Math Is Fun Forum

I've now tried it and it is very 'messy'.

a = 13i     b = 1     c = -101

Is that sufficient for an answer?  I checked with Wolfram alpha and that's the answer it gave.

If you've got another hour or so to spare, it is possible to find the complex square root of any number using De Moivre's theorem.

First find the modulus and argument of the complex number.

A square root will have a modulus that is the square root of the above mod. and an argument that is half the argument of the above.

That's where it gets messy. I had hoped it would come out to a 'nice, easy' number but it doesn't.

Bob

I've been thinking some more about this.

Keen astronomers get an 'equatorial mount' for their telescope.  The mount can rotate in two ways.  One of the axes is set to point at the pole star.  The other allows the astronomer to point at a star and then follow it as the Earth rotates. The polar axis rotation can automatically rotate at the right speed using a suitable motor.

Following the Sun would need a similar arrangement.  The reason the Sun moves across the sky is totally due to the rotation of the Earth so, once you are pointing at the Sun, you'll need to rotate your device around the polar axis.  The angle between the polar axis and the horizontal is the latitude.

There must be a reasonably simple formula for fixing the angle between the polar axis and a line pointing towards the Sun.  Don't know what it is yet but I'm hopeful I can work it out or look it up if the former fails. It will depend on the date.

Bob

hi KerimF

No to both.  I've made a simple diagram that will start to explain this.  It's definitely NOT to scale. It shows the Earth in orbit around the Sun going anticlockwise. The parallel lines mark the polar axis (ie the axis that the Earth rotates around) with North at the top.  (If you were looking down on the North Pole that rotation is anticlockwise too.)

This means that at mid-summer the northern hemisphere of the Earth is 'leaning' towards the Sun which is why it's warmer then for folks that live there. Sunlight strikes the southern hemisphere at a lower angle so it gets less hot there. 

6 months later the situation is reversed which is why it's summer then in the Southern hemisphere.

The angle between the polar axis and the plane of the Earth's orbit around the Sun (called the plane of the ecliptic) is about 67°.  This year the 20th Of March is the Spring equinox. The exact date varies a bit because of leap years. On that date and at noon the Sun will be exactly overhead the equator and the day length is 12 hours. After that date the Sun is overhead of places north of the equator, gradually 'moving' north to the summer solstice when it is overhead points on the line of latitude called the tropic of Cancer. Then it gradually 'moves' back towards the equator at the Autumnal equinox.  In winter the Sun at noon is lowest in the sky reaching its lowest at the Winter solstice .

So the angle between the Sun and the horizon depends on the observer's latitude and the date in the year.

If you're hoping to point a solar panel directly at the Sun for different dates then I need to know your latitude.

Bob

ps. I don't like the common practice of using the term 12 AM.  AM is an abbreviation of the Latin phrase ante meridiem which translates as 'before noon'. So how can 12 before noon make any sense.  We already have a perfectly good term for this moment in time. It's 12 noon.  PM stand for post meridiem which means after noon.  So for the next 11 hours and 59 minutes it's after noon and at exactly 12 hours later it's 12 midnight and definitely not 12 PM.  In some parts of the world 12 AM and 12 PM have the opposite meanings which also shows why neither term should be used.

Thanks.  Don't laugh ... I'm using Windows 11 on my laptop.

Bob

Every morning the Sun rises in the East and in the evening sets in the west. The Sun appears to move across the sky.  It's all relative.

If you set on a pendulum swinging to and fro you trace out a certain back and forth movement.  But if that pendulum is itself supported by another pendulum which is also swinging the resultant motion is surprising and I think you'll enjoy discovering what it is .

https://www.youtube.com/watch?v=uPbzhxYTioM

Bob

I check out all links that get posted. As far as I can remember my security software jumped in and blocked me from going there.  So I researched without following the link and found warnings about it on the net.

When I put in a link I deliberately don't disguise it at all so everyone can see where I'm sending them.

Almost every day the forum gets inappropriate posts and admin or a moderator has to deal with.  Annoying but it seems to be something we all have to cope with.  If you spot a 'dodgy' post click the 'report' link as it flags it up to us.

Bob

  Yes, that's right.  It's all to do with frames of reference.  From the plane you will see me apparently moving backwards along with the ground, but I think I'm stationary.

So I'll see the ball end up at B as well.

So "The ball starts at position a, and ends up at position b. Absolutely." is the correct version.

Bob

The latitude/longitude coordinate system was devised specifically for navigating on the surface of the Earth.  You can determine your latitude by observing the angle between the Pole star and the horizontal. Latitude is harder; if you can work out the right ascension amd declination of a star and you know the time and date, you can calculate the longitude, but not much help if you're on the Moon or Mars.  No one has devised a coordinate system for the Solar System yet. 

So maybe you should make one.  World fame awaits.

Bob

hi Nick

Welcome to the forum.

I'm thinking you'd want 7 at each venue, for each course.  So person A would meet 6 others each time, that's only 24.  So that won't work.

Can I do it with more folk at some venues and less elsewhere? I suspect not but I'll go back to my spreadsheet and try.

Trivial solution that you won't want: Everyone goes to venue 1, then all on to venue 2 etc etc.

Bob

That's great!  Wow, hasn't time flown by. I seem to remember you were a lot younger when you joined.

Anyhow, it'll be a few days before I get a chance to study it properly.  I'll let you know then how I got on.  Probably have to have a few things explained (to this old timer)

Bob

Just checking: so 3 = 3 = -3   ?

Bob

Cool indeed!  Thanks for posting.

Bob