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

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MLE of Integer Valued Normal Distribution

If Z is a normal random variable on $\mathbb{R}^d$ with parameters $(\mu,\Sigma)$ and we know that $\mu\in \mathbb{Z}^d$ and $\Sigma \in \mathbb{Z}^{d+}$; then how can we solve this MLE problem for ...
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5answers
96 views

How does 9 mod -7 = -5?

Forgive me if this question does not belong on this site for it is simplistic and this is my first post, however I do not seem to understand the modulo function when it comes to negative numbers. I'd ...
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2answers
54 views

An infinite square arithmetic progression? [duplicate]

How to prove that there does not exist and infinite arithmetic sequence that all of it's terms are distinct squares of integers?
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1answer
73 views

Time And Distance (Train Journey)

The average speed of a train in the onward journey is 25% more than that in the return journey. The train halts for one hour on reaching the destination. The total time taken for the complete to and ...
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5answers
129 views

Solving $ \sqrt{x - 4} + \sqrt{x - 7} = 1 $.

I have the equation $ \sqrt{x - 4} + \sqrt{x - 7} = 1 $. I tried to square both sides, but then I got a more difficult equation: $$ 2 x - 11 + 2 \sqrt{x^{2} - 11 x - 28} = 1. $$ Can someone tell me ...
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1answer
57 views

Time, Speed and Distance

A walks around a circular field at the rate of one round per hour while B runs around it at the rate of six rounds per hour. They start in the same direction from the same point at 7.30 a.m. They ...
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0answers
21 views

Percentage Increase and Dates

Suppose we have the following data: ...
43
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11answers
7k views

What exactly IS a square root?

It's come to my attention that I don't actually understand what a square root really is (the operation). The only way I know of to take square roots (or nth root, for that matter) it to know the ...
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3answers
135 views

Calculate fractional part of square root without taking square root

Let's say I have a number $x>0$ and I need to calculate the fractional part of its square root: $$f(x) = \sqrt x-\lfloor\sqrt x\rfloor$$ If I have $\lfloor\sqrt x\rfloor$ available, is there a ...
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0answers
50 views

Solving a (tricky) arithmetic congruence equation in the general case

Out of sheer curiosity, I am looking for the solutions of the congruence equation : $n^{n+km} \equiv n$ mod m for every k natural integer where $k,n,m \in N$ where m=$p_1^{\alpha_1}p_2^{\alpha_2} . ...
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1answer
14 views

A rescaled inner product inequality

I was wondering if the following inequality is true: Let $\xi_1,...,\xi_n$ be vectors in a Hilbert space $H$ and let $x_{i,j}$ be complex numbers such that $\prod x_{i,j}$ is real and $$\prod ...
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2answers
202 views

Parameterization of Natural Numbers

Suppose we have 4 positive integers $a<b<c<d$ such that $a+d=b+c=n$, i.e. $a,d$ and $b,c$ have the same average. Does there exist $p,q,r,s \in \mathbb Z$ such that \begin{equation*} ...
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1answer
31 views

Incommensurable units as ratios

I am having a bit of trouble understanding the concept of an incommensurable unit. From what I have gathered so far, it is simply a magnitude that cannot be expressed as the ratio of two natural ...
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0answers
37 views

Calculate shifted unit circle values

I have the black unit circle and I need to shift it by x degrees getting the red unit circle. How do I shift it, because just subtracting x degrees from the original black circle doesn't work. ...
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1answer
24 views

Find $f^{-1}(g(x))$ if $f(x) = 2x + 1$ and $g(x) = x^{2}$

Question: Let $f$ and $g$ be defined as: $$f(x) = 2x + 1, ~~~~x \in \mathbb{R}$$ $$g(x) = x^{2}, ~~~~~~~~~~~~x \in \mathbb{R}$$ Find a) $~~f^{-1}(x)$ b) $~~f(g(x))$ c) $~~g(f(x))$ d) ...
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1answer
60 views

Making the segment with given length $\sqrt[3]{2}$?

Using Pythagoras' Theorem we can make the segment with given length of Square root of natural numbers. For example the segment of given length The square root of 2 is equal to the length of the ...
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3answers
47 views

If I have the value of $\sqrt{1.3}$ could it be possible to find other square roots from that value? using the manipulation of surds?

If I have the value of $\sqrt{1.3}$ could it be possible to find other square roots from that value? using the manipulation of surds?
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11answers
2k views

Is there any way to define arithmetical multiplication as other thing than repeated addition?

