Ask Uncle Colin: Some number theory | Colin

Dear Uncle Colin,

I need to show that $\sqrt{7}$ is in $\mathbb{Q}[\sqrt{2}+\sqrt{3}+\sqrt{7}]$ and I don’t really know where to start.

We Haven’t Approached Tackling Such Questions

Hi, WHATSQ, and thanks for your message!

I am absolutely not a number theorist, although I must admit to getting a bit curious about it recently.
What does that notation mean?

In number theory, you can extend a field1 by throwing extra numbers or symbols into it. For example, the complex numbers can be written as $\mathbb{R} [i]$ – the real numbers, with the extra number $i$ added in such that $i^2 =…

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A moment of neatness | Colin

Working through an FP2 question on telescoping sums (one of my favourite topics – although FP2 is full of those), we determined that $r^2 = \frac{\br{2r+1}^3-\br{2r-1}^3-2}{24}$.

Adding these up for $r=1$ to $r=n$ gave the fairly neat result that $24\sum_{r=1}^{n} r^2 = \br{2n+1}^3 – 1 – 2n$.

Now, there are (at least) two ways to go here. The way I would habitually have done this, until fairly recently, was to expand the bracket and then factorise.

That’s not too awful (in fact, the expansion was part a) of the question), but it turns out to be more work than you need to do.


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Ask Uncle Colin: Powers | Colin

Dear Uncle Colin,

I’m ok with my basic power laws, but I don’t understand why $x^0$ is always 1, and I get mixed up when it’s a fraction or a negative power. Can you help?

Running Out Of Time

Hi, ROOT, and thanks for your message!

If it’s any consolation, you’re not alone — this is something that I struggled with when I was learning, and it’s a really common problem with my GCSE and A-level students.
First and biggest bit of advice

My first and biggest bit of advice, when you see a power you’re unfamiliar with: slow down and think. If I had a list of Important Mathematical…

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Ten great books to give an interested mathematician | Colin

Oh no! Your favourite mathematician has a birthday/Christmas/other present-giving occasion coming up and you don’t know what book to get them! They’ve already got Cracking Mathematics and The Maths Behind, obviously… so what can you give them this year?

Fear not, dear reader. I am at hand to list some of my favourite maths books (ordered alphabetically by author).

How Many Socks Make a Pair? by Rob Eastaway: Rob is an absolute hero of popular maths writing, and it’s always a delight to see him take an everyday phenomenon and turn it into a puzzle.

The Indisputable Existence Of…

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Ask Uncle Colin: Some Rigour Required | Colin

Integration by substitution, rigorously

Dear Uncle Colin,

Can you explain why integration by substitution works? I get that you’re not allowed to ‘cancel’ the $dx$s, but can’t see how it works otherwise.

– Reasonable Interpretation Got Our Understanding Ridiculed

Hi, RIGOUR, and thanks for your message.

First up, confession time: me and rigour go together like peanut butter and lawnmowers. We are not a natural combination, and any attempt to apply one to the other is likely to end in a mess – or at least anguished screams from @realityminus3. But, since you asked, I’ll do my…

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The Mathematical Ninja and the Cube Root of 4 | Colin

The student swam away, thinking almost as hard as he was swimming. The cube root of four? The square root was easy enough, he could do that in his sleep. But the cube root?

OK. Breathe. It’s between 1 and 2, obviously. What’s 1.5 cubed? The Mathematical Ninja isn’t going to like that – three-halves, that’s what we need. Cube that, it’s $\frac{27}{8}$, or 3.375. Quite a long way short. But – aha! (Breathe) $1.6^3 = 4.096$, should have got that straight away. That’s close enough for two lengths.

The student popped his head out of the water. “About 1.6, sensei.”

The Mathematical Ninja…

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Ask Uncle Colin: A Ridiculous Restriction | Colin

Dear Uncle Colin,

In a recent test, I was asked to differentiate $\frac{x^2+4}{\sqrt{x^2+4}}$. Obviously, my first thought was to simplify it to $\br{x^2+4}^{-\frac{1}{2}}$, but I’m not allowed to do that: only to use the quotient rule and the fact that $\diff {\sqrt{f(x)}}{x} = \frac{f'(x)}{2f(x)}$.

When Evaluating, Inappropriate Rules Demanded

Hi, WEIRD, and thanks for your message – that is indeed a weird restriction to put on this question! I would do it exactly the way you suggest. However, let’s play along.
Without any simplification

Using the quotient rule,…

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