, 250,000, 375,000, 500,000, 625,000, 750,000 or 875,000

**200,000**: The number ends in 000,000, 200,000, 400,000, 600,000 or 800,000

**250,000**: The number ends in 000,000, 250,000, 500,000 or 750,000

**500,000**: The number ends in 000,000 or 500,000

**1,000,000**: The number ends in 000,000

**Last 7 digits rules**

**128**: The last 7 digits are a multiple of 128

- The 1,000,000s digit is even and the last 6 digits are 000,000 or a multiple of 128
- The 1,000,000s digit is odd and the last 6 digits are 64 times an odd number

**640**: The last 7 digits are a multiple of 640

- The 1,000,000s digit is even and the last 6 digits are 000,000 or a multiple of 640
- The 1,000,000s digit is odd and the last 6 digits are 320 times an odd number

**3,200**: The last 7 digits are a multiple of 3,200

- The 1,000,000s digit is even and the last 6 digits are 000,000 or a multiple of 3,200
- The 1,000,000s digit is odd and the last 6 digits are 1,600 times an odd number

**16,000**: The last 7 digits are a multiple of 16,000

- The 1,000,000s digit is even and the last 6 digits are 000,000 or a multiple of 16,000
- The 1,000,000s digit is odd and the last 6 digits are 8,000 times an odd number

**78,125**: The last 7 digits are a multiple of 78,125

- The 1,000,000s digit is 0 or 5 and the last 6 digits are 000,000 or a multiple of 78,125
- The 1,000,000s digit is 1 or 6 and the last 6 digits are 15,625 times a number that ends in 1 or 6
- The 1,000,000s digit is 2 or 7 and the last 6 digits are 15,625 times a number that ends in 2 or 7
- The 1,000,000s digit is 3 or 8 and the last 6 digits are 15,625 times a number that ends in 3 or 8
- The 1,000,000s digit is 4 or 9 and the last 6 digits are 15,625 times a number that ends in 4 or 9

**80,000**: The last 7 digits are a multiple of 80,000

- The 1,000,000s digit is even and the last 6 digits are 000,000 or a multiple of 80,000
- The 1,000,000s digit is odd and the last 6 digits are 40,000 times an odd number

**156,250**: The last 7 digits are a multiple of 156,250

- The 1,000,000s digit is 0 or 5 and the last 6 digits are 000,000, 156,250, 312,500, 468,750, 625,000, 781,250, or 937,500
- The 1,000,000s digit is 1 or 6 and the last 6 digits are 093,750, 250,000, 406,250, 562,500, 718,750 or 875,000
- The 1,000,000s digit is 2 or 7 and the last 6 digits are 031,250, 187,500, 343,750, 500,000, 656,250, 812,500 or 968,750
- The 1,000,000s digit is 3 or 8 and the last 6 digits are 125,000, 281,250, 437,500, 593,750, 750,000 or 906,250
- The 1,000,000s digit is 4 or 9 and the last 6 digits are 062,500, 218,750, 375,000, 531,250, 687,500 or 843,750

**312,500**: The last 7 digits are a multiple of 312,500

- The 1,000,000s digit is 0 or 5 and the last 6 digits are 000,000, 312,500, 625,000 or 937,500
- The 1,000,000s digit is 1 or 6 and the last 6 digits are 250,000, 562,500 or 875,000
- The 1,000,000s digit is 2 or 7 and the last 6 digits are 187,500, 500,000 or 812,500
- The 1,000,000s digit is 3 or 8 and the last 6 digits are 125,000, 437,500 or 750,000
- The 1,000,000s digit is 4 or 9 and the last 6 digits are 062,500, 375,000 or 687,500

**400,000**: The last 7 digits are a multiple of 400,000

- The 1,000,000s digit is even and the last 6 digits are 000,000, 400,000 or 800,000
- The 1,000,000s digit is odd and the last 6 digits are 200,000 or 600,000

**625,000**: The last 7 digits are a multiple of 625,000

- The 1,000,000s digit is 0 or 5 and the last 6 digits are 000,000 or 625,000
- The 1,000,000s digit is 1 or 6 and the last 6 digits are 250,000 or 875,000
- The 1,000,000s digit is 2 or 7 and the last 6 digits are 500,000
- The 1,000,000s digit is 3 or 8 and the last 6 digits are 125,000 or 750,000
- The 1,000,000s digit is 4 or 9 and the last 6 digits are 375,000

