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

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Does $i = -\frac{(2\;W({\pi\over2}))}{\pi}$

Let $x = -\frac{(2\;W({\pi\over2}))}{\pi}$, where $W$ denotes the Lambert W-function. As $${\log(i^2)\over i} = \pi$$ and $${\log(x^2)\over x}=\pi$$ Does $x = i$?
2
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1answer
76 views

How to prove that $\frac{1}{n+1} = {\sqrt{n\over n+1}}\implies n = \Phi$?

Consider: $$\frac{1}{n+1} = {\sqrt{n\over n+1}}$$ How could one prove that $n$ is of such form that: $$\frac{1}{n+1} = {\sqrt{n\over n+1}}\implies n = {\sqrt{5\,}-1 \over 2} \implies n = \Phi$$ ...
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3answers
874 views

The limit of $\lim\limits_{x \to \infty}\sqrt{x^2+3x-4}-x$

I tried all I know and I always get to $\infty$, Wolfram Alpha says $\frac{3}{2}$. How should I simplify it? $$\lim\limits_{x \to \infty}\sqrt{(x^2+3x+4)}-x$$ I tried multiplying by its conjugate, ...
1
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2answers
177 views

What leads us to believe that 2+2 is equal to 4? [closed]

My professor of Epistemological Basis of Modern Science discipline was questioning about what we consider knowledge and what makes us believe or not in it's reliability. To test us, he asked us to ...
2
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2answers
118 views

2 questions regarding solutions for $\sqrt{a+b} - (a-b)^2 = 0$

Here's two questions derived from the following question: $\quad\begin{matrix} \text{Is there more than one solution to the following statement?} \\ \!\sqrt{a+b} - (a-b)^2 = 0 \end{matrix}$ ...
3
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2answers
178 views

property of equality

The property of equality says: "The equals sign in an equation is like a scale: both sides, left and right, must be the same in order for the scale to stay in balance and the equation to be true." ...
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2answers
292 views

When do we need parenthesis to change order of operations?

A few questions about order of operations: $1$) In an expression such as $1+3+3^2$, it is legal to simplify to $4+3^2$, a violation of the grade school order of operations. In this case, are we ...
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6answers
336 views

How can $4^x = 4^{400}+4^{400}+4^{400}+4^{400}$ have the solution $x=401$?

How can $4^x = 4^{400} + 4^{400} + 4^{400} + 4^{400}$ have the solution $x = 401$? Can someone explain to me how this works in a simple way?
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0answers
150 views

Is there a finite number of solutions to $\mathrm{Re}(a^n)+\mathrm{Im}(a^n)=b^n$, where $a$ is a Gaussian integer and $b \in \Bbb Z$?

Let $$E_n=\{(x,y,b) \in \mathbb{Z}^*\times \mathbb{Z}^*\times \mathbb{Z}^* ~|~ \gcd(x,y,b)=1 ~ \mathrm{and}~\mathrm{Re}((x+iy)^n)+\mathrm{Im}((x+iy)^n)=b^n \}$$ For $n \geq 3$, is $E_n$ finite or not ...
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3answers
149 views

IEEE-754 General Issues

I read the documents of the IEEE-754 and I could not understand some things 1) Is it possible to represent any number using this method? I know there are a number of final states, and there are ...
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2answers
243 views

Finding a 6 digit number

The sum of digits of a six digit number is $41$. At first we omit its leftmost digit and subtract it from the original number. Then we omit the two leftmost digits and add it with the result. Next we ...
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5answers
290 views

$1+1+1+1+1+1+1+1+1+1+1+1 \cdot 0+1 = 12$ or $1$?

Does the expression $1+1+1+1+1+1+1+1+1+1+1+1 \cdot 0+1$ equal $1$ or $12$ ? I thought it would be 12 this as per pemdas rule: $$(1+1+1+1+1+1+1+1+1+1+1+1)\cdot ...
1
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1answer
79 views

Proving that sequence $a_n = \sqrt{x \sqrt{x \sqrt{x \sqrt{\cdots}}}} = x^{1-2^{-n}}$

Let $x>0$. For sequence $a_n$, such that $n$ denotes the $n$th term: $$\begin{align} a_1&= \sqrt{x}\\ a_2&= \sqrt{x \sqrt{x}}\\ a_3&= \sqrt{x \sqrt{x \sqrt{x}}}\\ a_4&= \sqrt{x ...
1
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1answer
73 views

Is there a shorter way to write an equation with a clear pattern?

