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

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1
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2answers
36 views

How to evolve an expression with two denominators

The task is to simplify the expression: $\displaystyle\frac{f(x+h)-f(x)}{h}$ when $\displaystyle f(x) =\frac{1}{x}$. I don't know how to do this since I get to the step ...
33
votes
19answers
36k views

How do I explain 2 to the power of zero equals 1 to a child

My daughter is stuck on the concept that $$2^0 = 1,$$ having the intuitive expectation that it be equal to zero. I have tried explaining it, but I guess not well enough. How would you explain the ...
3
votes
2answers
88 views

How to replace addition with multiplication to find the next integer value?

Sorry in advance for my lack of mathematical knowledge, I am very new to it. Yesterday, I posed this question to myself: "In a world without addition or subtraction, how could we derive the next ...
0
votes
0answers
25 views

Order of evaluation of a simple term

If, for example, $a=14$, $b=2$ and $c=7$ and I'm working to (say) three decimal places, should I evaluate the term$$a\frac{b}{c},$$ in the order $$a\left(\frac{b}{c}\right)$$ ...
0
votes
2answers
35 views

Proportions manipulation [closed]

Knowing that \begin{equation}\frac{a}{b}=\frac{c}{d}\ \ , \ \ \frac{a'}{b'}=\frac{c'}{d'}\end{equation} find the condition that the sums \begin{equation}a+a',\ b+b',\ c+c',\ d+d',\end{equation} ...
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1answer
58 views

Find the number of six-digit numbers that can be formed

find the number of six-digit numbers that can be formed using the digits from the number 112 233. If these numbers are arranged in ascending order,find a.) the largest number. b.) the 30th number. ...
0
votes
1answer
165 views

Find the value of $k$ to satisfy the Arithmetic Progression

For what value of $k$ are $2k-7$, $k+5$ and $3k+2$ consecutive terms of an Arithmetic Progression?
1
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1answer
82 views

Prove that $\binom{p}{k}/p $ is integral for $k\in \{1,..,p-1\}$ with $p$ a prime number

I started by induction on $k$ For $k=1$ then : $1\in \mathbb{N}$ For $k=2$ then : $\frac{(p-1)}{2!} \in \mathbb{N}$ , indeed for all $p>{2}$, $p-1$ is even. (We still have $k<p$ it's ...
-2
votes
0answers
43 views

How to multiple and divide large decimal numbers? [closed]

I need to know how to multiple and divide large numbers especially decimal numbers on paper. It would help if you could demonstrate a simple calculation using a basic sure method then scaling it up to ...
2
votes
2answers
81 views

Simple Puzzle: A Matter Of Time

I am trying to solve a simple puzzle: Fifty Minutes ago if it was four times as many minutes past three O'clock, how many minutes is it to six O'clock. I tried solving it: Let x be the minutes past ...
1
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1answer
105 views

How does this card trick work?

Pick a card from the deck and keep it secret. Double the face value of the card (aces = 1, jacks = 11, queens = 12, and kings = 13). Add 3 to the result. Multiply this by 5. Add 1 if the card is a ...
1
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2answers
38 views

Confused about proof of division

I thought I was familiar with the regular Euclidean algorithm, but I am having trouble understanding a step in this proof from my notes, I am looking for any clarification. Theorem Let $a, b \in ...
1
vote
1answer
50 views

Finding remainder of negative number

Recently my colleague ask one mathematical question which is, What is the quotient and remainder of $(-29)/7$? and my answer was that quotient is $4$ and remainder is $-1$ and he told me I'm ...
3
votes
3answers
63 views

Negative roots of a cubic equation

Under what conditions will the cubic equation $ax^3 + bx^2 + cx + d$ where $a,b,c,d \in \mathbb R$ yield roots which have negative real parts? (All roots must have negative real parts) Motivation: I ...
0
votes
0answers
23 views

How many grams of coating is being yield per volume width?

The cylinder is 8.00 inches in diameter and 4.940 inches in width. 1.500 inches in width has a volume of 15. 3.440 inches in width has a volume of 12. The total grams of coating yield from the both ...
1
vote
1answer
52 views

(Visual) Intuition: Division and complex fractions

When treating division as "groups of the numerator" (sorry, I don't know the technical term -- see image), why does a complex fraction in the denominator get added together to produce a 1 (number of ...
1
vote
2answers
65 views

Does $k\cdot x < y$ imply that $x \ne y$?