**1,250,000**: The number ends in 0,000,000, 1,250,000, 2,500,000, 3,750,000, 5,000,000, 6,250,000, 7,500,000 or 8,750,000

**2,000,000**: The number ends in 0,000,000, 2,000,000, 4,000,000, 6,000,000 or 8,000,000

**2,500,000**: The number ends in 0,000,000, 2,500,000, 5,000,000 or 7,500,000

**5,000,000**: The number ends in 0,000,000 or 5,000,000

**10,000,000**: The number ends in 0,000,000

**Last 8 digit rules**

**256**: The last 8 digits are a multiple of 256

- The 10,000,000s digit is even and the last 7 digits are 0,000,000 or a multiple of 256
- The 10,000,000s digit is odd and the last 7 digits are 128 times an odd number

**1,280**: The last 8 digits are a multiple of 1,280

- The 10,000,000s digit is even and the last 7 digits are 0,000,000 or a multiple of 1,280
- The 10,000,000s digit is odd and the last 7 digits are 640 times an odd number

**6,400**: The last 8 digits are a multiple of 6,400

- The 10,000,000s digit is even and the last 7 digits are 0,000,000 or a multiple of 6,400
- The 10,000,000s digit is odd and the last 7 digits are 3,200 times an odd number

**32,000**: The last 8 digits are a multiple of 32,000

- The 10,000,000s digit is even and the last 7 digits are 0,000,000 or a multiple of 32,000
- The 10,000,000s digit is odd and the last 7 digits are 16,000 times an odd number

**160,000**: The last 8 digits are a multiple of 160,000

- The 10,000,000s digit is even and the last 7 digits are 0,000,000 or a multiple of 160,000
- The 10,000,000s digit is odd and the last 7 digits are 80,000 times an odd number

**390,625**: The last 8 digits are a multiple of 390,625

- The 10,000,000s digit is 0 or 5 and the last 7 digits are 0,000,000 or a multiple of 390,625
- The 10,000,000s digit is 1 or 6 and the last 7 digits are 78,125 times a number that ends in 2 or 7
- The 10,000,000s digit is 2 or 7 and the last 7 digits are 78,125 times a number that ends in 4 or 9
- The 10,000,000s digit is 3 or 8 and the last 7 digits are 78,125 times a number that ends in 1 or 6
- The 10,000,000s digit is 4 or 9 and the last 7 digits are 78,125 times a number that ends in 3 or 8

**781,250**: The last 8 digits are a multiple of 781,250

- The 10,000,000s digit is 0 or 5 and the last 7 digits are 0,000,000 or a multiple of 781,250
- The 10,000,000s digit is 1 or 6 and the last 7 digits are 156,250 times a number that ends in 1 or 6
- The 10,000,000s digit is 2 or 7 and the last 7 digits are 156,250 times a number that ends in 2 or 7
- The 10,000,000s digit is 3 or 8 and the last 7 digits are 156,250 times a number that ends in 3 or 8
- The 10,000,000s digit is 4 or 9 and the last 7 digits are 156,250 times a number that ends in 4 or 9

**800,000**: The last 8 digits are a multiple of 800,000

- The 10,000,000s digit is even and the last 7 digits are 0,000,000 or a multiple of 800,000
- The 10,000,000s digit is odd and the last 7 digits are 400,000 times an odd number

**1,562,500**: The last 8 digits are a multiple of 1,562,500

- The 10,000,000s digit is 0 or 5 and the last 7 digits are 0,000,000, 1,562,500, 3,125,000, 4,687,500, 6,250,000, 7,812,500 or 9,375,000
- The 10,000,000s digit is 1 or 6 and the last 7 digits are 0,937,500, 2,500,000, 4,062,500, 5,625,000, 7,187,500 or 8,750,000
- The 10,000,000s digit is 2 or 7 and the last 7 digits are 0,312,500, 1,875,000, 3,437,500, 5,000,000, 6,562,500, 8,125,000 or 9,687,500
- The 10,000,000s digit is 3 or 8 and the last 7 digits are 1,250,000, 2,812,500, 4,375,000, 5,937,500, 7,500,000 or 9,062,500
- The 10,000,000s digit is 4 or 9 and the last 7 digits are 0,625,000, 2,187,500, 3,750,000, 5,312,500, 6,875,000 or 8,437,500

