# Month: July 2021

## Equivalent Definitions of e

(A new question of the week) It is not unusual for mathematicians to define a concept in multiple ways, which can be proved to be equivalent. One definition may lead to a theorem, which another presentation uses as the definition, from which the original definition can be proved as a theorem. Here, in yet another …

## Finding a Function Value Recursively

(A new question of the week) May was a particularly good month for interesting questions! Here is one requiring us to find one value of a function, based on an unusual property: If $$a+b=2^x$$, then $$f(a)+f(b)=x^2$$. The problem turned out to be not as hard as it looked, yet the function itself is quite interesting …

## Separable Differential Equations

(A new question of the week) We received a couple different questions recently about solving differential equations by separation of variables, and why the method is valid. We’ll start with a direct question about it, and then look at an attempt at an alternate perspective using differentials.

## Long Division with Zero, Revisited

(A new question of the week) One of our first posts, in 2018, was about zeros in long division. But we still get many questions about this issue, and it’s time to dig in deeper. We’ll look here at two of them, answering the twin questions, “When do you put a zero in the quotient …

## Two Integration Puzzlers

Two recent questions (that came to us within two hours) dealt with apparent contradictions in integration. The first seems to give a result of zero that is clearly wrong; the second seems to give two different results for the same integral.