Ethan Bolker
Reputation
1,650
Next privilege 2,000 Rep.
 Oct 5 answered Prove rational sum and product of two irrational numbers Oct 4 answered Prove that Z/12Z is not isomorphic to Z/4Z × Z/6Z Oct 4 comment Explanation of $x\in\mathbb{N}^{+}_{0}$ You should consider using words rather than symbols to say what $x$ must or must not be. That will help both your reader and yourself. Sep 20 comment Proof formatting for an exam This is fine. I hope your professor doesn't require justification for routine algebra, which is the one step you seem worried about. As for your final comment: everything in sight is an integer and you can add and multiply those freely. Again I hope your professor does not require you to justify that. Sep 18 comment Is there more to explain why a hypothesis doesn't hold, rather than that it arrives at a contradiction? +1 I was planning to write an answer like this but now there's no need. The "why" in this particular question is indeed best answered by thinking about distributing a function over addition, which beginners want to invoke often - for squaring, exponentiating, ... Of course it works only rarely. I like to point out that the important place where it does is what they actually know as the "distributive law" for the function $f(x) = cx$, Of course you don't go on to discuss to formal linear operators ... Sep 15 comment How do I make a student understand contradiction? This is logically correct but I doubt its pedagogic value. In my experience truth table arguments don't really help students learn how to understand or construct proofs, Sep 11 comment Bijection between $\Bbb R$ and $S^1$? Continuous? No. Just a function, yes - both sets are uncountable. Sep 8 comment on selling 12 pens for a rupee a seller loses 12%. to earn a profit of 20% how many pens should be sold for a rupee? You can answer your own question - and tell us what you learned while figuring it out. Sep 6 comment Proof that a number and its multiplicative inverse have the same sign Hint. Can you see how to apply the rules you know (i.e. have proved) that tell you the sign of a product when you know the signs of the numbers being multiplied? Sep 5 comment Prove that $a \mid b$ iff $r_i \le s_i$ for every $i$. Hint. What you've proved is the opposite of what your title asks for. To do that one you can find the factor $c$ such that $ac=b$ - what must the powers of the primes in $c$ be? Work out some small numerical examples first if necessary. Sep 4 comment If $A$ is a commutative ring with unity, and let $a \in A$ be a nonzero element, is $\langle a \rangle$ necessarily an ideal of $A$? +1 for doing the research to answer your own question. Next time do the research first, save this site for more interesting things. Aug 23 comment Function of single variable $f(x)$, $f(x+y)=f(xy)$ and the exponential. Two hints, which someone will probably write as an answer before I've finished. 1. Set $x = y= 0$ in the equation $f$ satisfies. That tells you something about $f(0)$. 2. Use the definition of the derivative as the limit of $(f(x+h) - f(x))/h$ as $h$ approaches $0$. Aug 21 comment Calculate the Height of Christmas +1 for noting the importance of refraction. Aug 18 comment A Stupid Question About $O(3)$ Group I tell my students there's no such thing as a stupid question. Perhaps you're puzzled by something that ought to be easy - but for you it isn't, so ask (as you did). That's not stupid. Aug 16 comment If $3x^2 -2x+7=0$ then $(x-\frac{1}{3})^2 =$? You can learn from the two different kinds of solutions below. One strategy is to complete the square and hope to be lucky. That would occur to you if you had some experience knowing when and how to complete the square. The other is to start by multiplying out the sought for square, hoping to find something useful. That's a good strategy if nothing else comes to mind. Aug 16 comment Can we make a subgroup of a group by selecting exactly one element from each distinct left cosets of a subgroup of the given group? You can't for the even integers as a subgroup of the integers. Are you asking if this is ever possible? Aug 14 comment Formal proof of a simple fact, namely that $S$ has even cardinality if certain pairs could idenitifed I agree that formally it's equivalent. In that sense there's no harm in it. But to my taste the direct proof is cleaner. Since not all proofs by contradiction are constructive and students can't appreciate the distinction I want them to look for direct proofs when possible. Aug 14 comment Formal proof of a simple fact, namely that $S$ has even cardinality if certain pairs could idenitifed With this idea you can prove the theorem directly. You don't need the extra baggage of a proof by contradiction. Aug 2 comment What is the significance to our number and degrees systems? Duplicate of duplicate? math.stackexchange.com/questions/340467/… and math.stackexchange.com/questions/142735/…. Found these with a search for why 360 degrees Jul 31 comment Is ideal an “anti-field”? +1 for addressing the intuition behind the OPs question.