**3,125,000**: The last 8 digits are a multiple of 3,125,000

- The 10,000,000s digit is 0 or 5 and the last 7 digits are 0,000,000, 3,125,000, 6,250,000 or 9,375,000
- The 10,000,000s digit is 1 or 6 and the last 7 digits are 2,500,000, 5,625,000 or 8,750,000
- The 10,000,000s digit is 2 or 7 and the last 7 digits are 1,875,000, 5,000,000 or 8,125,000
- The 10,000,000s digit is 3 or 8 and the last 7 digits are 1,250,000, 4,375,000 or 7,500,000
- The 10,000,000s digit is 4 or 9 and the last 7 digits are 0,625,000, 3,750,000 or 6,875,000

**4,000,000**: The last 8 digits are a multiple of 4,000,000

- The 10,000,000s digit is even and the last 7 digits are 0,000,000, 4,000,000 or 8,000,000
- The 10,000,000s digit is odd and the last 7 digits are 2,000,000 or 6,000,000

**6,250,000:** The last 8 digits are a multiple of 3,125,000

- The 10,000,000s digit is 0 or 5 and the last 7 digits are 0,000,000 or 6,250,000
- The 10,000,000s digit is 1 or 6 and the last 7 digits are 2,500,000 or 8,750,000
- The 10,000,000s digit is 2 or 7 and the last 7 digits are 5,000,000
- The 10,000,000s digit is 3 or 8 and the last 7 digits are 1,250,000 or 7,500,000
- The 10,000,000s digit is 4 or 9 and the last 7 digits are 3,750,000

**12,500,000**: The number ends in 00,000,000, 12,500,000, 25,000,000, 37,500,000, 50,000,000, 62,500,000, 75,000,000 or 87,500,000

**20,000,000**: The number ends in 00,000,000, 20,000,000, 40,000,000, 60,000,000 or 80,000,000

**25,000,000**: The number ends in 00,000,000, 25,000,000, 50,000,000 or 75,000,000

**50,000,000**: The number ends in 00,000,000 or 50,000,000

**100,000,000**: The number ends in 00,000,000

4 Comments

2024/02/10

13:54 UTC

13:54 UTC

2

This varies from institution to institution as I am aware; but generally, what is the major difference between Introductory statistics and statistics for STEM majors? What topics are generally not included in the introductory course that will likely appear in the STEM statistics class?

1 Comment

2024/02/09

19:30 UTC

19:30 UTC

8

I was recently accepted to Budapest Semester in Mathematics for this upcoming fall semester and I’d love to hear some input about the program from people who’ve participated. How difficult is it? Are there any easier classes? Can you understand the professors? **Would you recommend it to someone who has recently realized they're not** **that****passionate about math**? What are the other students like? Is there any free time to travel? Anything else I should know?

Any input would be so greatly appreciated. Thank you so much!

0 Comments

2024/02/09

17:41 UTC

17:41 UTC

3

Why most of math that is applied uses the principles of bivalence (there are two truth values), non contradiction, identity etc.? I know that tertium non datur can be discarded maybe in part of quantum mechanics, but it’s kinda strange to me. Probably it’s a stupid question, because it’s just a contingency, and probably not that useful, more a “completeness” thing that we can ask ourselves, but would like answers from mathematicians, as I just started studying math and am probably too high on the iceberg of math and probably don’t realise why this question is or is not problematic and why…

31 Comments

2024/02/09

17:04 UTC

17:04 UTC

0

Number | Rule |
---|---|

101 | The difference between 10 times the last digit and the rest of the number is 0 or a multiple of 101 |

102 | The number is a multiple of 2 and 51 at the same time |

103 | The sum of 31 times the last digit and the rest of the number is a multiple of 103 |

104 | The number is a multiple of 8 and 13 at the same time |

105 | The number is a multiple of 5 and 21 at the same time |

106 | The number is a multiple of 2 and 53 at the same time |

107 | The difference between 32 times the last digit and the rest of the number is 0 or a multiple of 107 |