Sorry if the question is trivial, but I have trouble getting my head around it. To keep short, does $\forall k \in \mathbb Z^*, \forall (x, y) \in \mathbb Z^2, k\cdot x < y \implies x \neq y$? ...
5
votes
4answers
3k views

How much faster is the Trachtenberg system?

How much faster are various mathematical operations using a Trachtenberg method rather than a conventional method?
9
votes
6answers
529 views

How to prove that $7^{31} > 8^{29}$

How can I prove that $7^{31}$ is bigger than $8^{29}$? I tried to write exponents as multiplication, $2\cdot 15 + 1$, and $2\cdot 14+1$, then to write this inequality as $7^{2\cdot 15}\cdot 7 > ...
6
votes
4answers
631 views

Last 10 digits of the billionth fibonacci number?

I want to compute the last ten digits of the billionth fibonacci number, but my notebook doesn't even have the power to calculate such big numbers, so I though of a very simple trick: The carry of ...
24
votes
4answers
1k views

Prove that $\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$

show that $$\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$$ and I found $$LHs-RHS=0.017\cdots$$ I have post this interesting problem Prove $(\dfrac{2}{5})^{\frac{2}{5}}<\ln{2}$ can someone suggest ...
3
votes
10answers
405 views

Prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$

I'm trying to prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$ I've first written down $-5\lt a-b \lt5$ and have tried to add different things from all sides of the ...
0
votes
0answers
27 views

Properties of exponentiation proof

I'm trying to prove the following: "Let $x, y$ be non-zero rational numbers, and let $n,m$ be integers. Then we have $x^n x^m = x^{n+m}$." I've managed to prove by induction the case $n,m \geq 0$ ...
0
votes
0answers
9 views

Moving average where periods have unequal # of samples

I'm trying to compare a simple moving average approach to one that normalizes by the number of samples in a period to determine which is "more correct." Here's a representative piece of the data: ...
13
votes
4answers
18k views

Factorial, but with addition [duplicate]

Is there a notation for addition form of factorial? $$5! = 5\times4\times3\times2\times1$$ That's pretty obvious. But I'm wondering what I'd need to use to describe $$5+4+3+2+1$$ like the ...
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votes
2answers
65 views

Mathematical induction problem. Let $S_{n}=\left (3+\sqrt{5}\right)^{n}+\left(3-\sqrt{5}\right)^{n}$ [closed]

Let $S_{n}=\left (3+\sqrt{5}\right)^{n}+\left(3-\sqrt{5}\right)^{n}$then, by mathematical induction, show that $S_{n}$ is an integer. Also, prove that the next integer greater than ...
4
votes
3answers
444 views

Unusual result to the addition

Question: Prove that (666... to n digits)^2 + (888... to n digits)=(444... to 2n digits) My way: I just proved the given equation for three values of n and written at the bottom. "Since the ...
1
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1answer
72 views

Why ${(a^2)}^{\frac 12}=\sqrt {a^2}=|a| \neq a$?

Let $a\in \mathbb R$. It should be true that $\sqrt {a^2}=|a|$, since $\sqrt {(-2)^2}=\sqrt{2^2}=2$ and so on. But, it is also true that ${(a^2)}^{\frac 12}=a$, and by definition, ${(a^2)}^{\frac ...
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votes
1answer
56 views

Add or subtracts 1 to 9 numbers and get the answer 100 [closed]

so the question is that we have add or subtract numbers from 1 to 9 and the answer should be 100. (Note: The numbers shouldn't repeat). so what is the solution to this problem? Please answer this ...
0
votes
1answer
38 views

Does Conversion rates affect the profit earned?

About $~\$100~$ were given to me as an initial investment. I converted them to INR and traded with it in buying stocks and also selling stocks. I am now left with say $~$*Rs.*$6000~$ today. As all of ...
2
votes
2answers
91 views

How to show this fraction is not an integer

Suppose $k\geq 2$ is an integer. I want to show $$\frac{1+k+k(k-2)}{1+\frac{k-1}{k}+\frac{(-1-\sqrt{k-1} )^2}{k(k-2)}}$$ is not an integer. It is equal to $$\frac{(k-2) k (k^2-k+1)}{2 (k^2-2 ...
1
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5answers
148 views

If $a+b+c+d=1$ then why is the maximum value of $(a+1)(b+1)(c+1)(d+1)$ is ${\left(\frac{5}{4}\right)}^4$?

