# Tag: arithmetic

## arithmetic – Why is there no “remainder” in multiplication

The Question : 109 people think this question is useful 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 the feeling

## algebra precalculus – Why is negative times negative = positive?

The Question : 121 people think this question is useful 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 $(-x) \cdot (-y) = -xy$ Then divide both sides by

## algebra precalculus – Division by $0$

The Question : 126 people think this question is useful 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 divided by zero, but only

## arithmetic – Apparently sometimes $1/2 < 1/4$?

The Question : 141 people think this question is useful 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, at the risk of making the

## sequences and series – Why does an argument similiar to 0.999…=1 show 999…=-1?

The Question : 148 people think this question is useful 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 about? Here is

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

The Question : 150 people think this question is useful 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 follows. Using a single

## number theory – Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger.

The Question : 155 people think this question is useful 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 $n$. But

## algebra precalculus – Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$?

The Question : 165 people think this question is useful 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 \\ \\ \sqrt{\frac{-1}1} &= \frac1i \\ \\ \sqrt{-1} &=

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

The Question : 289 people think this question is useful 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

## algebra precalculus – How long will it take Marie to saw another board into 3 pieces?

The Question : 1015 people think this question is useful 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