108 | The number is a multiple of 4 and 27 at the same time |

109 | The sum of 11 times the last digit and the rest of the number is a multiple of 109 |

110 | The number is a multiple of 2, 5 and 11 at the same time |

111 | The difference between 11 times the last digit and the rest of the number is 0 or a multiple of 111 |

112 | The number is a multiple of 7 and 16 at the same time |

113 | The sum of 34 times the last digit and the rest of the number is a multiple of 113 |

114 | The number is a multiple of 2 and 57 at the same time |

115 | The number is a multiple of 5 and 23 at the same time |

116 | The number is a multiple of 4 and 29 at the same time |

117 | The difference between 35 times the last digit and the rest of the number is 0 or a multiple of 117 |

118 | The number is a multiple of 2 and 59 at the same time |

119 | The sum of 12 times the last digit and the rest of the number is a multiple of 119 |

120 | The number is a multiple of 3, 5 and 8 at the same time |

121 | The difference between 12 times the last digit and the rest of the number is 0 or a multiple of 121 |

122 | The number is a multiple of 2 and 61 at the same time |

123 | The sum of 37 times the last digit and the rest of the number is a multiple of 123 |

124 | The number is a multiple of 4 and 31 at the same time |

125 | The number ends in 000, 125, 250, 375, 500, 625, 750 or 875 |

126 | The number is a multiple of 2 and 63 at the same time |

127 | The difference between 38 times the last digit and the rest of the number is 0 or a multiple of 127 |

128 | The last 7 digits are a multiple of 128; the 1,000,000s digit is even and the last 6 digits are 000,000 or a multiple of 128, or the 1,000,000s digit is odd and the last 6 digits are 64 times an odd number |

129 | The sum of 13 times the last digit and the rest of the number is a multiple of 129 |

130 | The number is a multiple of 2, 5 and 13 at the same time |

131 | The difference between 13 times the last digit and the rest of the number is 0 or a multiple of 132 |

132 | The number is a multiple of 4 and 33 at the same time |

133 | The sum of 40 times the last digit and the rest of the number is a multiple of 133 |

134 | The number is a multiple of 2 and 67 at the same time |

135 | The number is a multiple of 5 and 27 at the same time |

136 | The number is a multiple of 8 and 17 at the same time |

137 | The difference between 41 times the last digit and the rest of the number is 0 or a multiple of 137 |

138 | The number is a multiple of 2 and 69 at the same time |

139 | The sum of 14 times the last digit and the rest of the number is a multiple of 139 |

140 | The number is a multiple of 4, 5 and 7 at the same time |

141 | The difference between 14 times the last digit and the rest of the number is 0 or a multiple of 131 |

142 | The number is a multiple of 2 and 71 at the same time |

143 | The sum of 43 times the last digit and the rest of the number is a multiple of 143 |

144 | The number is a multiple of 9 and 16 at the same time |

145 | The number is a multiple of 5 and 29 at the same time |

146 | The number is a multiple of 2 and 73 at the same time |

147 | The difference between 44 times the last digit and the rest of the number is 0 or a multiple of 147 |

148 | The number is a multiple of 4 and 37 at the same time |

149 | The sum of 15 times the last digit and the rest of the number is a multiple of 149 |

150 | The number is a multiple of 2, 3 and 25 at the same time |

151 | The difference between 15 times the last digit and the rest of the number is 0 or a multiple of 151 |

152 | The number is a multiple of 8 and 19 at the same time |

153 | The sum of 46 times the last digit and the rest of the number is a multiple of 153 |

154 | The number is a multiple of 2 and 77 at the same time |

155 | The number is a multiple of 5 and 31 at the same time |

156 | The number is a multiple of 4 and 39 at the same time |

157 | The difference between 47 times the last digit and the rest of the number is 0 or a multiple of 157 |

158 | The number is a multiple of 2 and 79 at the same time |

159 | The sum of 16 times the last digit and the rest of the number is a multiple of 159 |

160 | The number is a multiple of 5 and 32 at the same time |

161 | The difference between 16 times the last digit and the rest of the number is 0 or a multiple of 161 |

162 | The number is a multiple of 2 and 81 at the same time |

163 | The sum of 49 times the last digit and the rest of the number is a multiple of 163 |

