Im having trouble with this concept (1 Viewer)

[bos1234]

The Complex Numbers Form a Field (coroneos page 7)
Dear Sirs,
I have read this page in the coroneos book but I am stumped.

Could someone please explain to me these terms and what are the siginificance of these results?

Or is there any website which explains these ideas nicely

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What does E F mean? exists in a field?

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thank you in advancement

Regards,
Bos1234

 

[darkliight]

the wierd looking e means 'in'.

Think of F as a set. Much like the natural numbers are a set. The number 10 is 'in' the set of natural numbers, but the number -10 is not. instead of writing a whole lot of 'in's (they crop up everywhere) we just use that e symbol, but still just say it as in.

So, back to the fields, a+b in F means that given ANY a and b in F, then a+b is also in F. You might think "so what?" but this isn't always true. Going back to our natural numbers N, think 2 - 1 is in N, but 2 - 3 in not. So the natural numbers are NOT closed under subtraction. They are closed under addition though

I've not seen the funny 0, but since he is talking about a set G now, he might have moved onto a more abstract concept called a group (though they share alot in common). Maybe you can give us some more context? It looks like he has just defined his own operation (we are not restricted to the usual + and *). Did he not use c on purpose, or did you not copy it correctly?

Edit: some websites (wiki is pretty good)
* http://en.wikipedia.org/wiki/Field_(mathematics)
* http://en.wikipedia.org/wiki/Group_(mathematics)
* http://en.wikipedia.org/wiki/Naive_set_theory#Sets.2C_membership_and_equality

 

[bos1234]

the wierd looking e means 'in'.


So, back to the fields, a+b in F means that given ANY a and b in F, then a+b is also in F. You might think "so what?" but this isn't always true. Going back to our natural numbers N, think 2 - 1 is in N, but 2 - 3 in not. So the natural numbers are NOT closed under subtraction. They are closed under addition though

But how can natural numbers be closed under addition?

how about 2 + 1/2 ?

Thanks for your help once again.. Im slowly getting there

45 percent left

p.s those websites are good!

 

[darkliight]

but 1/2 isn't in N. Remember, its for all a, b in N For the subtraction example I gave above, 2 and 3 are both in N, but 2 - 3 isn't in N, hence, not closed under subtraction (it gives us something 'outside' of N).

 

[bos1234]

ahhhhhhh right right kkk

i should never question a premium member

 

[darkliight]

Premium member = $10 or something, don't accept anything I tell ya on blind faith. If it doesn't make sense, ask.

 

[bos1234]

Ok first ill try and understand "fields" with real numbers. Is it something like this?



So in this image R is a field
Q is a subset of R
and Z is a subset of R

Z will remain inside that small circle under subtraction and addition.. if you divide etc it will go onto the next circle?

 

[darkliight]

Yep. Z is also closed under multiplication. But as you say, as soon as you divide, you're not guaranteed to stay inside the circle.

As a point of interest, N is inside the Z circle, and C (the complex numbers) enclose all of these circles.

Q, R and C are all fields (you can try and prove that C is a field if you want). Z is not a field because there is no multiplicative inverse, and N is not a field because .. well a lot of reasons I'm sure you can think of at least one of the axioms that fail.

That is pretty much all the numbers you will come across in the HSC, so nice work getting a good idea of how they all fit together in one night!

 

[bos1234]

ok dear sir.. thanks very much for your help in making me understand

pretty interesting stuff

goodnight and goodluck to u

p.s is your uncle very good at maths?

cya bye!

 

[bos1234]

oh wait.. jsut ONE MORE QUESTION please

this is the last one and then i wont bother you

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whats the meaning of multiplicative inverse and additive inverse?



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This is the final of the final one..

The elements of a field F obey the following axioms

Does this sentence mean..

If you take a field R then the 11 axiom laws will be obeyed WITHIN that field?

 

[darkliight]

In the other thread, look at point number 3 in my post .. one for addition, another for multiplication.

An example, 1/2 is the multiplicative inverse of 2, because (1/2)*2 = 2*(1/2) = 1 (the multiplicative 'identity'), which satifies point 3

We only say additive inverse and multiplicative inverse because, usually, they are completly different numbers (the additive inverse of 2, for instance, is -2 because ....).

 

[Trebla]

Additive inverse is basically a complex number with signs reversed (i.e. the negative of a complex number). For example, -a - ib is the additive inverse of a + ib.
Multiplicative inverse is basically the reciprocal of a complex number. For example the multiplicative inverse of a + ib is 1/(a + ib) which by realising the denominator becomes (a - ib)/(a² + b²).
You don't really need to know these in the exam...

 

[bos1234]

ok thanks trbela and darkliight

thats enuf for one day

goodnight goodluck goodbye and God bless!

 

[milton]

the additive inverse of x is -x since x + -x = 0 (additive identity)
the multiplicative inverse of x is 1/x (this is also known as reciprocal) since x * 1/x = 1 (multiplicative identity)

most of this is really obvious stuff that you would have intuitively learnt in primary school but never really realised it. its just a formalisation of that.

 

[bos1234]

true

those big words confuse my brain

 

[dongypro]

to be honest.. who cares what they are.. theyre not even gonna be in the test

just look at simple definitions .. theyre not gonna be like classify this number i^7

u might have to evaluate it .. but u really only need to know what a complex number is compared to other numbers ie. real numbers, irrational, etc...

 

[bos1234]

i just like to know

 

[Slidey]

It's good that you want to know, bos1234. It comes up in first year mathematics. Maybe second year if your uni is slow. It's abstract algebra, although you'll probably cover it a bit in linear algebra.