Mathematics Stack Exchange
Mathematics Stack Exchange

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).

Arithmetic is defined as operations upon numbers using $4$ main operations along with many others:

addition - the sum of two numbers

subtraction - the difference of two numbers also defined as the addition of negative numbers

multiplication - the area of a rectangle with sides of lengths equal to the two operands

division - the number of times one number can be subtracted from another before equaling zero. Sometimes it will allow decimals and in other cases there will be a remainder left over when a number doesn't go into another evenly

See Arithmetic.

6376 questions
1100
votes
32 answers

If it took 10 minutes to saw a board into 2 pieces, how long will it take to saw another into 3 pieces?

So this is supposed to be really simple, and it's taken from the following picture: Text-only: It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long will it take for her to saw another board into $3$…
yuritsuki
  • 10,491
300
votes
18 answers

Why does this innovative method of subtraction from a third grader always work?

My daughter is in year $3$ and she is now working on subtraction up to $1000.$ She came up with a way of solving her simple sums that we (her parents) and her teachers can't understand. Here is an example: $61-17$ Instead of borrowing, making it…
user535429
  • 2,167
201
votes
14 answers

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

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}} &= \frac1i \\\\ \sqrt{\frac1{-1}} &= \frac1i \\\\…
Wilhelm
  • 2,213
177
votes
2 answers

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 show that $n < x < n+1$ for some integer $n$. …
MJD
  • 67,568
  • 43
  • 308
  • 617
166
votes
19 answers

Is there another simpler method to solve this elementary school math problem?

I am teaching an elementary student. He has a homework as follows. There are $16$ students who use either bicycles or tricycles. The total number of wheels is $38$. Find the number of students using bicycles. I have $3$ solutions as…
kiss my armpit
  • 3,963
155
votes
41 answers

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 this: Assume $(-x) \cdot (-y) = -xy$ Then divide both sides…
Sev
  • 2,151
154
votes
14 answers

Why does an argument similiar to 0.999...=1 show 999...=-1?

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the below argument that shows $999\ldots = -1$ is…
CommonToad
  • 1,605
151
votes
7 answers

Apparently sometimes $1/2 < 1/4$?

My son brought this home today from his 3rd-grade class. It is from an official Montgomery County, Maryland mathematics assessment test: True or false? $1/2$ is always greater than $1/4$. Official answer: false Where has he gone wrong? Addendum,…
SDiv
  • 2,610
144
votes
18 answers

How to put 9 pigs into 4 pens so that there are an odd number of pigs in each pen?

So I'm tutoring at the library and an elementary or pre K student shows me a sheet with one problem on it: Put 9 pigs into 4 pens so that there are an odd number of pigs in each pen. I tried to solve it and failed! Does anybody know how to solve…
zerosofthezeta
  • 5,710
133
votes
16 answers

Division by zero

I came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions. $\dfrac {x}{0}$ is Impossible ( If it's impossible it can't have neither infinite solutions or even one. Nevertheless, both $1.$ and $2.$ are…
danielsyn
  • 1,575
112
votes
22 answers

Why is there no "remainder" in multiplication

With division, you can have a remainder (such as $5/2=2$ remainder $1$). Now my six year old son has asked me "Why is there no remainder with multiplication"? The obvious answer is "because it wouldn't make sense" or just "because". Somewhat I have…
topskip
  • 1,131
105
votes
4 answers

Factorial, but with addition

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 factorial $5!$ way. EDIT: I know about the formula. I want to…
akinuri
  • 1,393
  • 4
  • 10
  • 17
103
votes
8 answers

Why is $\frac{987654321}{123456789} = 8.0000000729?!$

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 then:) My questions are: Why is this so? What…
marty cohen
  • 110,450
100
votes
1 answer

$4494410$ and friends

The number $4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number. $3883544142410_{10}=3883544E24A_{16}$ is another. These numbers are in…
Ross Millikan
  • 383,099
99
votes
14 answers

Formal proof for $(-1) \times (-1) = 1$

Is there a formal proof for $(-1) \times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed?
tvamsisai
  • 971
1
2 3
…
99 100