April, 2018

Eigenvalues

I remember, with a faint feeling of dread, having to calculate the eigenvalues of a matrix. It became routine in the end, but I was recently reminded of the pain when a student asked if there was a shortcut. For a 2-by-2 matrix? Yes. It is up to you, though,

Ask Uncle Colin: A Fractional Kerfuffle

Dear Uncle Colin, I was trying to work out $\frac{\frac{3}{7+h}-\frac{3}{7}}{h}$, and I got it down to $\frac{\frac{3}{h}}{h}$ - but that's not the answer in the book! What have I done wrong? - Likely I've Mistreated It Terribly Hi, LIMIT, and thank you for your message! I'm afraid you're right, you

Wrong, But Useful: Episode 55

This month, it's Gathering4Gardner special, largely recorded in Atlanta with Adam Atkinson and @dragon_dodo. We discuss: Our favourite talks of the event, including: The Juggler Problem The Taxman Problem Ramanujan Sums Optical illusions Doris Schattschneider's talk on Marjorie Rice, an amateur mathematician who made huge leaps in the study of

The Involution of Polynomials

Last time out, I looked at a problem unearthed by @mathsjem - to find the cube root of a degree-six polynomial. This led (unsurprisingly) to a quadratic: $3 + 4x - 2x^2$. When checking whether this was indeed the answer, I hit a problem: is there a simple way to

Ask Uncle Colin: Tangents to a circle

Dear Uncle Colin, I'm told that two lines through $(0,12)$ are tangent to the circle with equation $(x-6)^2 + (y-5)^2 = 17$ and I need to find their equations - but I'm getting in a muddle. Can you help? - Terribly Awkward Numbers, Getting Equations Not Trivial Hi, TANGENT, and

The Evolution of Polynomials

It's always fascinating to see what's going on in textbooks of the olden days, and National Treasure @mathsjem recently found a beauty of its type. Look at those whences! Check out the subjunctives! It thrills the heart, doesn't it?1 What caught my attention, though, was evolution - in this context,

Ask Uncle Colin: A Troublesome Triangle

Dear Uncle Colin, I couldn't make head nor tail of this geometry problem: "If $a:b=12:7$, $c=3$, and $B\hat{A}C = 2 B\hat{C}A$, find the length of the sides $a$ and $b$." - Totally Rubbish In Geometry Hi, TRIG, and thank you for your message! (And don't put yourself down like that,

Using Units to Deal With Density

Glancing over sample papers for the new GCSE, I stumbled on this: Zahra mixes 150g of metal A and 150g of metal B to make 300g of an alloy. Metal A has a density of $19.3 \unit{g/cm^3}$. Metal B has a density of $8.9 \unit{g/cm^3}$. Work out the density of

Ask Uncle Colin: A modulus power

Dear Uncle Colin, How would you work out $9^{41} \pmod{61}$? - Funky Exponential Result, Missed A Tutorial Hi, FERMAT, and thanks for your question! I think the answer is "ponderously"! There are only 61 possible answers (in fact, 60, because you know 61 is not a factor of $9^{41}$). I'd

Sprinkle on the sugar, eat the lot

A Christmas Pudding Puzzle I swear, this one came up in real life! My partner made a Christmas pudding for the most recent festive season. Delicious, it was. When it was about half-eaten, I went to microwave a portion. "Hang on," she said: "there might be a coin in there."