Questions on basic arithmetic, e.g. addition, subtraction, multiplication, division, powers, roots, etc.
23
votes
0answers
817 views
$4494410$ and friends
$4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number. ...
20
votes
0answers
196 views
+100
Manual proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ is a noninteger
Conor McBride asks for a fast proof that $$x = {\left(\pi^\pi\right)}^{\pi^\pi}$$ is not an integer. It would be sufficient to calculate a very rough approximation, to a precision of less than 1, and ...
7
votes
0answers
111 views
Numbers of the form $(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$
I'm looking for numbers of the form
$$(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$$
where $p_{i}$ are prime numbers, ...
3
votes
0answers
79 views
Square and reverse reading of an integer
For all $n=\overline{a_k a_{k-1}\ldots a_1 a_0} := \sum_{i=0}^k a_i 10^i\in \mathbb{N}$, where $a_i \in \{0,...,9\}$ and $a_k \neq 0$,
we define $f(n)=\overline{a_0 a_1 \ldots a_{k-1} a_k}= ...
3
votes
0answers
44 views
How to find an expression whose value is 190
Given a set of numbers (in this case):
3, 7, 7, 100, 50
Either:
prove it is impossible to form the number k = 190 using ( ) + - * / operators between sub set of the these numbers ex: 1000 = ((3 + ...
3
votes
0answers
43 views
Efficiency in factoring lists of consecutive numbers
Suppose I'm looking at prime factorizations of numbers in the vicinity of this one:
$$
1354 = 2 \times 677
$$
The smallest prime appears here, and the next prime after that does not.
Going one step ...
3
votes
0answers
121 views
Arithmetic mean sum
Let $$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^ng(k)=A $$
Then for what functions $f(x)$ does $$\lim_{n\to\infty}\frac{\sum_{k=1}^n f(k)g(k)}{\sum_{k=1}^nf(k)}=A$$
3
votes
0answers
123 views
Is there any recursive definition, using only addition, of the set of values of $x^2+y^2$?
There is a recursive definition of the set of squares which uses only addition:
$CS(x,y) := IS(x) \wedge IS(y) \wedge x \lt y \wedge \forall z: (x
\lt z) \wedge (z \lt y)⇒\neg IS(z)$
$IS(x)⇔ x=0 ...
3
votes
0answers
94 views
Least characters in a numerical representation of integers
I was wondering what the shortest way to represent any given number is. For example, $387420489=9^9$. So, for this case, the smallest representation is of order 2 (2 numbers). Alternatively, ...
3
votes
0answers
508 views
Is there an equivalent to the distributive law for division over subtraction and/or addition?
I understand that the the distributive law cannot be applied to division over addition/subtraction, but is there an equivalent law to expand it out. For example, I know:
$$100 \times (5 + 3) = (100 ...
2
votes
0answers
40 views
Clarification of variable values in Arithmetic Coding algorithm
I have been trying to follow this video to implement my own Arithmetic Coding algorithm in Java. I am having a bit of trouble figuring out what some of the variables in the video should be.
For ...
2
votes
0answers
28 views
Need to determine the formula to work out a reduction percentage
We currently calculate a value for a business function as follows;
Inputs:
A list of values such as;
12
12
6
12
We then add 1 to each of these values and we have values as follows;
13
13
7
13
...
2
votes
0answers
39 views
How to prove that $G_3>0$ in this case?
Let $\Lambda=\{a+be^{2\pi i/3}|a,b\in Z\}$, then $G_{3}(\Lambda)=\sum_{\omega\in\Lambda-\{0\}}\frac{1}{\omega^{6}}$ should be real and nonzero, but how can one prove that it's positive?
Moreover, in ...
2
votes
0answers
15 views
Decomposability in a context of size constraints on intervals
Let $F$ be a finite set of pairs of positive integers. Say that a set $A \subseteq {\mathbb Z}$ is $F$-admissible iff its intersection with any integer interval of length $a$ has cardinality at most ...
2
votes
0answers
91 views
Is there a term used to describe both an equation and inequality?
Is there a term used to describe both an equation and inequality?
The closest thing I can think of is "relation".
2
votes
0answers
113 views
1/3+2/3 in double precision
When I add 1/3 and 2/3 in double precision, I ended up with $1.\boxed{111\ldots1}1\times2^{-1}$, where the boxed part is the 52-bit mantissa. By the rounding to even rule, I should round it up, right? ...
1
vote
0answers
35 views
Gold Ring Calculation
I have problem with gold ring weight calculation
I have below details to calculate gold ring calculation
...
