Questions on basic arithmetic, e.g. addition, subtraction, multiplication, division, powers, radicals, etc.

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4
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2answers
119 views

If a sum is one, the sum of all products are also one.

Let $p_1,\ldots,p_s$ be $s$ number in the unit interval such that $$p_1+\ldots+p_s=1.$$ Is it then true, that for every $n\geq 1$ we have $$ \sum_{(k_1,\ldots,k_n)\in \{1,\ldots,s \}^n} p_{k_1}\cdot ...
1
vote
2answers
66 views

why $\sum_{k=0}^{\infty}(10^{-2})^k = \frac{1}{1-10^{-2}}$

i was reading a book and suddenly saw this step: $\sum_{k=0}^{\infty}(10^{-2})^k = \frac{1}{1-10^{-2}}$ i am actually not bad at calculation and also i am okay in precalculus, but i am really stuck ...
3
votes
0answers
260 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 $$\lim_{n\to\infty}\frac{\sum_{k=1}^n f(k)g(k)}{\sum_{k=1}^nf(k)}=A$$
1
vote
1answer
39 views

using >>, -, + to make a number X times the constant K for the following numbers

if were only allowed to use >>, -, + to make a number X times the constant K lets assume K is 17 and K is 20 how to make an expression for each I was trying to think of ways to do this but I cant.
1
vote
4answers
333 views

To prove an Arithmetic Progression

If the $p^{th}$ term of an arithmetic progression is $\alpha$ and the $q^{th}$ term is $\beta$, prove that the sum of its $p+q$ term is ...
0
votes
1answer
50 views

Confusion with respect to definition of addition using sections (Hardy)

I am having difficulty interpreting the definition of addition provided in Hardy' Course of Pure Mathematics. (i) Addition. In order to define the sum of two numbers α and β, we consider the ...
3
votes
1answer
54 views

Lifting étale sections

Let $X$ be a scheme, $F$ and $G$ be sheaves in groups on $X$ for the étale topology and $f:F\rightarrow G$ a morphism of group sheaves. Assume that there exists an étale covering $i:U\rightarrow X$ ...
0
votes
1answer
58 views

hyperpower modular

How can I calculate this? ${(p-1)}^{{(p-2)^{{(p-3)}^{(p-4)...}}}} (mod {.p}) $ and so on till 1. I don't know how to write it with a Knuth or Ackerman or more compact notation. I've tried to find a ...
3
votes
1answer
65 views

Does this pattern of arithmetic exist?

I have not studied mathematics very deeply, and I'm not familiar with the terminology. However, I have thought about a kind of pattern and wonder if anything like that exists or is being used today. ...
1
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1answer
96 views

When to use significant figures

Often I am not sure when to make use of significant figures in a calculation. For example at the moment. I have to calculate a certain equilibrium point which is given by: $$m^* = ...
3
votes
2answers
237 views

How to solve $(a+\sqrt{b})^n - (a-\sqrt{b})^n = x$?

Consider equation $(a+\sqrt{b})^n - (a-\sqrt{b})^n = x$ How do I properly solve for $n$ given $x$?
2
votes
2answers
129 views

Prove that gcd(n, mp) = gcd (n, m) if n and p are relatively prime

Let n, m and p non-zero natural integers, with n and p relatively prime. Prove that gcd(n, mp) = gcd (n, m). This problem had three questions. First, to prove that if d divides n then d and p are ...
1
vote
1answer
16 views

problem with pupils and find the number of the pupils -arithmetic

In a classroom there are 28 pupils. 19 pupils know to speak English ad 16 pupils speak French. Knowing that every children is speaking at least one language(English or French) find out the number of ...
1
vote
7answers
2k views

Simplifying a dividing surd?

Can anyone explain how I would simplify this dividing surd: $$\frac{3\sqrt{14}}{\sqrt{42}}$$ As far as I can see $\sqrt{14}$ and $\sqrt{42}$ can't be simplified, right? So how does the division and ...
12
votes
4answers
441 views

How many powers of 2 are easy to double? [duplicate]

Possible Duplicate: Is 2048 the highest power of 2 with all even digits (base ten)? Numbers written in base $10$ are easiest to double when their digits lie in the range $0, \ldots, 4$, so ...
3
votes
3answers
2k views

Is there one method of adding and subtracting without a calculator?

