Questions tagged [arithmetic]

Questions on basic arithmetic involving numerical quantities only. Questions involving variable values (other than the result of the operation) should be placed under the (algebra-precalculus) tag. Questions about number theory (sometimes called "arithmetic") should not use this tag and should instead use (number-theory) or (elementary-number-theory).

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Finding the number of terms in an arithmetic sequence

The formula for finding the length ( Number of terms ) in a finite arithmetic series is the following: ( The last term - First term ) / ( Common difference ) and finally adding 1. What is the ...
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Where is the error: $-3=(-27)^{\frac{1}{3}}=(-27)^{\frac{2}{6}}=[(-27)^2]^{\frac{1}{6}}=[(27)^2]^{\frac{1}{6}}=27^{\frac{1}{3}}=3$? [duplicate]

Where is the error is the following operations? I guess is the second equality, but why? Can someone point out a theorem or definition that giustifies the mistaken step is not correct? $-3=(-27)^{\...
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How high is one billion pennies stack on top of eachother? A penny is about 1/16 or .0625 of an inch thick.

One Billion Pennies A penny is 1/16 or .0625 of an inch thick 1,000,000,000 x .0625 = 62,500,000 It takes 16 pennies to make an inch stack 62,500,000 / 16 = 3,906,250 inches So how many feet? 3,906,...
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2 answers
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What is the relation between playback speed and real time?

I am struggling to understand the relation between playback speed versus real time, for example, on YouTube. If there is 10 minute duration video. If I listen it on 1.5 then as per my understanding 1....
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4 votes
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I found a pattern in consecutive squares: $(a^2-b^2)-(b^2-c^2)$ is always $2$.

I was working on squares of numbers then found out that the difference of difference between two consecutive numbers is $2$. Saying this with an example like $$2^2=4\qquad\qquad 3^2=9\qquad\qquad 4^2=...
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-3 votes
2 answers
48 views

Easy way to convert from one fiat to another without using a calculator

I'm trying to convert from one fiat to another without using a calculator. In my case, I was trying to convert 25 TRY to BGN. Let's assume that 1 TRY = 0.13 BGN. That means 25 TRY = 25 * 0.13 BGN = 3....
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1 answer
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Prove $2^n + 2^m ≠ 2^p$ [duplicate]

Prove : $2^n + 2^m ≠ 2^p$ For any $n,m$ ($n≠m$) and $p$ being positive integers. I hadn't yet studied very much mathematics yet but I came across it when I had to build a python code. Any help is ...
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60 views

Division with comma and zero [closed]

Could you give me a detailed explanation Why you add a comma and a 0 in a decimal division ,help me whith this decimal point and zero
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30 views

Can the quotient of two vectors, multiplied by same vector, be expressed using dispersion

I am trying to reformulate the following normalised vector equation: $$ L = \frac{X^T Z}{Y^TZ} $$ where $||X||_1 = ||Y||_1 = ||Z||_1 = 1$ and $X, Y, Z \in \mathcal{R}^n$, into some kind of dispersion ...
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How would you go about using $ \pi $ as a number-writing base?

I read here that any real number can be used as a base in a numerical system. This does seem to me to raise a number of problems. If you take $\pi$, for example, how would you write the number $2$ in ...
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Finding sum borrowed when two unequal annual instalments are given

A sum needs to be repaid in two annual instalments of $2227$ and $2023$ rs., given the rate of interest as 19%, then the problem is to find what sum was borrowed at the starting of first year. The way ...
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2 answers
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How to "model" $\frac{6}{x-9}$

I'm working as a tutor and I got a question (from grade 9 no less) that reads "model: $\frac{6}{x=9}$. The only example I could get from the students was: Which...dosn't help - how do you ...
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Strong and Simple induction are equivalent taken with Peano axioms 1,2,3,4

What is the proof of strong and weak induction are equivalent? What book on formal arithmetic is a source of such proofs? (There is at least one similar question on Math.SE, but it doesn't have an ...
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Fraction substraction arithmetic

$$ 3\frac38 - \frac56-1\frac23={}? $$ Can you solve this fraction substraction? I’m getting the wrong answer, on my book the answer is $\frac78$.
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Induction with an existential quantifier

Take as starting point the following "strong and weak" forms of the induction schema $\forall n. \Big(\big(\forall(k< n). \phi(k)\big)\to \phi(n))\Big)\ \to\ \forall m. \phi(m)$ $\Big(\...
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How get the cost price? [closed]

I have the following spreadsheet: For column B, the formula is simple: B1 + (B1 * A2) The question is: how to get the numbers in column C? I manually typed all values on column C but the idea is if I ...
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3 votes
3 answers
106 views

Show that $a^3+5a$ is an integer

I've been given the following task. Let $$a = \sqrt[3]{1+\sqrt{\frac{152}{27}}}-\sqrt[3]{-1+\sqrt{\frac{152}{27}}}$$ Show that $a^3+5a$ is an integer. I tried calculating it by hand but the small page ...
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-2 votes
1 answer
19 views

Ways to split water bill based on person and days

Here is a real-world math problem that I am trying to find a solution. It's kind of similar to Real life water bill problem (want to check myself), but with some additional stuffs. We have 8 ...
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5 votes
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Why counting the number of 1 digits that appear in all integers in 0-9, 0-99, 0-999, 0-9999 follow an arithemtic-geometric sequence?