1
vote
0answers
25 views
integer S transform
I found the following integer transform, where all variables $x_0,x
_1$ and $x_2$ are integers:
$
y_0=x_1+floor(\frac{1}{4}(x_0+2x_1+x_2))\\
y_1=x_2-x_1\\
y_2=x_0-x_1
$
The inverse transform is ...
1
vote
0answers
68 views
Question on the proof of $\mathrm{Var}(X) = \lambda$ for the Poisson distribution - dropping undefined variables
Here's a proof of $\mathrm{Var}(X) = \lambda$ for the Poisson distribution.
Proof:
First we work out $E(X(X-1))$
$$E(X(X-1)) = \sum_Xx(x-1)f(x)$$
$$= \sum_{x = ...
1
vote
0answers
61 views
Need help simplifying an equation.
I'm trying to speed up the following code:
sum = 0
for (k = 1 ... N) {
f = Fibonacci(k);
for (a = 1 ... 24)
for (b = 1 ... 24)
for (c = 1 ... 24) {
sum = sum + m(a, b, c) // ...
1
vote
0answers
36 views
Calculation Error or?
Is it a calculation error or am I missing something? A popular company like Plimus that handles thousands of payments daily can't make this mistake, I thought.
1
vote
0answers
72 views
quicker way of doing this in your head?
I have a question. For $x,y,n \in\Bbb N$, $y$ a power of two, being given $x$ and $y$ is there a faster way to mentally calculate $ny$ where $ny ≤ x< (n+1)y-1$ other than $\lfloor x \div y \rfloor ...
1
vote
0answers
42 views
Calculate minimum value based on constraints
This seems like a bit of a simple question in amongst all the 'proper' maths, but I'm a bit rusty...
If I have the following constraints:
$$ A = c(T - x) $$
$$ x - A > 0 $$
$$ 0 < c < 1 ...
1
vote
0answers
66 views
History of operator precendence
I have seen a lot of debates over operator precedence but what is the history of operator precedence and how it evolved over time? Why multiplication precedes addition; Is it just to be definitive?
...
1
vote
0answers
78 views
Does the shifting square root method work for non-integer bases?
Under "methods of computing square roots", Wikipedia states that the digit-by-digit calculation method, of which the shifting $n^{th}$ root algorithm is a generalization, works for all bases, but the ...
1
vote
0answers
50 views
base change and constructible sheaves
let $f:X\to S$ be a proper morphism of schemes with $S=Spec(A)$ affine. Consider $F$ a constructible sheaf on $X$. I am interested to know for which ring $B$ with morphism $Spec(B)\to Spec(A)$ is it ...
1
vote
0answers
139 views
square root of s domain transfer function
I have a question about calculating the square root of the magnitude of a transfer function. When you take the square root, what is happening? My initial idea of a magnitude is a single value, but in ...
1
vote
0answers
49 views
Valuations, simplicial complexes and arithmetics
I am currently working on an Introduction to geometric probability (Klein, Rota, 1997). This book is very stimulating, and I find myself toying with the subject, and a note in particular (pp 95-97 for ...
1
vote
0answers
249 views
How can I use an abacus to teach concepts to a toddler?
My 18-month old son got a $10\times10$ abacus as a Christmas present, and he enjoys it as a toy. I'm fine with him just playing with it, but I don't want to miss an opportunity to introduce ...
1
vote
0answers
211 views
How do I calculate cost savings and profits in this example?
How are profits and cost savings calculated in this table?
Is there enough given data to calculate these two?
Here is what I ...
1
vote
0answers
654 views
Teach me a simple, efficient division algorithm
I want to implement arbitrary-precision arithmetic in JavaScript for non-negative integer numbers. Long division isn't efficient if instead of the usual 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) there ...
1
vote
0answers
111 views
most comprehensive coverage of arithemetic and elementary algebra?
To put it shortly, I never learned mathematics. I never learned the algorithms for multiplication and division, &c. (I had always just mentally scaled by powers of 10, making it unnecessary for me ...
0
votes
0answers
37 views
Calculate the distance between the points (1, 2, …, n) and (2, 3, … n, 1)
I know that the operation to find the distance between two vectors is:
$$\sqrt{(b_1-a_1)^2+(b_2-a_2)^2+...+(b_n-a_n)^2}$$
So:
The distance between $(7, 5, 3, 1)$ and $(1, 3, 5, 7)$ is:
...
0
votes
0answers
52 views
Arithmetic Coding Example
I have a question for static arithmetic coding which I have done in class but I can't seem to figure out where I got the answers. It's a bit hard to get in touch with my lecturer at the moment so I'm ...
