# Browsing category ask uncle colin

## Ask Uncle Colin: Why Do I Have To Learn All This?

Dear Uncle Colin, Why do I have to learn all these theorems? Why is it important that we don’t know the value of $\pi$? Does that have any application in, say, rocket science? What can I do to learn things in a less repetitive way? - Ridiculously Onerous, Can’t Know

## Ask Uncle Colin: An Inequality Implication

Dear Uncle Colin, Does $a > b$ imply $\frac{1}{b} > \frac{1}{a}$? - Inequality: Mightily Perplexing Logically, Yeah? Hi, IMPLY, and thanks for your message! On the face of it, that seems sensible, doesn’t it? It’s tempting to say something like $1 > \frac{b}{a}$, so $\frac{1}{b} > \frac{1}{a}$. With an equation,

## Ask Uncle Colin: a surprising sum

Dear Uncle Colin, Apparently, $\sum_1^\infty \left(\frac{n^2}{2^n}\right) = 6$, which surprised me. Can you explain why? - Some Intuition Gone Missing, Aaargh! Hi, SIGMA, and thanks for your message! I’m not sure I can give you intuition, but I can explain where the result comes from. Let’s start from the binomial

## Ask Uncle Colin: Do You *Have* To Be Such A Smart-Arse?

Dear Uncle Colin, Why do you insist on giving smart-arse answers to simple puzzles on Twitter? - Malice Establishes Almost Nothing Hi, MEAN, and thanks for your message! First up, please don’t confuse my cheerful ragging of questions with malice; on the contrary, coming up with alternative approaches, unexpected ways

## Ask Uncle Colin: What shape is this?

Dear Uncle Colin, The question says that $z_1$, $z_2$, $z_3$ and $z_4$ are distinct complex numbers representing the vertices of a quadrilateral ABCD, in order. Further, $z_1 - z_4 = z_2 - z_3$ and $\arg\left( \frac{z_4 - z_1}{z_2 - z_1}\right) = \piby 4$. The shape is supposed to be one

## Ask Uncle Colin: How did they do this so quickly?

Dear Uncle Colin, At a Mathscounts competition, a contestant was asked “how many six-digit positive integers are divisible by 1000 but not by 400?”. Within four seconds, they correctly answered 450 – how on earth could they do that so quickly? - Mathematical Evaluations Normally Take A Lot Longer, Yes?

## Ask Uncle Colin: A Bogus Proof

Dear Uncle Colin, I’ve been shown a proof that goes like this: To show: $a^n = 1$ for all non-negative integers $n$ and for all non-zero real numbers $a$. Proceed by induction. Base case: $a^0 = 1$ by definition, so the base case holds. Inductive step: Suppose $a^j = 1$

Dear Uncle Colin, I’m told there are two circles that touch the x-axis at the origin and are also tangent to the line $4x-3y+24=0$, but I can’t find their equations. Any ideas? - A Geometrically Nasty Example Seems Impossible Hi, AGNESI, and thanks for your message! I’m going to start

## Ask Uncle Colin: Angles and roots

Dear Uncle Colin, In my non-calculator paper, I’m told $\cos(\theta) = \sqrt{\frac{1}{2}+ \frac{1}{2\sqrt{2}}}$ and that $\sin(\theta) = -\left(\sqrt{\frac{1}{2}-\frac{1}{2\sqrt{2}}}\right)$. Given that $0 \le \theta \lt 2\pi$, find $\theta$. I’ve no idea how to approach it! - Trigonometric Headaches Evaluating This Angle Hi, THETA, and thanks for your message! My third thought

## Ask Uncle Colin: A Calculator Error

Dear Uncle Colin, I have to work out $\cot\left( \frac{3}{2}\pi \right)$. Wolfram Alpha says it’s 0, but when I work out $\frac{1}{\tan\left(\frac{3}{2}\pi\right)}$, my calculator shows an error. What’s going on? - Troublesome Angle, No? Hi, TAN, and thanks for your message! The cotangent function is slightly unusual in that it