I noticed that 0-9 = has only 1 '1' 0-99 = has 20 '1's [1,10,11,12,13,14,15,16,17,18,19,21,31,41,51,61,71,81,91] 0-999 = 300 0-9999 = 4000 It follows the formula of n = number of digits in the ...
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Is absolute value an algebraic operation?

What is an absolute value operation in terms of math classification? We study absolute value in algebra class along with other algebraic operations, but if I understand correctly, it doesn't seem to ...
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-1 votes
1 answer
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Is there an algebra equivalent to the matrix cookbook, IE a collection of properties or formulas that you can look at for reference? [closed]

I'm looking for a collection of algebraic theorems, formulas, properties, or "cheat sheet" style shortcuts that are similar to, for example: $$\frac{a-b}{b-a}=-1$$ $$\frac{a}{b}+\frac{f}{g}=\...
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Show that $\forall k,j,m,n \in \mathbb{N}^: 3kn^2 +3jn +3m+2 \notin S=\{x^2 \mid x\in \mathbb{N}\}$

Show that : $$\forall \hspace{2mm} (k,j,m,n) \in \mathbb{N}^4 : 3kn^2 +3jn +3m+2 \notin S=\{x^2 \mid x\in \mathbb{N}\}$$ Clearly this is solvable using modular arithmetic because claiming the ...
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0 votes
1 answer
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If $1/16$ of $x$ is $120$, what is the $1/4$ of $x$? [closed]

If $1/16$ of a pie is $120$ calories, how many calories are consumed if I have $1/4$ of the pie? Would it be $120 \times 4 = 480$?
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3 votes
2 answers
91 views

How to determine what integer value of $j$ returns an integer when plugged in to $\sqrt{5j + 1}$ without trial and error?

I am given $\sqrt{5j + 1}$ and I need to find what integer value of j evaluates this equation to an integer. I could only think of try and error by trying out different values of j incrementally to ...
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1 vote
0 answers
82 views

How to replace two binary additions by one ternary

Consider the "sum" of two intervals of natural numbers $\sigma_i = \langle n_i, k_i \rangle := \{ n_i,n_i+1,\dots,n_i + k_i\} $ being the set of numbers $\sigma = \sigma_1 \oplus \sigma_2 $ ...
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2 votes
0 answers
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Multiplication and Equivalent Geometrical Figure

Just like we visualize multiplication of two numbers as a rectangle, multiplication of three numbers as a cube, how can we visualize the multiplication of four and higher numbers? Is there any ...
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7 votes
4 answers
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Express $|\pi - \dfrac{23}{7}|$ without the absolute value symbol

Express $|\pi - \dfrac{23}{7}|$ without the absolute value symbol. I know I have to check if $\pi - \dfrac{23}{7}$ is greater than (or equal to) zero, but how can I do it analytically (without a ...
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1 answer
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Why is -4 x -4 = +16 and when added its -16 [closed]

When we multiply (-4)(-4) the answer is positive 16 but when we add -4+-4+-4+-4 the answer is negative 16. So is there some logic behind this or there’s something wrong with my calculator.
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1 answer
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Billing distribution

What would be the best mathematical approach to devise a payment distribution method among three people for two separate billable services and if one of the three people did not use the services for ...
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3 answers
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Why does $-\frac{1}{17-x}$ equal $\frac{1}{x-17}$?

Why does $-\frac{1}{17-x}$ equal $\frac{1}{x-17}$? Is there any simple computation to make this seem a little bit more intuitive? Right now, I cannot wrap my head around the fact that I can just ...
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1 vote
2 answers
66 views

Calculating a discount when getting better currency rates [closed]

I have a hard time wrapping my head around this: A particular item costs $10€$ in both shops. Shop A and Shop B accept my currency $C$ as the payment method. In Shop A, $1C$ is currently worth $10€.$ ...
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1 vote
1 answer
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Mercari says something like, "Your purchase is actually half price". But I don't think so. (Keyword: Shopping)

I am very sorry. I forgot an important condition. This campaign is to give back 50% of the points for the purchase amount for only one order during the period. Mercari is a well-known Japanese ...
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1 answer
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A question on compound interest?