0
votes
0answers
48 views
Can all programs reducible to ones with only arithmetic operations on inputs be simulated with polynomial overhead by arithmetic machine?
In Can all programs be modeled as operations of elementary arithmetic operations on inputs? and computabiltiy theory, I
asked:
we treat all inputs and intermediate results and
final outputs as ...
0
votes
0answers
35 views
How to represent probability as Integer - Arithmetic coding
I am doing an assignment for a class in college where we have to write an arithmetic encoder / decoder in Java.
In this video, it shows how to set up / define all the variables required for the ...
0
votes
0answers
19 views
Iterated Vandermonde identity
Let $r,s \in \mathbb{C}$ or more generally elements of a commutative $\mathbb{Q}$-algebra $R$ and $n \in \mathbb{N}$. Then we have the following "iterated Vandermonde identity":
$$\binom{r\cdot s}{n} ...
0
votes
0answers
30 views
summation formula involving mertens function
from the residue theorem
$$ M(x) = \sum_\rho \frac{x^\rho}{\rho \zeta'(\rho)} - 2+\sum_{n=1}^\infty \frac{ (-1)^{n-1} (2\pi )^{2n}}{(2n)! n \zeta(2n+1)x^{2n}} $$
asssuming there are no multiple ...
0
votes
0answers
41 views
How can we prove that the set of integer has an associative law about addition?
I know that the definition of the addition of two integer.
But I don't know how the associative law about addition is proved. Please teach me.
0
votes
0answers
69 views
When $(\sum_{i=1}^nk_i < \prod_{i=n}^ni^{k_i}k_i!)$?
Consider $\Omega \subset \mathbb{N}$ a finite subset of $\mathbb{N}$, $\phi: \Omega \rightarrow \mathbb{N}$ an enumeration of $\Omega$ such that $\phi(\omega)=i$ and $|\Omega|=n$,
$$
...
0
votes
0answers
16 views
Get a representative vector from a large set, and compare it with samples.
We're a small team of programmers and we're triying to solve a little problem, but we think we need some advices from professional mathematicians.
We want to know if a picture of a card is an ...
0
votes
0answers
34 views
Get formula for percentage
I have these values:
x = 0.81
and this:
y = 0.83
The difference is 0.02, which is 1.79%.
I need to get 1.79 from 0.81 and ...
0
votes
0answers
68 views
How to solve these equations
I'm considering a coursera astronomy course and two of the prerequisites are listed below :
Could provide me with an explanation of how to solve points 2 & 3 above ?
0
votes
0answers
19 views
Addition of mixtures
You are required to make 200g of a 5% w/w mixture and hold a 2% w/w mixture in stock. How extra active much do you need to add to 200g of 2% w/w mixture to produce a final mixture with a 5% w/w ...
0
votes
0answers
96 views
Square root using simple arithmetic shift, inversion etc
Suppose we have a function which is sampled by a sampling time 10ms. This function comes in to the computer, then this computer should calculate square root (for every sampling time) from that ...
0
votes
0answers
22 views
Missing steps in proof of MVT for complex field integrals
I am reading through Bak and Newman's Complex Analysis, where typically the authors are pretty telegraphic in their proofs. However, I am a little confused by their proof of MVT for complex numbers, ...
0
votes
0answers
131 views
Puzzle: Representing age using digits from birth-year in order. Impossible cases?
I recently wrote my friend a birthday card and thought it would be fun to write her age using mathematical operations on the digits of her birth-year in order. For example she turned 36 and was born ...
0
votes
0answers
30 views
how does the following expression behave?
Let $b$ be a positive number, and $n$ a large natural number.
We know that if $b < 1/2$ then
$$(1 + b/n)^n - 1 \le 2b$$
because
$$(1 + b/n)^n \le 1 + 2b$$.
Is there a similar statement, or what ...
0
votes
0answers
65 views
Basic arithmetic question
I have to find 80%, 60%, 40%, 20% and it's values of a range of scores.
For one range, my values go from 2 to 8 (mean will be 5 which means, 50% of that range's value = 5)
For second range, my ...
0
votes
0answers
96 views
When can $a^m-b^m$ divide $a^m+b^m$. where $a$, $b$, $m$ are natural numbers, $a\gt b$
When can $(a^m-b^m)$ divide $(a^m+b^m)$, where $a$, $b$, $m$ are natural numbers, $a
\gt b$.
I approached this way:
Let $(a,b)=d$ and $\frac{a}{A}=\frac{b}{B}=d$, so $(A,B)=1$ and $A>B$.
...