I have this simple expression: $$\left(\begin{array}{l} 1\\ +(A)\\ +(A+B)\\ +(A+B+C)\\ +(A+B+C+D)\\ +(A+B+C+D+E)\\ +(A+B+C+D+E+F)\\ +(A+B+C+D+E+F+G)\\ +(A+B+C+D+E+F+G+H)\\ +(A+B+C+D+E+F+G+H) ...
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2answers
696 views

Pages 6 and 27 are on the same (double) sheet of a newspaper, how many pages are there in the news paper altogether

I was never really good at maths, trying to get back into it. I have this question: ...
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3answers
862 views

Does half-life mean something can never completely decay?

Caffeine has a half-life of approximately six hours. I understand this to mean that every six hours, the amount of caffeine in the body is half of what it was six hours prior. Does that mean that ...
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4answers
50 views

Correlation between multiplied numbers?

I do not have a strong math background, but I'm curious as to what this pattern is from a mathematical standpoint. I was curious how many minutes there were in a day, so I said "24*6=144, add a 0, ...
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2answers
93 views

Two questions on clock arithmetic

I have two questions on clock arithmetic, both of which I have solved, but I am looking for neater proofs. Let us suppose we have a circle named $\mathbb{Z}_n$ with $n$ equally spaced points on it ...
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2answers
246 views

A proof of $n*0=0$?

The only proof I've seen for this assumes that $0$ follows all the rules of arithmetic. How can we make that assumption when dividing by $0$ is a problem? I know that some people don't agree that all ...
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1answer
471 views

Is it possible to rationalize a denominator containing two cube roots?

The fraction in question is $$-\frac{12}{\sqrt[3]{12\sqrt{849} + 108} - \sqrt[3]{12\sqrt{849} - 108}}$$ And was reached in calculating the solution to $x^4 - x - 1 = 0$. I've tried all the standard ...
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1answer
66 views

Math arithmetic

So I got this question in my exam yesterday, and only a few people from my school could solve this. What's so hard about it? I have learned the simple arithmetic questions, but I don't really ...
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4answers
364 views

Why are the order-of-operations conventions good?

Children are sometimes taught silly mnemonics like "PEMDAS" to remember conventions on order of operations. (I never heard of "PEMDAS" until long after graduating from college, as far as I can ...
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8answers
241 views

What is $2 - 1 + 1$? [duplicate]

$2-1+1$; a fairly straightforward question, but I (well, not me, but Henry Reich) found something strange. Most people would evaluate it as $2+(-1)+1 = 2$; however, this goes against the famed, and ...
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2answers
373 views

How to derive the sum of an arithmetic sequence?

I'm attempting to derive a formula for the sum of all elements of an arithmetic series, given the first term, the limiting term (the number that no number in the sequence is higher than), and the ...
3
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2answers
247 views

real mechanism behind addition, subtraction, multiplication, division

we all know the basic rules for operations of addition, subtraction, multiplication and division. but what i don't know is why these rules (of addition, subtraction, multiplication and division) ...
61
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5answers
17k views

Why is 987654321/123456789 = 8.0000000729?

Many years ago, I noticed that $987654321/123456789 = 8.0000000729\ldots$. I sent it in to Martin Gardner at Scientific American and he published it in his column!!! My life has gone downhill since ...
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6answers
2k views

Order of operations (BODMAS)

$$40-20/2+15\times1.5\\\hspace{.1in}\\40-20/2+15\times1.5=\\ 40-10+22.5=7.5$$ I'm studying and this is from an example. In BODMAS, aren't addition and subtraction have same level? So, in the ...
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4answers
765 views

Calculating $\sqrt{28\cdot 29 \cdot 30\cdot 31+1}$

Is it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid? I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work.
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3answers
550 views

How to sum numerator and denominator of a fraction?

I want to do sum over this. Can apply the summation to top and bottom separately? $$\sum\limits_{i=1}^{n} \frac{-a(x_i-\mu)^2}{x_i}$$ ...
6
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4answers
414 views

Definition(s) of rational numbers

The definitions of rational numbers are somewhat confusing for me. The definition of rational numbers on wikipedia and most other sites is: In mathematics, a rational number is any number that ...
0
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1answer
28 views

(Follow Up) Checking the solutions of a quadratic polynomial

I'm following up from this question: Solve a polynomial involving geometric progression? I have had trouble with this question: "Solve the equation $8x^3−38x^2+57x−27=0$" if the roots are in ...
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2answers
48 views

Prove that nearly all positive integers are equal to $a + b + c$ where $a | b$ and $b | c$, $a \lt b \lt c$