164 | The number is a multiple of 4 and 41 at the same time |

165 | The number is a multiple of 5 and 33 at the same time |

166 | The number is a multiple of 2 and 83 at the same time |

167 | The difference between 50 times the last digit and the rest of the number is 0 or a multiple of 167 |

168 | The number is a multiple of 8 and 21 at the same time |

169 | The sum of 17 times the last digit and the rest of the number is a multiple of 169 |

170 | The number is a multiple of 2, 5 and 17 at the same time |

171 | The difference between 17 times the last digit and the rest of the number is 0 or a multiple of 171 |

172 | The number is a multiple of 4 and 43 at the same time |

173 | The sum of 52 times the last digit and the rest of the number is a multiple of 173 |

174 | The number is a multiple of 2 and 87 at the same time |

175 | The number is a multiple of 7 and 25 at the same time |

176 | The number is a multiple of 11 and 16 at the same time |

177 | The difference between 53 times the last digit and the rest of the number is 0 or a multiple of 177 |

178 | The number is a multiple of 2 and 89 at the same time |

179 | The sum of 18 times the last digit and the rest of the number is a multiple of 179 |

180 | The number is a multiple of 4, 5 and 9 at the same time |

181 | The difference between 18 times the last digit and the rest of the number is 0 or a multiple of 181 |

182 | The number is a multiple of 2 and 91 at the same time |

183 | The sum of 55 times the last digit and the rest of the number is a multiple of 183 |

184 | The number is a multiple of 8 and 23 at the same time |

185 | The number is a multiple of 5 and 37 at the same time |

186 | The number is a multiple of 2 and 93 at the same time |

187 | The difference between 56 times the last digit and the rest of the number is 0 or a multiple of 187 |

188 | The number is a multiple of 4 and 47 at the same time |

189 | The sum of 19 times the last digit and the rest of the number is a multiple of 189 |

190 | The number is a multiple of 2, 5 and 19 at the same time |

191 | The difference between 19 times the last digit and the rest of the number is 0 or a multiple of 191 |

192 | The number is a multiple of 3 and 64 at the same time |

193 | The sum of 58 times the last digit and the rest of the number is a multiple of 193 |

194 | The number is a multiple of 2 and 97 at the same time |

195 | The number is a multiple of 5 and 39 at the same time |

196 | The number is a multiple of 4 and 49 at the same time |

197 | The difference between 59 times the last digit and the rest of the number is 0 or a multiple of 197 |

198 | The number is a multiple of 2 and 99 at the same time |

199 | The sum of 20 times the last digit and the rest of the number is a multiple of 199 |

200 | The number is a multiple of 8 and 25 at the same time |

1 Comment

2024/02/09

14:54 UTC

14:54 UTC

10

Hi I teach maths at highschool level and one of my students was talking to me about how he felt that eulers formula felt forced. His problem generally seems to be between the relationship between e^itheta and sin/ cos mostly because he feels that the concept of the real and imaginary plane seems like a forced concept and he takes issue with argand diagram itself and I can kind of see what he means. I was wondering if anyone had any explanations, resources or videos to share that would either help me to quell his doubts on the argand diagram and hence eulers formulas mcclaurin series etc. or any criticisms of the use of the argand diagram that suggests he is correct! I did some research already but a lot of stuff that's coming up is just people accepting it as truth and using it. There's not much debate of how we know it works etc.

22 Comments

2024/02/09

12:14 UTC

12:14 UTC

2

I'm studying mathematics for my engineering degree.

I'm doing a calculus course and I just am clueless seeing any question, I don't know limits function derivatives and integration,I am very weak at mathematics and can't seem to figure out how and where to start a question from.

Right now my prof is teaching partial derivatives and they gave a question where tan inverse was written and the only thing I could think of was how I don't know inverse trigonometric functions and how I have no idea to solve them.

I am facing a lot of resistance in my head, I just keep beating myself down for not knowing things.

I feel like I need to relearn how to learn maths and what to think when solving a question.

