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