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

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113
votes
18answers
11k views

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 ...
92
votes
1answer
11k views

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 ...
84
votes
12answers
11k views

How to convince a math teacher of this simple and obvious fact?

I have in my presence a mathematics teacher, who asserts that $$ \frac{a}{b} = \frac{c}{d} $$ Implies: $$ a = c, \space b=d $$ She has been shown in multiple ways why this is not true: $$ ...
82
votes
24answers
6k views

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 ...
78
votes
1answer
3k views

$4494410$ 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. ...
76
votes
18answers
8k views

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 ...
71
votes
14answers
4k views

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?
70
votes
5answers
2k views

Root Calculation by Hand

Is it possible to calculate and find the solution of $ \; \large{105^{1/5}} \; $ without using a calculator? Could someone show me how to do that, please? Well, when I use a Casio scientific ...
57
votes
5answers
13k views

Why is 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 ...
48
votes
8answers
3k views

The last digit of $2^{2006}$

My 13 year old son was asked this question in a maths challenge. He correctly guessed 4 on the assumption that the answer was likely to be the last digit of $2^6$. However is there a better ...
47
votes
10answers
3k views

Division by $0$

I thought it was elementary to me, but I started to do some exercises and came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions. $\dfrac {x}{0}$ is ...
45
votes
9answers
8k views

$\sqrt a$ is either an integer or an irrational number.

I got this interesting question in my mind: How do we prove that if $a \in \mathbb N$, then $\sqrt a$ is an integer or an irrational number? Can we extend this result? That is, can it be shown ...
43
votes
11answers
21k views

Can a piece of A4 paper be folded so that it's thick enough to reach the moon?

While procrastinating around the web I stumbled on a page that contained the image below, from cracked.com. I can't help but believe that this is false… Even though the article header says: ...
43
votes
7answers
2k views

Computing $999,999\cdot 222,222 + 333,333\cdot 333,334$ by hand.

I got this question from a last year's olympiad paper. Compute $999,999\cdot 222,222 + 333,333\cdot 333,334$. Is there an approach to this by using pen-and-paper? EDIT Working through on paper ...
43
votes
15answers
13k views

Why is negative times negative = positive?

Someone recently asked me why a negative * a negative is positive, and why a negative * a positive is negative, etc. I went ahead and gave them a proof by contradiction like so: Assume $(-x) * (-y) ...
43
votes
11answers
4k views

How to round 0.4999… ? Is it 0 or 1?

If you want to round the repeating decimal 0.4999... to a whole number what is the right answer? Is it 0 or 1? On the one hand you only look at the first digit when you round numbers which in this ...
41
votes
12answers
5k views

Is it wrong to tell children that 1/0 = NaN is incorrect, and should be ∞?

I was on the tube and overheard a dad questioning his kids about maths. The children were probably about 11 or 12 years old. After several more mundane questions he asked his daughter what 1/0 ...
38
votes
11answers
3k views

Why are addition and multiplication commutative, but not exponentiation?

We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not. My question is why. As background there are plenty of mathematical schemes that ...
33
votes
10answers
5k views

Is it possible to simulate a floor() function with elementary arithmetic?

I'm using a "programming language" that only allows basic operations: addition, subtraction, multiplication, and division. Is it possible to emulate a floor function (i.e. drop the decimals a number) ...
33
votes
7answers
1k views

Bad Fraction Reduction That Actually Works

$$\frac{16}{64}=\frac{1\rlap{/}6}{\rlap{/}64}=\frac{1}{4}$$ This is certainly not a correct technique for reducing fractions to lowest terms, but it happens to work in this case, and I believe there ...
30
votes
6answers
2k views

$-1$ is not $ 1$, so where is the mistake?

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}} ...
26
votes
14answers
6k views

Logic behind dividing negative numbers

I've learnt in school that a positive number, when divided by a negative number, and vice-versa, will give us a negative number as a result.On the other hand, a negative number divided by a negative ...
26
votes
13answers
5k views

How to avoid arithmetic mistakes?

When dealing with several numbers and long equations, it's common to make careless arithmetic mistakes that give the wrong answer. I was wondering if anyone had tips to catch these mistakes, or even ...
25
votes
17answers
21k 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 ...
25
votes
4answers
1k views

What is to geometric mean as integration is to arithmetic mean?

The arithmetic mean of $y_i ... y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i $$ For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and using ...
23
votes
5answers
4k views

In primary school I was showed this. Why does it work?

When I was in primary school a teacher showed us the following exercise in arithmetic. Take any 3 digit number between 201 and 998 provided that the hundreds digit is bigger than the ones digit and ...
23
votes
2answers
864 views

A strange little number - $6174$.

Take a 4 digit number such that it isn't made out the same digit $(1111, 2222, .. . $ etc$)$ Define an operation on such a four digit number by taking the largest number that can be constructed out of ...
22
votes
7answers
1k views

Sum of the sum of the sum of the first $n$ natural numbers

I have here another problem of mine, which I couldn't manage to solve. Given that: $$x_n = 1 + 2 + \dots + n \\ y_n = x_1 + x_2 + \dots + x_n \\ z_n = y_1 + y_2 + \dots + y_n $$ Find ...
22
votes
5answers
949 views

a big number that is obviously prime?

I once heard it asserted that $91$ is the smallest composite number that is not obviously composite. The reasoning was that any composite number divisible by $2$, $3$, or $5$ is obviously composite, ...
22
votes
1answer
2k views

How would one be able to prove mathematically that $1+1 = 2$?