One can on a sheet of paper, without a calculator, add two numbers or subtract two numbers, each with it's own method. This is second grade maths. However, is it possible to solve both these with a ...
1
vote
1answer
127 views
3
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5answers
103 views

Sum of some place digits in a product

If $x$ is the tens place digit and $y$ is the ones place digit of the product $725278\times 67066$, what is $x+y$? I have no idea how to even approach this.
0
votes
1answer
80 views

Please help with this word problem involving trigonometry [closed]

A small aircraft has a glide ration of 15:1. (The glide ratio means the plane moves 15 units horizontally for every one unit elevation). You are exactly in the middle of a 3.0 mile diameter lake at ...
6
votes
3answers
183 views

Simple problems that can be carried around in your head

What are some problems that can easily be carried around in your head and require no need of ink and paper? Problems like irrationality of $\sqrt2$ or infinitude of primes or Gauss's first initiation ...
1
vote
1answer
330 views

A question about orders of magnitude

I'm reading a book in which the author compares two pairs of numbers $(0.31, 0.39)$ and $(6.10,0.39)$ and multiplies the second member of each pair by a factor $R_1 = 2$ and $R_2 = 20$ so that both ...
1
vote
2answers
56 views

simple question related to the arithmetic of binomial coefficient

Bonjour, The problem i have follows from the definition of the binomial coefficient: $\frac{n(n-1)...(n-k+1)}{k!} = {n \choose k}$ For 0$\leq{i}$ and i less than k, we observe that: ...
2
votes
5answers
284 views

Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k$?

Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k\;$? I'm aware that $3 = 3^1$ but I would expect $3\cdot 3^k\;$ to be $\;9^k$ or $\;9^{k+1}$.
2
votes
3answers
647 views

Use three 11's and various math symbols to make an equation equal to 6

The puzzle is to use the following symbols $$+,\;-,\;*,\;/,\;(\;,\;),\;!, \;\sqrt(\cdot)$$ in order to make a valid equation out of $$11~~~~~~11~~~~~~~11 = 6.$$ (There are three elevens with space in ...
1
vote
1answer
26 views

Calculate the Mean earnings of a person

There are $12$ employees who earn $\$750$ collectively. I need to calculate the mean salary earned by a single employee. My workings so far: I tried to divide $750 / 12$, but I am not sure if this ...
0
votes
1answer
17 views

A step with respect to showing equivalence of displacements

I am currently studying Hardy's A Course of Pure Mathematics and am on Chapter 3, Section 35, Equivalence of displacements. Multiplication of displacements by numbers. In this chapter, displacement's ...
1
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0answers
728 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 = ...
0
votes
1answer
49 views

A walk with mom.

A mother takes two strides to her daughter's three. If they set out walking together, each starting with the right foot, when will they first step together with the left? Is there a general solution ...
1
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0answers
88 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) // ...
3
votes
1answer
138 views

Name of a simple equivalence relation on real numbers?

Define the relation $x \sim y$ where $x$ and $y$ are real numbers to hold if and only if there exist natural numbers $n$ and $m$ such that $x^n = y^m$. It is easy to see that $\sim$ is an equivalence ...
1
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1answer
260 views

$\frac 00=$ ? Why It is un answerable? [duplicate]

Possible Duplicate: Division by $0$ What will be the answer if zero$(0)$ divided by zero$(0)$ Why It is un answerable? $$\frac 00=\space ?$$ Thanks in Advance, Vicky
16
votes
1answer
952 views

Given a set of digits, what is the biggest number we can make using exponentiation - numberphile noodle quiz

The question is motivated by a question on a can of number noodles. Each item is a digit between $0$ and $9$. Clearly, if you form a string and consider it to represent a base $10$ integer, then ...
15
votes
13answers
2k views

How to explain that division by $0$ yields infinity to a 2nd grader

How do we explain that dividing a positive number by $0$ yields positive infinity to a 2nd grader? The way I intuitively understand this is $\lim_{x \to 0}{a/x}$ but that's asking too much of a child. ...
3
votes
2answers
1k views

How do I get the integer part of a number by using basic arithmetic?