Is there any way to define arithmetical multiplication as other thing than repeated addition? For example, how could you define $a\cdot b$ as other thing than $\underbrace{a+a+\cdots+a}_{b ...
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1answer
34 views

How to convert an interval to different one

I have a variable x in the interval [-30; 30]. I need to convert this interval so x would be in the interval [0; 1]. What I mean is like this: ...
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5answers
51 views

canceling double fractions how?

I had this example: $$ \frac{\frac{11}{5}}{2} = \frac{11}{10} $$ then: $$ \frac{2\frac{1}{5}}{2} = \frac{11}{10} $$ $$ \frac{1}{5} \not= \frac{11}{10} $$ is this right canceling of double ...
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1answer
22 views

Combinations in Arithmetic progression?

My question actually come from this: Intuitively understanding $\sum_{i=1}^ni={n+1\choose2}$ I was once examining the sum of an A.P series with first term $a$ and common difference $d$ . And this ...
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1answer
22 views

Dust coating a length of road that is 1.25 miles long.

A length of road is 1.25 miles. 8 landowners along the road would like to split the cost of dust coating the road. The cost of dust coating is at $53 per 100 feet. The county will reimburse the ...
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4answers
3k views

Subtraction when second number is bigger than first number

I'm a bit new to this. I'm trying to figure out how subtraction works pen and paper wise. I have a bit of a program where I can't seem to find any answers online. What I want to do is use the ...
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2answers
369 views

Continuum between addition, multiplication and exponentiation?

I noticed this old post which attempts to find the shades of grey between a linear and log scale where results are between zero and one. However, I was looking for the more general case where we find ...
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1answer
50 views

Proof that $a^n \ne b^{n+1}, (a, b) \in \mathbb{Z}$ [closed]

I was wondering if there exists a proof that: $$a^n \ne b^{n+1},$$ $$a, b \in \mathbb{Z}$$ It would really help with a project that I am working on. Thank you very much!
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1answer
45 views

What is meant by “decimal adjustment” in this problem?

I know how to do addition in different bases, and I know how to convert between bases, but I have never heard of "adjustments" between bases while adding. I would solve this by simply getting a ...
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6answers
242 views

Why do we invert then multiply when dividing fractions?

I have looked up various sites online. They only explain it in a very basic arithmetic way. Would someone explain to me why is $\displaystyle\frac{x}{\frac{1}{2}}$ is $\displaystyle\frac{2x}{1}$?
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2answers
954 views

Find P + Q + R?

I was doing questions from previous year exam paper and I'm stuck on this question. It seems easy but I am not getting the point from where I should start. Suppose $P, Q, R$ are positive integer, ...
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2answers
35 views

Simple Adding and Subtracting algorithm to get a current amount

So this is my first post and its probably a really dumb question but I cannot get my head around a proper algorithm to handle the following situation. I have a dataset that contains employees who ...
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1answer
24 views

Figure out attendance needed for remaining days to reach attendance goal

Trying to come up with attendance percentage needed to reach a goal. To simplify a student to date has gone to school 140 of the 150 days and therefore their current attendance is .93 There are 70 ...
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1answer
46 views

Finding the solution to an equation using trial and improvement. [closed]

Using trial and improvement to find this solution to 2 decimal places. The equation $x^3=10-3x$ has a solution such that $1 \le x\le 2$.
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1answer
33 views

Weighted mean of an object (Centers of mass)

I am having trouble understanding the concept. Usually when I calculate the center of mass of an object when given area and dimensions I'd multiply corresponding distances with areas etc then ...
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1answer
42 views

Confusion about exponents like ${x^m}^{(1/n)}$.

I've been reading this post. It says that $\sqrt[m]{x^n} = x^{n\frac 1m}=x^{\frac mn}=x$ if $m=n$. Let's take $x=-2$, and $m=n=2$. Now we have, $\sqrt[2]{(-2)^2}=\sqrt[2]{4}=2$ But according to that ...
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1answer
41 views

How to find square and cubes

I want to find the smallest positive integer A in which $$10A$$ is a perfect square and $$6A$$ is a perfect cube Thanks for the hint, I can see now I just needed $$2^5,3^2 , 5^3$$
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1answer
73 views

Quotients of $\mathbb{Z}[i]$

Let $\mathbb{Z}[i]$ be the ring of Gauss integers. For a simple representation it is all the complex numbers of the form $a+ib$ such that $a,b \in \mathbb{Z}$. It is known that $\mathbb{Z}[i]$ is a ...
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0answers
20 views