What I know is that for equations of type $x+y=8$, $xy$ attains its maximum value when $x=y$ and this can be proved by either solving the quadratic equation with completing the squares or finding the ...
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3answers
38 views
4
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4answers
97 views

$(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$?

The question given is Show that $(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$. What I tried is suppose $a=(y+z-x),\ b=(z+x-y)$ and $c=(x+y-z)$ and then noted that $a+b+c=x+y+z$. So the ...
1
vote
3answers
123 views

Square root and principal square root confusion

A few months ago I asked a question about the $\pm$ symbol because I was confused about it... I still carry the same confusion (which really bugs me) but I think the real confusion has to do with the ...
4
votes
1answer
132 views

How do I find two integers - $x$ and $y$ - whose values satisfy the expression $x^2 + y^2 = z$, where $z$ is a perfect square?

I watched a YouTube video of an episode of Who Wants To Be A Millionaire?, in which a contestant was presented with a list of perfect squares. He was asked to choose the number that was also the sum ...
3
votes
2answers
78 views

If $\frac{(b−c)}{a} + \frac{(a+c)}{b} + \frac{(a−b)}{c}=1$ and $a-b+c \neq 0 $, then prove that $\frac 1a = \frac 1b + \frac 1c$

The question given is If $\dfrac{(b−c)}{a} + \dfrac{(a+c)}{b} + \dfrac{(a−b)}{c}=1$ and $a-b+c \neq 0 $ then prove that $\dfrac 1a = \dfrac 1b + \dfrac 1c$ I tried to take $abc$ on the right ...
1
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2answers
35 views

Significant figures reduction of solution

I have a problem which has to be answered using two significant figures from the solution value. My solution value is x = 303.385789245434541 What should my answer be? Thanks
0
votes
1answer
25 views

Summing Bases and Comparing

Let $b$ be an integer greater than 2, and let $N_b = 1_b + 2_b + \cdots + 100_b$ (the sum contains all valid base $b$ numbers up to $100_b$). Compute the number of values of $b$ for which the sum of ...
1
vote
1answer
54 views

Euler and probability - a $\zeta$-distributed random variable

Let's consider a random variable $X$ on $\mathbb{N}^*$ such as $\mathbb{P}[X=n]=n^{-s}\zeta(s)$. Thanks to that random variable we can prove that $\zeta(s)= ...
0
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1answer
65 views

Is multiplication commutative?

Say you have 3 apples and 2 oranges, and you want to multiply these two groups of fruits together to obtain a desired result, for instance: A. You want 3 apples for each orange, so you have 6 apples ...
2
votes
0answers
25 views

Determine if the members of a set can be made to equal a given number

Is there an easy way to determine if some combination of addition, subtraction, multiplication, and division will enable the numbers in a set to equal a given number? For example, if I have the ...
5
votes
1answer
59 views

What are examples of cases where floating-point $aaaa\ne(aa)(aa)$?

As explained in answers to this question on SO, due to non-associativity of floating-point arithmetic repeated multiplication like $aaaa$ can't be optimized to $(aa)(aa)$. Of course, aside from just ...
3
votes
1answer
78 views

Adding $inches^2 + inches$?

At our school we have a "summer packet" we have to complete. In this packet was the following problem: At first the "sum" part puzzled me. I had no idea why they would quiz us on basic arithmetic ...
2
votes
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 ...
1
vote
0answers
112 views

The minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time. How much a day does the clock gain?

The question in the textbook is: The minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time. How much a day does the clock gain? My method: The correct ...
6
votes
1answer
88 views

A runs 7/4 times as fast as B. If A gives B a start of 84m, how far must the winning post be…?

The problem statement in the book is: $A$ runs $7/4$ times as fast as $B$. If $A$ gives $B$ a start of $84$m, how far must the winning post be so that $A$ and $B$ might reach it at the same time? ...
3
votes
1answer
87 views

Interval arithmetic for open intervals

I found a detailed paper which outlines the rules of interval arithmetic for closed intervals, including unbounded closed intervals, but it makes no mention whatsoever about open intervals. I'm ...
11
votes
1answer
183 views

Representing predicate logic as arithmetic

Summary Since the below is quite long, I thought I'd add this summary. Given the following: A statement in proposition logic can be converted to an arithmetic expression over the integers modulo ...
2
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2answers
45 views