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

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5
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4answers
4k views

How much faster is the Trachtenberg system?

How much faster are various mathematical operations using a Trachtenberg method rather than a conventional method?
0
votes
4answers
85 views

Fastest way to perform arithematic calculations

This is one of the questions asked for Junior Trader position at prop. trading firm. Perform the following operation. 4.3 * 0.58 + 2.0E-5 - 0.9 How can one ...
1
vote
1answer
39 views

Determine what when multiplied with $180$ gives a perfect cube

Recently, at a math competition, I was given the following question: Determine the smallest number that gives a perfect cube when multiplied by $180$ . I had thirty seconds to solve this question and ...
0
votes
3answers
68 views

$13\mid(3^{n+2}+4^{2\cdot n+1})$. Is my arithmetical proof using induction correct?

The exercise 2.b of my textbook ask me to prove that: $$\text{(P): }\;\forall n\in \mathbb{N}, 13\;|\;(3^{n+2}+4^{2\cdot n+1})$$ I would like to know if my proof is correct and if not what I need to ...
7
votes
3answers
317 views

The number $90$ is a polite number, what is its politeness?

The number $90$ is a polite number, what is its politeness? A. $12$ B. $9$ C. $6$ D. $14$ E. $3$ How did you get that answer? I tried Wikipedia to figure out what a polite number was ...
-2
votes
1answer
85 views

Kindergarden help arithmetic problem [closed]

What is 20 + 25? I can't seem to count with my fingers... I only have 10 :( Mummy and daddy won't help me:(
0
votes
1answer
25 views

Find original cost based on fractional purchase

How would I go about finding the original cost of bitcoin knowing that $20 purchased .0531401 of bitcoin? I would like to know what the cost of 1 bitcoin was at the time of purchase? ...
0
votes
2answers
31 views

Why not a simple division in this case?

n represents total number of persons. max represents the maximum number of persons by department. To know the accepted number ...
1
vote
2answers
28 views

Inequality problem

$5\geq X \geq 2$ $0.8 \geq Y\geq 0.5$ Find range of the values of A) $\sqrt{XY}$ B) $(\frac{X}{Y})^2$
1
vote
1answer
93 views

Converting division by a constant into multiplication

Given a small integer constant $N$, how can I find a fast to compute expression for $\lfloor \frac xN \rfloor$, working for each integer $x \in [-2^{31}, +2^{31})$? Here, "fast to compute" means that ...
4
votes
3answers
57 views

How to solve the following arithmetic radical problem?

$$ 2(4\sqrt{7} + 1 + 3\sqrt{7} + 2) $$ I distribute first right? $$ 8\sqrt{14} + 2 + 6\sqrt{14} + 4$$ $$ 14\sqrt{14} + 6$$ BUT IT LOOKS LIKE ITS SUPPOSED TO BE $$14\sqrt{7} + 6$$ I also have a ...
6
votes
4answers
4k views

Adding and Removing Non-Compounded Percentages does not produce the same result?

If I take the value 100 and I want to and 10% tax to it and then a 7% tax to it, I am doing the following: $$\begin{align*} 100 \times \left(1 + \frac{10}{100}\right) &= 110\\ 100 \times ...
1
vote
3answers
76 views

How to derive that for every real $y > 0$, for every positive real $z \neq 1$, there is a $x \in \mathbb{R}$ such that $y=z^x$.

I am not sure on how to derive the following statement concerning the reals (that I think should be true). For every real $y > 0$, for every positive real $z \neq 1$, there is a $x \in ...
57
votes
8answers
3k views

Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\ \\ \frac1{\sqrt{-1}} &= \frac1i \\ \\ \frac{\sqrt1}{\sqrt{-1}} ...
3
votes
1answer
39 views

Partitioning the natural numbers to finite number of arithmetic progressions

Some already asked on this site whether the natural numbers can be partitioned to finite number of arithmetic progresions with distinct differences, with the condition that the intercection between ...
2
votes
3answers
572 views

Profit sharing among investors

Abigail, Bart, and Cathy invested \$2,400, \$3,600, and \$6,000 respectively to start a partnership business. At the end of the first year, the business earned a profit of 40% on the initial ...
2
votes
1answer
41 views

What is the area of the square given the following circumstances?

