That's helpful to hear because I often feel the need to be alone to process my thoughts, and I feel guilty about it. It's like I'm saying my loved ones can't help but it's not that.
I did 15 miles yesterday and the headphones were in the whole time! I know I'm lucky to have beautiful places close by.
Not maths but I'd appreciate input from my mathematical friends.
What do you do when life is tough, what brings you out of a period low mood (or at least helps you manage it)?
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Unauthorised?
Love it!
Inset does have some perks! 😋
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Great to see @jamesmunromaths.bsky.social and @oxfordmathematics.bsky.social offering TMUA prepartion sessions. Private tuition can make a big difference with these tests and anything to help level the playing fields is welcome.
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Great to see this, it means lots of maths applicants, not only those brave enough to go for Oxford, will get additional free support. Thanks @jamesmunromaths.bsky.social 😁
I thought I was the only one banging on about socks and shoes 😂
Without knowledge of complex roots coming in conjugate pairs, is it clear that a quartic can be decomposed into a product of quadratic factors with coefficients over the reals?
This was going to be my original approach if hints needed. I'm on the fence as to whether to wander off track and talk about complex numbers. Perhaps I'll stick to my original plan and save the complex number fun! Thanks for engaging, I don't always get to engage with colleagues about this stuff.
It won't be a straightforward grind for the class because they will need to think about how to deal with the algebraic fraction after substitution. On reflection, I'm tempted to use this as an introduction to defining i^2=-1. Will be nice to factorise x^4+1.
This is brilliant! I can't believe I hadn't thought of using that particular antiderivative of x to simplify things.
Today's tricky integral is a beast...
Integrate sqrt(tan(x)).
Easier if you have knowledge of complex numbers but at this stage my class don't, so that's the real challenge!
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For my class, 2 was much easier. It helped we integrated arctan(x) the day before! With 3 they didn't know what substitution to go for.
I like to the FM back because it forces me to make the single maths more interesting!
Yep, we do single maths in Year 12 and so these are set just as we finish integration. I'm not teaching FM content explicitly but sprinkling it in where it's a natural fit. I've also been a fan of mastering single maths before doing any FM.
The pupils laughed!
Some integration fun with Year 12 FM pupils today!
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I've said it before, if I wasn't tied to Devon I'm certain I'd love working with @sxpmaths.bsky.social
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(z^n-1)/(z-1)=1+z+z^2+...+z^(n-1) for z not equal to one. It's clear that as z approaches 1, this quotient approaches 1+1+...+1=n.
Interesting that a limiting argument is needed for a point on the circle.
We want the product of |1-z_k| and we know that each z_k is a root of z^n-1=0. The expression z^n-1 factorises as f(z)=(z-1)(z-z_1)...(z-z_(n-1)). The product of the relevant lengths is the value of |f(z)/(z-1)|when z=1. L'Hopitals gives the limit as n.
On reflection, you can avoid L'Hopital's...
Love the link to L'Hopital's, great question!
I'm after variety so this is great, thanks.
Is that a STEP 3 question?
Does anyone please have a stash of Complex Number Geometry problems (likely involving roots of unity) they can share with me? Unusually, Madas has none and I've exhausted recent past paper questions. STEP 3 has some but they are too hard for what I want!
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It's wild that P(X=x)=0 for all x. Without any measure theory, how are they supposed to accept that!? I jest, some geometrical probability problems tend to help. Firing an arrow at a target with concentric regions is always a nice example. I like linking probability to length, area and then volume.
My point is that if the coefficients are the same in the denominator the the factorisong shortcut will always work. I don't know why the author of those examples have used two different methods. They didn't need to use different methods for these two questions.
So, for the two examples you give the factorisong trick works but for the below you have to 'realise'. I'm not sure why they've 'realised' in your second example because there is no need.
It's to do with the coefficients in the denominator, if they are equal then the factorisong trick works but if they are different then you need to 'realise' the denominator. Funny, I was just talking to a collegue about this!
