meaning of ordered pairs? (1 Viewer)

[ronaldinho]

in a book they have

(a,b)+(c,d) =(a+c,b+d)

(a,b) X (c,d) = (ac-bd,bc+ad)

how did they do the one in bold?? y are the adding and subtracting etc?

 

[Affinity]

Well it's defined to be that way.

It's the formal definition of complex numbers,
one can't use sqrt(-1) = i to define the complex numbers.

Definition (complex numbers) a complex number is a pair of real numbers (a,b). We Define the following for complex numbers:
1.) (a,b) @ (c,d) = (a+c,b+d)
2.) (a,b) & (c,d) = (ac-bd,bc+ad)
3.) Mod((a,b)) = sqrt(a^2 + b^2)

And by tradition @ is called "addition" and will be denoted by +, & is called multiplication and will be denoted by X.
we also identify each real number r with the complex number (r,0)

and for convenience we write (a,b) as a+ib



By the way which book is it?

 

[Affinity]

Remember + and x are just symbols, they mean what one intends.

 

[Affinity]

Compare with this:

we have order pairs of Integers such that:

(a,b)+(c,d) = (ad + bc,bd)
(a,b)x(c,d) = (ac,bd)

(a,b) Is considered same as (c,d) if and only if ad = bc

what does (a,b) represent?
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Here's something similar, see if you can figure out what it is:

A "__________" is a sequence of real numbers:
(a[0], a[1], a[2], ... )
such that only finitely many are non zero.

assume P=(a[0], a[1], a[2], ... ) and Q=(b[0], b[1], b[2], ... )
Define the following:

P+Q = (a[0] + b[0], a[1] + b[1], a[2] + b[2],...)
P*Q = (a[0]*b[0] , a[0]*b[1] + a[1]*b[0], a[0] * b[2] + a[1]*b[1] + a[2]*b[0],
a[0]*b[3]+a[2]*b[1]+a[1]*b[2]+a[3]*b[0],...)


Now to test your understanding,
for __________ (a[0], a[1], a[2], ... ), a[0] is called the ___________ _______
and the highest n such that a[n] is not zero is called the _____________ of the _______________ and in such case, a[n] is called the ______________. If a[n] is 1, the ______________ is called a ___________ ____________.