Browsing category algebra

A STEP expansion

A STEP question (1999 STEP II, Q4) asks: By considering the expansions in powers of $x$ of both sides of the identity $(1+x)^n (1+x)^n \equiv (1+x)^{2n}$ show that: $\sum_{s=0}^{n} \left( \nCr{n}{s} \right)^2 = \left( \nCr{2n}{n} \right)$, where $\nCr{n}{s} = \frac{n!}{s!(n-s)!}$. By considering similar identities, or otherwise, show also that: (i)

Read More

Further Decimal Curiosities

This post is based on work by Mark Ritchings; I know of no finer1 maths tutor in Bury. A few weeks ago, I pointed in the vague direction of a few decimal curiosities -- fractions that spit out lovely patterns in their decimal expansions. Having found one that generated the

Read More

Ask Uncle Colin: Is my friend crazy?

Dear Uncle Colin, A friend of mine told me that $1 + 2 + 4 + 8 + ... = -1$. Is he crazy, or is there something going on here? -- Somehow Enumerating Ridiculous Infinitely Extended Sum Dear SERIES, There are a couple of 'proofs' of this non-fact that

Read More

Ask Uncle Colin: A logarithmic coincidence?

Dear Uncle Colin, I noticed that $2^{\frac{1}{1,000,000}} = 1.000 000 693 147 2$ or so, pretty much exactly $\left(1 + \frac{1}{1,000,000} \ln(2)\right)$. Is that a coincidence? Nice Interesting Numbers; Jarring Acronym Dear NINJA, The easiest way to see that it's not a coincidence is to check out $3^{\frac{1}{1,000,000}} $, which

Read More

Ask Uncle Colin: a disguised quadratic

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to and Uncle Colin will do what he can. Dear Uncle Colin, How do I solve $3x^{\frac{2}{3}} + x^{\frac{1}{3}}-2

Read More

Getting your $x$s in one basket

A reader asks: I need to solve $\frac ac \frac {NP}{N_0 + N} = mP$ for $N$, and I don't know where to start. Help! I had a maths teacher in the early 90s who loved nothing more than making the class groan with bad jokes. If she showed up

Read More


A student asks: How do I simplify horrible product and quotient rule expressions? The example they gave was differentiating: $f(x) = (2x - 3)^4 (x^2 + x + 1)^5$ First up, a careful bit of product rule: $u = (2x - 3)^4$, so $\diff{u}{x} = 8(2x- 3)^3$ $v = (x^2

Read More

The Mathematical Ninja and The $n$th Term

Note: this post is only about arithmetic and quadratic sequences for GCSE. Geometric and other series, you're on your own. Quite how the Mathematical Ninja had set up his classroom so that a boulder would roll through it at precisely that moment, the student didn't have time to ponder. He

Read More

The Attack of the Mathematical Zombie: $(a+b)^2$

An occasional series highlighting common errors that refuse to die. “It just… won’t stay dead!” he said, as the Mathematical Zombie moved closer. “$(a+b)^2 = a^2 + b^2$”, it said. “Brains! $(a+b)^2 = a^2 + b^2$.” “But… it doesn’t!” he said. “You have to multiply out the brackets!” “$(a+b)^2 =

Read More

A multiplication pattern

Long ago on Wrong, But Useful, my co-host @reflectivemaths pointed out the ‘coincidence’ that $7\times8 = 56$ and $12 \times 13 = 156$ - a hundred more. In fact, it works for any pair of numbers that add up to 15: $x(15-x) = 15x - x^2$, and $(x+5)(20-x) = 100

Read More

Sign up for the Sum Comfort newsletter and get a free e-book of mathematical quotations.

No spam ever, obviously.

Where do you teach?

I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.

On twitter