12-240/Classnotes for Thursday September 20

In this class, the professor completes the lecture about complex number and then introduces vector space.

Complex number

Definition and properties

C={(a,b): a, b $\in \!\,$ R}

1 ( of C) = (1,0); 0 ( of C)= (0,0)

i=(0,1)

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

i^2=-1

C contains R as {(a,0)} ( actually, this is not the set of real number but a copy of it )

Political statement

The professor totally disagrees with the name complex number because, indeed, the construction of C is much easier than the construction of R.

From Q ( set of quotient number) we can also construct a set containing i, which has a square equal to -1, and this construction is considered relatively easy Meanwhile, from Q, the construction of R is extremely hard and hence, of course, much more complicated.

Interpretation of complex number

Since complex number has two elements, it can be express in geometric form in coordinate plane

Lecture 4, scanned notes upload by Starash

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