Friday was a much-needed quiet day at home ๐
We started off with a list of things to do and we actually managed to tick more than half of them off ๐
One of them was collecting the parcel we failed to collect yesterday, but that was complicated by the fact that another slip had appeared on Thursday saying that a different parcel had failed to be delivered and that they would try again on Friday morning, so we were hoping that that one would arrive early enough for us to get to the depot before it closed at 1:00. It arrived at 3 ๐ so we still have a parcel waiting for us, which K is convinced will be a birthday present for him, as it should have arrived on his big day.
There was lots of good schools TV on, so we watched that whilst emptying and reloading the dishwasher (actually J supervised the d/w emptying while I was feeding and changing A; I came back to find it all done except for a few plates and mugs whose homes the children could not reach) and putting on and hanging out washing.
Double Maths and English happened without (much) fuss, as promised on Thursday (it was a condition of going to play with Z ๐ ) and we also finally got a folder sorted out with stuff from the Terracotta Army trip we did a few weeks ago. J, K and L each dictated while I typed so we got a page on each of their chosen topics: technology, power and animals respectively. Then they drew things to cut out and stick on to decorate the folder (I still need to do the actual sticking ๐ณ ) and we put everything safely inside ๐ With almost impeccable timing, the parcel which finally arrived in the afternoon, having been promised in the morning, included three little terracotta warriors to dig out of plaster blocks, so that was Saturday morning’s activity sorted ๐
Also managed to package up and post off the nappies I am swapping for a frontbackandsides carrier, which was just as well, as the mei tai arrived just before my side of the bargain left ๐ณ It looks good though (much nicer irl than in the photo I had seen) and might just be a keeper. Must get rid of a few others though… Anyone need a sling or two? I seem to have exchanged a cloth nappy addiction for a carrier addiction ๐ณ
J’s English included riddles to work out, which made him think of a question he had seen in Murderous Maths and assumed to be a riddle or a joke: what number is the square root of -1? I said I didn’t know and he told me with glee that it was i, then waited for me to explain why it was funny. I didn’t. I couldn’t! We waited for Daddy to get home… Teatime conversation included a long chat, including paper and pens, about i…
Most of Saturday was spent doing sundry bits of DIY, such as putting up new coat hooks in the hall (Bob, supervised by A and me) and having fun creating clouds of plaster dust in the garden (J, K and L) whilst digging out little warriors. Photos hopefully on flickr soon.
Today we went to local, rather than old, church, although this was not what we had intended. It should have been an all age service so we had intended to go back for it, but the preacher didn’t realise, so had not prepared. Meanwhile there was no Sunday School planned because it was all age worship, so a steward had phoned round on Sat evening to see if anyone was coming with children, we were the last regular family and the only ones planning to come and therefore said we would make things simpler by not going ๐ Actually the service we did go to was very good and fellowship good also, so I think it was perhaps as well that we were there today – except that as well as Bob doing the website I have now been reminded about local preaching (good, I guess), asked to go to meetings about the future direction of worship at the church (again, I guess that’s good really, but a little eek!) and also asked to consider taking over the church newsletter/diary (big eek!) because I look organised (huh?) the children are nice (well, mostly) and I come recommended (? by whom???) – the woman who currently does it (but is leaving in July) said it takes her a few hours a month and she comes in at 10 one Friday a month to print and collate, which takes a while but it’s okay because you can do it while the children are at school – er, when would that be then? I think we’re being sucked in and I’m not sure how I feel about that! At least it’s not Sunday School – I’ve done that at every church so far and I’ve had enough!
Oh and J lost a tooth. In the middle of the children’s address, which was about waiting – so he gave the example of waiting for a tooth to come out ๐ Which reminds me, I must go and check he’s left it somewhere easy for the tooth fairy to get to ๐
This afternoon we went to a fab birthday party at the gym where the boys do gymnastics. The equipment was all out as usual, but with a few extra soft play bits as well, and 30 children were let loose for an hour! It was great! L was able to have a go at all the things she normally only gets to see the boys do and Bob and I were able to watch them doing things I normally only see from the balcony and Bob not at all ๐ It was lovely to see how confident and daring J and K have become in such a short time.