The perimeter of square HJKL is 2 times the perimeter of square WXYZ The SO if perimeter of HJKL is 2 times the perimeter the WXYZ than $$ \text{Perimeter of } HJKL = 2\cdot2(l+w)$$ So the L and ...
0
votes
0answers
13 views

Calculating inflation using CPI having m-o-m and y-o-y data

inflation Hello guys, I am having trouble calculating inflation from CPI - I would like to know what was the CPI in the yellow box... so far I have not been able to solve this... I would be extremely ...
1
vote
1answer
82 views

Prove that $(n!)!$ divisible by $(n!)^{(n-1)!}$ [duplicate]

I was trying to think of a situation and use combinatorics to solve the problem. Any other arithmetic solution is also appreciated.
9
votes
9answers
17k views

What's the formula for the 365 day penny challenge?

You might have seen the viral posts about "save a penny a day for a year and make $667.95!" The mathematicians here already get the concept while some others may be going, "what"? Of course, what the ...
-9
votes
3answers
105 views

$-25+55+(85+65)= 180$ or $120$ ?And Why? [closed]

I mean this question can have both the anwers but which one is correct and why? $-25+205=180$ And $-30+150=120$ But which answer is correct ?
4
votes
3answers
473 views

Landau's “Foundations of Analysis” - Addition of natural numbers

At the beginning of his Foundations of Analysis book (translated from German), Landau writes in his Preface for the Teacher : Peano defines $x+y$ for fixed $x$ and all $y$ as follows : $$x+1 = ...
-4
votes
3answers
113 views

BODMAS riddle - which is correct? [closed]

I fully understand the order of BODMAS - It's the order of operations in maths equations. So lately this little puzzle/riddle has been going around on social media - ...
4
votes
1answer
85 views

What was babylonians estimation for square root 3?

We see a lot of papers and talk about ancient Babylonians exactness of calculating the value of square root of 2. For example: ...
1
vote
2answers
38 views

Struggling with a task with brackets

The task is to change brackets (operations, numbers and their positions should be remained the same) to make $F$ equal to $850$. $F = (1 + 2) \times (3 + 4) \times (5 + 6) \times (7 + 8) \times (9 ...
0
votes
1answer
30 views

Binary subtraction with borrowing vs. 2's complement

Consider the following two binary numbers which are using sign-magnitude representation. Therefore, both are positive numbers. ...
2
votes
2answers
220 views

Should BODMAS not be BODMSA?

Let me and blindly follow for a second BODMAS. $$ \begin{align*} 1-3+2 &=1-(3+2)\\ &=1-5\\ &=-4 \end{align*} $$ (The brackets are just to make the error clear - I wouldn't write them in ...
1
vote
1answer
41 views

Prove that two arithmetic grammars generate the same language

In my university's automata theory book it is claimed that the following two arithmetic grammars generate the same arithmetic language but no proof is shown. $G_1=(\{E\}, \{a,+,*,(,)\}, \{E\to ...
0
votes
3answers
4k views

What is the difference between a quadratic equation and a quadratic function?

I cannot dicepher the difference between a quadratic equation and a quadratic function. I read the following "A quadratic equation can tell us a lot about the graph of a quadratic function." I see the ...
11
votes
9answers
596 views

Make the number $100$ out of $1,2,3,$ and $4$ digits, without repeats

How can we make the number $100$, using only the following digits: $1,2,3,4$. You cannot repeat any of them.
-2
votes
5answers
12k views

How to reverse percentage?

If I have a value of 25%, and I want the value as it would be on the opposite side of the halfway point (75% in this case), what is a formula that can calculate this? I don't know the appropriate ...
0
votes
2answers
28 views

How to solve a word problem when given width and height of the following?

The width of a room is 4 feet shorter than its length, and its height is 3 feet less than its length. The area of four walls is larger than the sum of the areas of the floor and ceiling by 134 square ...
2
votes
1answer
67 views

Follow-on to exponents in Birthday Paradox

Follow on to discussion on Birthday Paradox I read the Scientific American article again Please explain why they want to find the probability of NOT matching birthdays, and then subtracting that ...
2
votes
2answers
333 views

Not understanding division in Birthday Paradox

I am reading Scientific American's explanation for birthday paradox here I understand everything in the article up to Every one of the 253 combinations has the same odds, 99.726027 percent, of ...
3
votes
2answers
89 views

Show $\frac{(p^d - 1)(p^{d-1} - 1)}{(p-1)(p^2 - 1)} \equiv 1 \pmod{p}$.

