# June, 2014

## Why the Maclaurin series gives you Pascal’s Triangle

The Mathematical Ninja, some time ago, pointed out a curiosity about Pascal's Triangle and the Maclaurin1 (or Taylor2 ) series of a product: $\diffn{n}{(uv)}{x} = uv^{(n)} + n u'v^{(n-1)} + \frac{n(n-1)}{2} u'' v^{(n-2)} + ...$, where $v^{(n)}$ means the $n$th derivative of $v$ - which looks a lot like Pascal's

## Which is larger?

A MathsJam classic question asks: Without using a calculator, which is bigger: $e^\pi$ or $\pi^e$? It's one of those questions that looks perfectly straightforward: you just take logs and then... oh, but is $\pi$ bigger than $e\ln(\pi)$? The Mathematical Ninja says "$\ln(\pi)$ is about 1.2, because $\pi$ is about 20%

## A numerical curiosity

A numerical curiosity today, all to do with $\i$th powers. Euler noticed, some centuries ago, that $13({2^\i + 2^{-\i}})$ is almost exactly $20$. As you would, of course. But why? And more to the point, how do you work out an $\i$th power? It's all to do with the exponential

## Why is $\arcosh$ the positive root?

A student asks: I know the method for finding the hyperbolic arcosine1 - but I get two roots out of my quadratic formula. Why is it just the positive one? A quick refresher, in case you don't know the method Hyperbolic functions are the BEST FUNCTIONS IN THE WHOLE WIDE

## Complex mappings

Just for a change, an FP3 topic. I've been struggling to tutor complex mappings properly (mainly because I've been too lazy to look them up), but have finally seen - I think - how to solve them with minimal headache. A typical question gives you a mapping from the (complex)

"But I don't liiiiike fractions," said the student. He also didn't like the look of the poker in the Mathematical Ninja's hand, which was beginning to glow red. "Sure you do," said the Mathematical Ninja. "Do I?" "How do you do percentages?" He swished the poker around a bit, as

## There’s More Than One Way To Do It: Arithmetic Series

You've got the formulas in the book, of course. $u_n = a + (n-1)d$ $S_n = \frac n2 \left(a + L\right) = \frac n2 \left(2a + (n-1)d\right)$ This is somewhere the book and I have a serious disagreement: as a mathematical document, it ought to define its terms. $a$ is

## Wrong, But Useful: Episode 15

It's the (slightly delayed) monthly chat between @reflectivemaths (Dave Gale) and me on whatever maths has caught our eyes. This month: Why protractors and set squares? You can find centres of rotation. Why constructions? Colin launches an impassioned defence and compares them to Killer Sudoku Dave has some great ideas

## A student asks: Why is there a $+c$ when you integrate?

A student asks: We've just started integration and I don't understand why there's always a $+c$ - I understand it's a constant, I just don't understand why it's there! Great question! The simple answer is, because constants vanish when you differentiate, they have to appear when you integrate - it's

## Secrets of the Mathematical Pirate: Switcheroos

"Don't tell the Mathematical Ninja," said the Mathematical Pirate. The student shook his head enthusiastically. "Narr!" "You've got $\frac {7}{x} = 14$. Ask yourself: what would the Mathematical Ninja do?" "The Mathematical Ninja would do something that looked extremely dangerous and terrifying, but was completely under control." "Correct!" said