If a positive integer $n$ is equal to $a + b + c$ where $a | b$, $b | c$ and $a \lt b \lt c$, let it be called "faithful". Prove that nearly all numbers are faithful and list the non-faithful ...
0
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1answer
44 views

4 equations with specific set of numbers

I must make 1 addition (x+y=z), 1 subtraction (x-y=z), 1 multiplication (x*y=z), and 1 division (x/y=z) equation with the following numbers. All the numbers must be used to fill x, y, and z of each ...
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2answers
105 views

Counting Number of Objects - When to Add One Back

I know this might be a very basic question. Sometimes to count objects, we just subtract. For example -- If there are 5 apples and I take away 1, then remaining are $5 - 1 = 4$ apples. But other ...
6
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1answer
284 views

Find the value of $x^3-x^{-3}$ given that $x^2+x^{-2} = 83$

If $x>1$ and $x^2+\dfrac {1}{x^2}=83$, find the value of the expression$$x^3-\dfrac {1}{x^3}$$ a) $764$ b) $750$ c) $756$ d) $760$ In this question from given I tried to ...
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1answer
71 views

4 equations with set of numbers

I must make 1 addition (x+y=z), 1 subtraction (x-y=z), 1 multiplication (x*y=z), and 1 division (x/y=z) equation with the following numbers. All the numbers must be used to fill x, y, and z of each ...
2
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1answer
74 views

Age is fraction of year man dies

My friend sent me a question from an olympiad and im not sure that we have followed the right method, we both did the same thing: The age of a man was 2/61 of the year in which he died. How old would ...
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3answers
90 views

finding nth term

Let 3,8,17,32,57 . . . . . be a pattern.How do we find the nth number?My brains are completely jammed,I am tired.I do not even recognize the pattern.I calculated a few ways,but all I want is a ...
0
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1answer
119 views

Finding how many times one number fits into the other

Is there a way to tell - for example - how many $60$s are there in a number that cannot be divided by $60$, like $183$? I know that we should remove all numbers after the decimal point but how can we ...
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3answers
206 views

sum of ten squares

You are given an unlimited supply of $1\times 1,2\times 2,3\times 3,4\times 4,5\times 5,6\times 6$ squares.Find a set of ten squares whose areas add up to $48$.If not the whole solution,even a little ...
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0answers
175 views

Gold Ring Calculation

I have problem with gold ring weight calculation I have below details to calculate gold ring calculation ...
102
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1answer
11k views

Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) 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 ...
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5answers
3k views

Finding two numbers when having their sum and product

I have two numbers, their sum is 41 and their product is 238. What are the numbers? I got during this far in my calculations: $a+b=41,\quad ab=238,\quad 238=41-b.$ I appreciate answers or tips to ...
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6answers
161 views

I have difficulty counting

Take this example: $$84 + 87 + 90 + 93 .. + 180 + 183$$ If we want to use Gauss' way of finding this sum, we have to find the number of elements in this. What I do is just take the difference of the ...
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4answers
673 views

Simple Math Problem

A bat and ball cost \$1.10. The bat costs a dollar more then the ball. How much does the ball cost? If this is not the correct place to ask a question like this please tell me and I will remove it ...
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1answer
246 views

Calculate the distance between the points $(1, 2, \dots, n)$ and $(2, 3, \dots, n, 1)$

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: ...
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3answers
71 views

What have I done wrong in solving this problem with indices rules?

The question asks to simplify: $$\left(\dfrac{25x^4}{4}\right)^{-\frac{1}{2}}.$$ So I used $(a^m)^n=a^{mn}$ to get $$\dfrac{25}{4}x^{-2} = \dfrac{25}{4} \times \dfrac{1}{x^2} = \frac{25}{4x^2} = ...
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2answers
100 views

when the numerator is less than the denominator

when the numerator is less than the denominator the result is always between 0 and 1? for example if I have a number like x/y where x<y then the result will be ...
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5answers
97 views

Why do this expressions evaluate to different results?

$$\frac{12}{15} = 0.8\\ \frac{15}{12} = 1.25$$ If $15$ divided by $12 = 1.25$ shouldn't $12$ divided by $15$ be the same as $\frac{15}{12}$ and have the same result? Can someone please explain this ...
0
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1answer
87 views

Length of DNA strand

The DNA molecule has a double helix structure. The radius of each helix is approximately $10$ angstroms ($1$ angstrom $=10^{-8}$cm). Each helix goes up by approximately $34$ angstroms every ...