7 Comments

2024/02/09

11:30 UTC

11:30 UTC

0

2 Comments

2024/02/09

06:21 UTC

06:21 UTC

0

Google provided me with no answers just telling me there's 4 branches of mathematics

15 Comments

2024/02/09

03:12 UTC

03:12 UTC

7

Number | Rule |
---|---|

1 | Any number is a multiple of 1 |

2 | The number ends in 0, 2, 4, 6 or 8 |

3 | The sum of the digits is a multiple of 3 |

4 | The last 2 digits are a multiple of 4; the 10s digit is even and the last digit is 0, 4 or 8, or the 10s digit is odd and the last digit is 2 or 6 |

5 | The number ends in 0 or 5 |

6 | The number is a multiple of 2 and 3 at the same time |

7 | The difference between twice the last digit and the rest of the number is 0 or a multiple of 7 |

8 | The last 3 digits are a multiple of 8; the 100s digit is even and the last 2 digits are 00 or a multiple of 8, or the 100s digit is odd and the last 2 digits are 4 times an odd number |

9 | The sum of the digits is a multiple of 9 |

10 | The number is a multiple of 2 and 5 at the same time |

11 | The difference between the last digit and the rest of the number is 0 or a multiple of 11 |

12 | The number is a multiple of 3 and 4 at the same time |

13 | The sum of 4 times the last digit and the rest of the number is a multiple of 13 |

14 | The number is a multiple of 2 and 7 at the same time |

15 | The number is a multiple of 3 and 5 at the same time |

16 | The last 4 digits are a multiple of 16; the 1,000s digit is even and the last 3 digits are 000 or a multiple of 16, or the 1,000s digit is odd and the last 3 digits are 8 times an odd number |

17 | The difference between 5 times the last digit and the rest of the number is 0 or a multiple of 17 |

18 | The number is a multiple of 2 and 9 at the same time |

19 | The sum of twice the last digit and the rest of the number is a multiple of 19 |

20 | The number is a multiple of 4 and 5 at the same time |

21 | The difference between twice the last digit and the rest of the number is 0 or a multiple of 21 |

22 | The number is a multiple of 2 and 11 at the same time |

23 | The sum of 7 times the last digit and the rest of the number is a multiple of 23 |

24 | The number is a multiple of 3 and 8 at the same time |

25 | The number ends in 00, 25, 50 or 75 |

26 | The number is a multiple of 2 and 13 at the same time |

27 | The difference between 8 times the last digit and the rest of the number is 0 or a multiple of 27 |

28 | The number is a multiple of 4 and 7 at the same time |

29 | The sum of thrice the last digit and the rest of the number is a multiple of 29 |

30 | The number is a multiple of 2, 3 and 5 at the same time |

31 | The difference between thrice the last digit and the rest of the number is 0 or a multiple of 31 |

32 | The last 5 digits are a multiple of 32; the 10,000s digit is even and the last 4 digits are 0,000 or a multiple of 32, or the 10,000s digit is odd and the last 4 digits are 16 times an odd number |

33 | The sum of 10 times the last digit and the rest of the number is a multiple of 33 |

34 | The number is a multiple of 2 and 17 at the same time |

35 | The number is a multiple of 5 and 7 at the same time |

36 | The number is a multiple of 4 and 9 at the same time |

37 | The difference between 11 times the last digit and the rest of the number is 0 or a multiple of 37 |

38 | The number is a multiple of 2 and 19 at the same time |

39 | The sum of 4 times the last digit and the rest of the number is a multiple of 39 |

40 | The number is a multiple of 5 and 8 at the same time |

41 | The difference between 4 times the last digit and the rest of the number is 0 or a multiple of 41 |

42 | The number is a multiple of 2 and 21 at the same time |

43 | The sum of 13 times the last digit and the rest of the number is a multiple of 43 |

44 | The number is a multiple of 4 and 11 at the same time |

45 | The number is a multiple of 5 and 9 at the same time |

46 | The number is a multiple of 2 and 23 at the same time |

47 | The difference between 14 times the last digit and the rest of the number is 0 or a multiple of 47 |

48 | The number is a multiple of 3 and 16 at the same time |

49 | The sum of 5 times the last digit and the rest of the number is a multiple of 49 |

50 | The number is a multiple of 2 and 25 at the same time |

51 | The number difference between 5 times the last digit and the rest of the number is 0 or a multiple of 51 |

52 | The number is a multiple of 4 and 13 at the same time |

53 | The sum of 16 times the last digit and the rest of the number is a multiple of 53 |

