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February 1st, 2014, 11:37 AM   #1
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Meaning of + in x+iy???

Very simple question on very difficult topic.
What does mean + in x+iy???
If you to say some kind of formality then I would say
"Oh, real? And what about multiplicity?@
Please, try to find out the answer, fear I don't know either.
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February 1st, 2014, 12:08 PM   #2
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Re: Meaning of + in x+iy???

Let's decypher my thoughts
On one of the Russian lection course books on high math book 1 page 40
was written: before use any graph, don't forget about the mastab! (Or something alike )
And what about our case?
For me seems impossible to plus to numbers one with mas 1 and another wih mas i = sqrt(-1).
And how about you, my friends (if you let me )?
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February 1st, 2014, 12:09 PM   #3
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Re: Meaning of + in x+iy???

Let's decypher my thoughts
On one of the Russian lection course books on high math book 1 page 40
was written: before use any graph, don't forget about the mastab! (Or something alike )
And what about our case?
For me seems impossible to plus to numbers one with mas 1 and another wih mas i = sqrt(-1).
And how about you, my friends (if you let me )?
mac_alleb is offline  
February 1st, 2014, 12:34 PM   #4
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Re: Meaning of + in x+iy???

By the way, lay out mas and so on.
You about integrals and Jordan's Lemma? (ha-ha!, sorry)
Draw for yourself circle (very round (zic!)) and cross it by diameter (for instance).
Conclusions are all after you.
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February 2nd, 2014, 05:05 AM   #5
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Re: Meaning of + in x+iy???

It's hard to decipher what you are saying. I don't know what "mastab" means or what "mas 1" and "mas i" mean. Is that "plus"?

But you seem to be asking what the "+" in the notation for a general complex number, a+ ib, could mean since we have not, when that notation is first introduced, have a way of "adding" a real and an imaginary number.

Part of your difficult is that the way complex numbers are introduced in "elementary" courses is NOT the way they are rigorously defined by mathematicians. Rather, we define the complex numbers as pairs of real numbers, (a, b), with addition defined by (a, b)+ (c, d)= (a+ c, b+ d) and multiplication defined by (a, b)*(c, d)= (ac- bd, ad+ bc). Then we have (a, 0)+ (b, 0)= (a+b, 0) and (a, 0)*(b, 0)= (ab, 0) so we can think of the real numbers as "imbedded" in the complex numbers by x--> (x, 0). We also have (a, 0)*(c, d)= (ac-0*d, ad+ 0*c)= (ac, ad) so we can think of the single real number, x, times a complex number, (a, b), as meaning (ax, bx). Further (0, 1)*(0, 1)= (0*0- 1*1, 1*0+ 0 *1)= (-1, 0) so (0, 1) squared is equal to the complex number we identify with -1. If we label that pair "i" we are saying that "[tex]i^2= -1[/iex]. Using those definitions, we can write the general complex number (a, b) as (a, 0)+ (0, b)= a(1, 0)+ b(0, 1)= a+ bi. The "+" is in that sense: defined as (a, b)+ (c, d)= (a+ c, b+ d).
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