Let $p$ be prime and $d \ge 2$. I want to show that $$ \frac{(p^d - 1)(p^{d-1} - 1)}{(p-1)(p^2 - 1)} \equiv 1 \pmod{p}. $$ I have a proof, but I think it is complicated, and the statement appears in ...
3
votes
3answers
84 views

What is value of $a+b+c+d+e$?

What is value of $a+b+c+d+e$? If given : $$abcde=45$$ And $a,b, c, d, e$ all are distinct integer. My attempt : I calculated, $45 = 3^2 \times 5$. Can you explain, how do I find the distinct ...
2
votes
6answers
100 views

Prove that $5^n + 2\cdot3^{n-1} + 1$ is multiple of $8$

Prove that $5^n + 2\cdot3^{n-1}+ 1$ is multiple of $8$. I've tried using induction (it isn't): For $n=1$: $$5^1 + 2\cdot3^{n-1} + 1 = 8$$ If it is true for $n$, then $n+1$? \begin{align} 5^{n+1} ...
36
votes
4answers
2k views

Property of 111,111

Whilst playing on my calculator, I noticed the following pattern. $1^2-0^2=1$ $6^2-5^2=11$ ${20}^2-{17}^2=111$ ${56}^2-{45}^2=1{,}111$ ${156}^2-{115}^2=11{,}111$ To me, this is where it gets ...
2
votes
4answers
1k views

On the commutative property of multiplication (domain of integers, possibly reals)

$ab = ba$ This is, inherently, true. Some texts drop it like an axiom without any justification. But I'm a bit curious where it stems from or basically why/how it works. If anyone could enlighten me ...
2
votes
4answers
143 views

How come $\left(\frac{n+1}{n-1}\right)^n = \left(1+\frac{2}{n-1}\right)^n$?

I'm looking at one of my professor's calculus slides and in one of his proofs he uses the identity: $\left(\frac{n+1}{n-1}\right)^n = \left(1+\frac{2}{n-1}\right)^n$ Except I don't see why that's ...
-3
votes
2answers
98 views

Can $60+60\times0+1$ be both $1$ and $61$ [closed]

The expression under debate is $60+60\times0+1$. I'm in a debate on Facebook and people that are saying $61$ and that the people that are saying $1$ are and vice versa but I'm saying they are both ...
-2
votes
1answer
30 views

Numbers $65x1y$ multiples of 12 [closed]

Find all the five digit numbers in the form $65x1y$ multiples of $12$
2
votes
1answer
33 views

How could I simplify this algebra expresion?

I had been solving an equation with complex numbers: $z = \frac{x - iy}{x + iy}$ I solved it up to the point where I get: $z = \frac{x^{2} - y^{2}}{x^{2} + y^{2}}$. But I have no idea how to simplify ...
63
votes
34answers
23k views

Why is negative times negative = positive?

Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc. I went ahead and gave them a proof by contradiction like so: Assume ...
8
votes
3answers
248 views

What does $23_4$ mean?

I just saw this on a mathematical clock for $11$, i.e $23_4=11$: $\qquad \qquad \qquad \qquad \qquad$ I guess it is some notation from algebra. But since algebra was never my favorite field of ...
1
vote
2answers
45 views

Ratios with A.P.

Two A.P.’s have the same number of terms. The ratio of the last term of the first progression to first term of the second progression is equal to the ratio of the last term of the second progression ...
2
votes
1answer
52 views

How do I calculate $1.496\,\text{E}11$? [closed]

Sorry for that noobie question but how do I calculate this type of number $1.496\,\text{E}11$?
2
votes
9answers
12k views

Can $\frac {100-100}{100-100}=2$?

\begin{align*} \frac{0}{0} &= \frac{100-100}{100-100} \\ &= \frac{10^2-10^2}{10(10-10)} \\ &= \frac{(10+10)(10-10)}{10(10-10)} \\ &= \frac{10+10}{10} \\ &= \frac{20}{10} \\ &= ...
5
votes
2answers
54 views

Arithmetic: Prove that is multiple of 30

Prove that $n^{19}-n^7$ is multiple of $30$ I've seen $6$ can divide it because $$n^{19}-n^7=n^7(n^{12}-1) = n^7(n^6+1)(n^6-1)=n^4(n^6+1)(n^3-1)n^3(n^3+1)$$ And there are three consecutive ...
1
vote
2answers
43 views

Question about something I found in a proof

I came across in a proof I was reading in my textbook that $ab - a'b' = ab - ab' + ab' - a'b'$. I was wondering why/how that equality is true.