Mathematical operations on tolerances

This question is about manufacturing tolerances; for instance, a manufactured component expressed as "$5 \pm 0.3\,\mathrm{in}$" may acceptably be no larger than 5.3 in and no smaller than 4.7 in. I ...
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2answers
69 views

Least integer function and Greatest Integer Function Without using ceil() and Floor()

I was wondering if there is any mathematical way to calculate Least Integer and Greatest integer without using predefined Ceil() and Floor() Function of Programming Language.
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1answer
35 views

Link between two products

Could someone help me to solve this problem : Let's denote by $A_i$ the following product, $$ A_i = \prod_{\substack{k=1 \\ k\neq i}}^n (a_k - a_i) $$ Is there any link or simple formula between ...
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1answer
79 views

How can $n^5+4$ be a perfect square?

How can one find all $n \in \mathbb{N}$ such that $n^5+4$ is a perfect square? I see that $n^5=(x+2)(x-2)$ here im suck can someone help ?
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4answers
329 views

Compare three numbers, expressed as powers: $4^{68}$, $5^{51}$ and $12^{23}$

So I have these numbers: $4^{68}, 5^{51}, 12^{23}$ They need to be ordered from the smallest to greatest. I have no idea how to solve this. Typically, one should break down the exponents somehow to ...
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1answer
75 views

How to define multiplication in $\mathbb{Z}$ with divisibility and addition?

Q: Show that $(\mathbb{Z},|,+,0,1)$ defines multiplication in $\mathbb{Z}$. I know how to do this in $\mathbb{N}$, but I'm stuck trying to do this is $\mathbb{Z}$. The idea I have is to define lowest ...
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3answers
61 views

Comparing the size of $(\sqrt{5})^e$ and $e^{\sqrt{5}}$

So I have to figure out which one is bigger between $(\sqrt{5})^e$ and $e^{\sqrt{5}}$. After some trial and error I've come to the conclusion that $(\sqrt{5})^e > e^{\sqrt{5}}$. But of course I ...
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0answers
13 views

Finding the frequency of a fan using a slow motion camera

This problem has me a little stumped. I'm not sure if my answer is correct and would just like to check: The camera shoots at 187 frames per second. The fan takes 33 frames to complete one revolution ...
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1answer
30 views

$b^{\frac{m}{n}}=(b^{\frac{1}{n}})^m=(b^m)^{\frac{1}{n}}$ except $b$ is not negative when $n$ is Even.

The following property, known as Rational number property, is taken from the book (I am following now a days) College Algebra by Raymond A Barnett and Micheal R Ziegler I restate, ...
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2answers
66 views

How to find minutes?

Need help solving this real life problem, I have an SD Card of $4$GB(gigabyte), and a $32$ second video occupies $6.12$MB(megabyte), I need to know how many minutes or seconds can this $4$GB SD Card ...
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2answers
63 views

Variations in the successor fuction from Peano's axioms

Concerning the successor function in Peano's axioms, what prevents me from defining it in the following way: 0 to 2, 2 to 1, 1 to 4, 4 to 3, 3 to 6, 6 to 5, 5 to 8, 8 to 7 ... and so on. It seems ...
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2answers
55 views

Simplifying Square Roots Frustration

Okay, I'm really frustrated with this. So, when you have $3 \sqrt 5 + 5 \sqrt 5$, you get $8\sqrt5$, right? Okay, so what do I do for here: $\sqrt{11} - 3 \sqrt{11}$ Is it just $-3 \sqrt{11}$ ? ...
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1answer
28 views

Interval arithmetic - faster version

As per the below question picked from self training exercise: Q4: In passing, Ben also cryptically comments, "By testing the signs of the endpoints of the intervals, it is possible to break ...
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1answer
43 views

order of operations in different cultures?

Are there any cultures or countries around the world that use a different convention for order of operations than the BEDMAS convention? i.e.: Parentheses Exponents & Roots Multiplication & ...
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2answers
59 views

How many numbers smaller that $N$ can be written as a sum of two square numbers?

We define $$a_N =\# \{ n \leq N, \exists (n_1,n_2) \in \mathbb{N}^2, n = n_1^2 + n_2^2 \}.$$ Can we have the exact value of $a_N$, or at least an asymptotic behavior such as $$ \alpha N \leq a_N \leq ...