Is it possible to prove that $1+1 = 2$? Or rather, how would one prove this algebraically or mathematically?
22
votes
3answers
988 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 ...
22
votes
1answer
812 views

Is it possible $n(n+1)(n+2)…(n+k)$ is a square?

Let $n,k$ two integers greater than $1$, is it possible that $n(n+1)(n+2)...(n+k)$ is a square $m^2$, with $m$ an integer ? Thanks in advance.
22
votes
6answers
824 views

If $(\sqrt{y^2-x}-x)(\sqrt{x^2+y}-y)=y$ then $x+y=0$.

Let $x,y$ be real numbers such that $$\left(\sqrt{y^{2} - x\,\,}\, - x\right)\left(\sqrt{x^{2} + y\,\,}\, - y\right)=y$$ Show that $x+y=0$. My try: Let ...
21
votes
9answers
4k views

What is the purpose of Stirling's approximation to a factorial?

Stirling approximation to a factorial is $$ n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n. $$ I wonder what benefit can be got from it? From computational perspective (I admit I don't ...
20
votes
16answers
3k views

Why is $n$ divided by $n$ equal to $1$?

I've understood how to apply this operation for many years but I recently was reminded that: $n^0 = 1$ Which in high school I accepted and just "solved for", but I'm now curious. Why is that true? ...
20
votes
10answers
3k views

Get $5$ by doing any operations with four $7$s

How can one combine four sevens with elementary operations to get $5$? For example $$\dfrac{(7+7)\times7}{7}$$ (though that does not equal $5$). I am not able to do this. Can you solve it or prove ...
20
votes
5answers
8k views

How can I find the square root using pen and paper?

Okay, I know this is very basic question. I learned 2 methods in school. But now, I forget one. Here is a simple method that I know. Find the prime divisors of the number Omit the half of numbers ...
19
votes
7answers
2k views

Subtracting Quarters of Squares Equals Multiply?!

Can anyone explain to me how/why this works (hopefully in mostly layman's terms)? It seems pretty magical to me at the moment. $${{(a+b)^2\over4} - {(a-b)^2\over4}} = a b.$$
19
votes
10answers
2k views

Does there exist a system such that the additive identity is non-zero?

I am trying to explain how although the additive identity is written as $0$, it is not the same as the number $0$. For example for a $2\times 2$ matrix the additive identity is $\begin{pmatrix} 0 ...
19
votes
6answers
426 views

$\lim_{n\to\infty}\sqrt{6}^{\ n}\underbrace{\sqrt{3-\sqrt{6+\sqrt{6+\dotsb+\sqrt{6}}}}}_{n\text{ square root signs}}$

We have the following representation of pi: $$\pi=\lim_{n\to\infty}2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\dotsb+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}}}}}}_{n\text{ square root ...
18
votes
6answers
2k views

If multiplication is not repeated addition

What is it? How do you define multiplication? All of us had to memorize the multiplication table in elementary school, but how did they come up with it if so many people claim that it is wrong to ...
18
votes
5answers
1k views

Rule for multiplying rational numbers

The rule for multiplying rational numbers is this: $\space\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$ Can the rule be proven or is it meant to be taken as a given? Edit: Where $b\neq 0$ and $d\neq ...
18
votes
3answers
256 views

What combinatorial quantity the tetration of two natural numbers represents?

Tetration is a generalization of exponentiation in arithmetic and a part of a series of other generalized notions, Hyperoperators. Consider $m\uparrow n$ denotes the tetration of $m$ and $n$. i.e. ...
17
votes
11answers
3k views

Why is $\frac{1}{\frac{1}{X}}=X$?

Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$ And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't ...
17
votes
10answers
2k views

Is there any simple method to calculate $\sqrt x$ without using logarithm

Suppose that we don't know logarithm, then how we would able to calculate $\sqrt x$, where $x$ is a real number? More generally, is there any algorithm to calculate $\sqrt [ n ]{ x } $ without using ...
17
votes
4answers
1k views

Why not to extend the set of natural numbers to make it closed under division by zero?

We add negative numbers and zero to natural sequence to make it closed under subtraction, the same thing happens with division (rational numbers) and root of -1 (complex numbers). Why this trick ...
17
votes
7answers
2k views

Why do we stop at exponentiation stage in arithmetic of natural numbers?

In natural numbers the unary successor operator $S$ is the most natural function which maps each number to the next one. Furthermore we may consider the binary relation $+$ as an iteration of $S$. ...
17
votes
2answers
640 views

How to define addition through multiplication?

One might define multiplication $\bullet$ on $\mathbb Z$ as follows: $\bullet: \mathbb Z\times \mathbb N\ni (a,b) \mapsto a+\cdots+a\in \mathbb Z$ where we add $b$ times. But suppose we are in a ...
17
votes
2answers
652 views

Multiplication tables with all entries distinct

Let positive integers $\alpha$ and $\beta$ be given. It is easy to find sets $A$ and $B$ of positive integers such that: $|A|=\alpha$ and $|B|=\beta$ The set $P = \{ab : a\in A, b\in B\}$ contains ...
16
votes
5answers
4k views

Is 'no solution' the same as 'undefined'?

Today in class my teacher wrote something along the lines of: $6^x = 0$ And proceed to heed a response from the class. A few people shouted undefined. So the teacher then writes: no solution ...