54 | The number is a multiple of 2 and 27 at the same time |

55 | The number is a multiple of 5 and 11 at the same time |

56 | The number is a multiple of 7 and 8 at the same time |

57 | The difference between 17 times the last digit and the rest of the number is 0 or a multiple of 57 |

58 | The number is a multiple of 2 and 29 at the same time |

59 | The sum of 6 times the last digit and the rest of the number is a multiple of 59 |

60 | The number is a multiple of 3, 4 and 5 at the same time |

61 | The difference between 6 times the last digit and the rest of the number is 0 or a multiple of 61 |

62 | The number is a multiple of 2 and 31 at the same time |

63 | The sum of 19 times the last digit and the rest of the number is a multiple of 63 |

64 | The last 6 digits are a multiple of 64; the 100,000s digit is even and the last 5 digits are 00,000 or a multiple of 64, or the 100,000s digit is odd and the last 5 digits are 32 times an odd number |

65 | The number is a multiple of 5 and 13 at the same time |

66 | The number is a multiple of 2 and 33 at the same time |

67 | The difference between 20 times the last digit and the rest of the number is 0 or a multiple of 67 |

68 | The number is a multiple of 4 and 17 at the same time |

69 | The sum of 7 times the last digit and the rest of the number is a multiple of 69 |

70 | The number is a multiple of 2, 5 and 7 at the same time |

71 | The difference between 7 times the last digit and the rest of the number is 0 or a multiple of 71 |

72 | The number is a multiple of 8 and 9 at the same time |

73 | The sum of 22 times the last digit and the rest of the number is a multiple of 73 |

74 | The number is a multiple of 2 and 37 at the same time |

75 | The number is a multiple of 3 and 25 at the same time |

76 | The number is a multiple of 4 and 19 at the same time |

77 | The difference between 23 times the last digit and the rest of the number is 0 or a multiple of 77 |

78 | The number is a multiple of 2 and 39 at the same time |

79 | The sum of 8 times the last digit and the rest of the number is a multiple of 79 |

80 | The number is a multiple of 5 and 16 at the same time |

81 | The difference between 8 times the last digit and the rest of the number is 0 or a multiple of 81 |

82 | The number is a multiple of 2 and 41 at the same time |

83 | The sum of 25 times the last digit and the rest of the number is a multiple of 83 |

84 | The number is a multiple of 4 and 21 at the same time |

85 | The number is a multiple of 5 and 17 at the same time |

86 | The number is a multiple of 2 and 43 at the same time |

87 | The difference between 26 times the last digit and the rest of the number is 0 or a multiple of 87 |

88 | The number is a multiple of 8 and 11 at the same time |

89 | The sum of 9 times the last digit and the rest of the number is a multiple of 89 |

90 | The number is a multiple of 2, 5 and 9 at the same time |

91 | The difference between 9 times the last digit and the rest of the number is 0 or a multiple of 91 |

92 | The number is a multiple of 4 and 23 at the same time |

93 | The sum of 28 times the last digit and the rest of the number is a multiple of 93 |

94 | The number is a multiple of 2 and 47 at the same time |

95 | The number is a multiple of 5 and 19 at the same time |

96 | The number is a multiple of 3 and 32 at the same time |

97 | The difference between 29 times the last digit and the rest of the number is 0 or a multiple of 97 |

98 | The number is a multiple of 2 and 49 at the same time |

99 | The sum of 10 times the last digit and the rest of the number is a multiple of 99 |

100 | The number is a multiple of 4 and 25 at the same time |

1 Comment

2024/02/08

22:10 UTC

22:10 UTC

1

For complete self study, I have seen a few, which is the best among them?

Joseph_Gallian Contemporary_Abstract_Algebra

David S. Dummit, Richard M. Foote Abstract Algebra

John B. Fraleigh, Neal E. Brand - A First Course in Abstract Algebra

Michael Artin - Algebra

W. Keith Nicholson - Introduction to Abstract Algebra

Or, If you want to suggest anything else ......

1 Comment

2024/02/07

19:28 UTC

19:28 UTC

6

Hello everyone, I’ll br applying for a Phd in math next year and I’m interested in Operator Theory. Do you know any good place to do it?

5 Comments

2024/02/08

15:45 UTC

15:45 UTC