Home for bath and early bed – all round I think ๐
I think you could have explained here too! Dredging the depths of my memory didn’t work so googled and found a comparison of “i” and God. Have now decided to buy: “Imagining Numbers (Particularly the Square Root of Minus Fifteen)” by Barry Mazur
“At one point in Ars Magna, Cardano found it necessary to invoke a square root of -15. รขโฌหYou will have to imagine รขหลก-15,รขโฌโข he tells the reader. Mazurรขโฌโขs book is a fascinating and clearly written introduction to imaginary numbers, and an exploration of the relationship between maths and the imagination.”
It’s a read blogs and buy books day today.
sounds lovely.
funny — i’ve been roped in to do our parish newsletter.
excuse typing — cuddling at keyboard!
I’m aware of quite how many people with maths degrees might read this, but what the hey. I also had to refer to my A level Further Maths text book, as I was rusty on all but the basics.
i is defined as รขหลก-1, so i*i = -1. This may seem a bit of a cop out, but with this single cop out you get quite a lot. E.g. รขหลก-4 = 2i because 2 * i * 2 * i = (2*2) * (i*i) = 4 * -1 = -4.
Also, that number line you may have taught / be teaching to your children is actually just the x axis of a number plane. The x axis describes the real numbers, and the y axis describes the imaginary numbers – so you can have the normal number 2 as actually 2 + 0i (or (2,0) as co-ordinates on the plane), 7 lots of i is 0 + 7i (or (0,7) as co-ordinates) and also things like 3 + 2i – numbers that have a non-zero value for the real and imaginary parts are called complex.
The wacky bit, but that makes maths nice, is when i forms a bridge between trigonometry and raising numbers to powers. This may make your head hurt if it isn’t doing so already. But it does produce a lovely little equation at the end that contains lots of mathematical goodness.
I’m going to wave my hands over the detail a bit here, but take my word that the following 2 equations are true:
1. cos x = 1 – (x^2)/2! + (x^4)/4! – (x^6)/6! + …
2. sin x = x – (x^3)/3! + (x^5)/5! – (x^7)/7! + …
Where x^y is x to the power y, and e.g. 5! = 5 * 4 * 3 * 2 * 1 = 120 and 3! = 3 * 2 * 1 = 6 etc. You also have to make sure that x is expressed in terms of radians (2รโฌ radians is 360 degrees) or these equations aren’t true.
If you add them together you get
sin x + cos x = 1 + x – (x^2)/2! – (x^3)/3! + (x^4)/4! + (x^5)/5! – …
which is a lot like
e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! …
If you’ve not come across e before (in the context of maths, rather than drugs or expressions of Northern delight), it’s a number like รโฌ that is a never-ending decimal number that starts 2.71828183. It’s a fundamental constant for describing things like compound interest and so on (like รโฌ is for describing circles).
Anyway, if you just twiddle things a bit you get
e^ix = 1 + ix + ((ix)^2)/2! + ((ix)^3)/3! + …
= 1 + ix – (x^2)/2! -i(x^3)/3! + (x^4)/4! + i(x^5)/5! – …
= cox x + i sin x
the last line is by pulling out alternate things i.e. all the things that start with i go together, and then all the other things go together.
Given that
sin (90 degrees) = sin (รโฌ radians) = 1
and
cos (90 degrees) = cos (รโฌ radians) = 0
then you get
e^(iรโฌ) = 1
with a bit of rearranging this is
e^(iรโฌ) – 1 = 0
which gives you all the constants you need to do an awful lot of maths in one simple package. If you have a maths degree, go and breath deeply until my naive mistakes don’t annoy you any more. If you don’t have a maths degree, go and lie in a darkened room until your head doesn’t hurt any more. ๐
Sorry – I forgot to say that with J I stopped before the wacky bit. Also, I sprung the e^x=… thing from nowhere – another thing to just take my word for or look up. ๐
i was wondering when J glazed over!