$10\%$ compounded quarterly, what is the equivalent rate of interest with monthly compounding? Equivalent rate of interest$= (1+\frac{0.1}{3})^3 -1 =(1+0.033333)^3 -1 =0.1(nearly) =10\%$ Is this ...
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does the minimal $\sf Z_2$ $\omega$-model exist? What is it's recursive axiomatization?

I know the theorem that $\sf Z_2$ is axiomatizable as $\Pi_{\infty}^1-{\sf CA}_0$ and this axiomatization is easily seen to be recursive. Without this theorem, what is the most natural recursive ...
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subsytems of second order arithmetic

Consider the 5 prominent subsystems of second order arithmetic. I would like to know which of these subsystems of second order arithmetic have computably axiomatizable theory.
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2 votes
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A Peano system with an infinite initial segment

Let $T$ be a binary tree with the lexicographic order. And $f:T→T$ be the successor operation. We denote the empty sequence in $T$ by $i$. Also Suppose that: For all $X⊂T$ if these two conditions hold:...
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Understanding Complex Fractions (eg, $\;8/(1/4)\;$) without Simplification

Normal teachings for thinking about fractions like 4/8 or 8/4 goes something like: Use the denominator to tell you the amount of equally sized parts in a whole, and use the numerator to tell you the ...
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Finding Next Occurence of Event

A math exercise of sorts, not work, not school, just some idea I have rolling in my head that I'm trying to get a concrete answer for. A revolutionary item was introduced in 1990, first of its kind. ...
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2 answers
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Proof addition is commutative in $ \mathbb R$

I want to formally prove that $ + $ is commutative in $ \mathbb{R} $. I thought about the following proof so far: Given $ n_1, n_2 \in \mathbb{N} $. $ A = \{x \in \mathbb{N} \mid 1 \le x \le n_1\}$ $ ...
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Am I allowed to add and subtract expressions whose domain might not be $ \mathbb{R} $

We all know the trick of adding and subtracting an expression, for example to calculate: $ \int \frac{2x}{2x + 1} \, dx = \int \frac{2x + 1 - 1}{2x + 1} \, dx = \int \frac{2x + 1}{2x + 1} \, dx + \int ...
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Solve for $x$ in $(x + (a<<s)-(b<<s))>>s = v$

Is there a solution for $x$ in $$(x + (a<<s)-(b<<s))>>s = v$$ for $a,b,s,v$ positive integers? If not exact, is there an approximate solution? I tried $$(x+(a<<s)-(b<<s)) ...
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$a-b = (e + (a<<s)-(b<<s))>>s$, why it does not work for some $e$?

If I subtract 2 numbers shifted left to a number and shift right, I get the original number. Example: $$30 - 1 = ((30<<7) - (1<<7)) >> 7 = 29$$ or in other words $$a-b = ((a<<s)...
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  • 111
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1 answer
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find equation for x halved, y gets doubled

How can I calculate a $y=f(x)$ formula for data like this: x y 10.000 0.016 5000 0.032 2500 0.064 so if $x$ is halved, $y$...
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A common, ambiguous matrix operation (flanking matrices)

There is a certain matrix operation that I've seen performed in several proofs/derivations, but I've never been able to find a clear rule for it. I'm not even sure how to search for it online. It ...
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3 votes
3 answers
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Which one is the larger : $20!$ or $2^{60}$?

Which one is the larger : $20!$ or $2^{60}$ ? I am looking for an elegant way to solve this problem, other than my solution below. Also, solution other than using logarithm that uses the analogous ...
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1 vote
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Deriving addition from various other operations

problem I need to derive addition and/or subtraction from a limited set of mathematical operations: limitations I can do arithmetic with constant values e.g. x * 2 ...
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1 vote
2 answers
157 views

Which branch of math studies this problem?

If we want to calculate the same arithmetic operation on 2 numbers, to say something, the square of $5$ and $7$, we can calculate the square of each one, or we can do this: $$a= 5*1,000,000+7 = 5,000,...
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3 answers
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Do we not need to prove that Terence Tao's definition of the addition of natural numbers is true?

Definition 2.2.1 (Addition of natural numbers). Let $m$ be a natural number. To add zero to $m$, we define $0$ $+$ $m$ := $m$. Now suppose inductively that we have defined how to add $n$ to $m$. Then ...
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2 votes
2 answers
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How do I solve $(68!+68^{86} ) \mathrm{ \ mod \ }75$?

I feel like it's really easy and i'm just misunderstanding something. It's obvious that I need to use Wilson' and Fermat's theorems: $(p-1)!≡-1\mathrm{ \ mod \ }p$ $a^{p-1}≡1\mathrm{ \ mod \ }p$ if $...
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-2 votes
1 answer
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Divide y into x parts

If I have 2 natural numbers x and y. First, I want to know if I can break the number y into x distinct natural numbers. Furthermore, if the answer is yes, then how to find the x distinct natural ...
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