While it is trivial to simply remove the fractional part of an irrational or rational number, and in programming I could just use the floor() or ...
2
votes
1answer
149 views

double digit sums 1-99 * 1-99

How many unique answers are there to all the natural whole numbers 1 - 99 multiplied by all the natural whole numbers 1-99? For instance all the single digits 1-9 multiplied by all the single digits ...
-1
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3answers
414 views

Arithmetic sequence explicit formula.

Which arithmetic sequence explicit formula would yield the following: $1$, $-1$, $1$, $-1$.
0
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1answer
1k views

how many 2x2 matrices are invertible in mod p

I am trying to solve this problem for homework but unable to get anything. The question is to find the number of invertible 2x2 matrices in mod p? Each entery can bee from the set ...
0
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1answer
87 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 ?
1
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1answer
522 views

Prove that given any rational number there exists another greater than or equal to it that differs by less than $\frac 1n$

I am currently attempting to prove a claim in Hardy's Course of Pure Mathematics and am currently stuck. I was hoping that someone would be able to provide some assistance on how to go about this. ...
0
votes
5answers
267 views

How to explain that (a^b)^c is not equal to a^(b^c) [duplicate]

Possible Duplicate: Can anyone explain why $a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(b*c)}$ Its been a while since I worked much with exponents, and I got confused and thought that $(a^b)^c = ...
0
votes
3answers
126 views

Limit of a n-root

I'm trying to find out why: $$\lim_{n \rightarrow \infty}\sqrt[n]{\frac{4^nx^{2n}}{n^2}} = 4x^2$$ Seems to me that it go $\rightarrow\infty$ because of the ...
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0answers
42 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.
0
votes
1answer
113 views

Find the set of all natural number that make $(n+1)/(n+3)$ reducible

Lets assume $d$ is a natural number which makes $(n+1)/(n+3)$ reducible, then $d|n+1$ and $d|n+3$. $d|[n+3-(n+1)] = d|2$ which means $d=1$ or $d=2$. $n+1$ and $n+3$ must be divisible by $2$ so all ...
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0answers
81 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 ...
4
votes
1answer
171 views

Is every φ above the second level of the arithmetical hierarchy independent of PA?

If I am not wrong, every $\Sigma_n$ (or $\Pi_n$ ) statement $\phi$ is equivalent to a statement that says that a given Turing machine halts (or doesn't halt) on input $C$ using a ...
9
votes
7answers
2k views

Why is $\sqrt{8}/2$ equal to $\sqrt{2}$?

I am trying to help my daughter on her math homework and I am having some trouble on some equation solving steps. My current major concern relies on understanding why $\sqrt{8}/2$ equal to $\sqrt{2}$. ...
1
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1answer
70 views

Calculating how long it will take for the half-life of X amount to fall below Y amount

I am trying to determine how long it will take ($t$) for the half life of 500 amount of substance to fall below 100 (to be $\le$ 99 -- I am only concerned with integers) when it has a half life of 5. ...
5
votes
3answers
543 views

How to calculate $3^{45357} \mod 5$?

I wrote some code, here is what it gives: \begin{align*} 3^0 \mod 5 = 1 \\ 3^1 \mod 5 = 3 \\ 3^2 \mod 5 = 4 \\ 3^3 \mod 5 = 2 \\\\ 3^4 \mod 5 = 1 \\ 3^5 \mod 5 = 3 \\ 3^6 \mod 5 = 4 \\ 3^7 \mod 5 = 2 ...
23
votes
7answers
1k views

Sum of the sum of the sum of the first $n$ natural numbers

I have here another problem of mine, which I couldn't manage to solve. Given that: $$x_n = 1 + 2 + \dots + n \\ y_n = x_1 + x_2 + \dots + x_n \\ z_n = y_1 + y_2 + \dots + y_n $$ Find ...
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vote
1answer
65 views

Show that a certain point lies outside a ball, might be simple but i am stuck…

Consider the ball $$ B(0, R) := \{ x | ||x|| \le R \} $$ and consider a point $x$ outside of the ball, that is $||x|| > R$. Now i construct another ball of radius $\frac{1}{2}(||x|| - R)$ around ...