# 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 **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