### Week 2 Complex Numbers system

Definition: Complex Numbers

A complex number is an object of the form

a+bi(5.1.1)(5.1.1)a+bi

where aa and bb are real numbers and i2=1i2=−1 .

The form a+bia+bi , where a and b are real numbers is called the standard form for a complex number. When we have a complex number of the form z=a+biz=a+bi , the number aa is called the real part of the complex number zz and the number bb is called the imaginary part of zz . Since i is not a real number, two complex numbers a+bia+bi and c+dic+di are equal if and only if a=ca=c and b=db=d .

There is an arithmetic of complex numbers that is determined by an addition and multiplication of complex numbers. Adding and subtracting complex numbers is natural:

(a+bi)+(c+di)=(a+c)+(b+d)i(5.1.2)(5.1.2)(a+bi)+(c+di)=(a+c)+(b+d)i

(a+bi)+(c+di)=(a+c)+(b+d)i(5.1.3)(5.1.3)(a+bi)+(c+di)=(a+c)+(b+d)i

That is, to add (or subtract) two complex numbers we add (subtract)their real parts and add (subtract) their imaginary parts. Multiplication is also done in a natural way – to multiply two complex numbers, we simply expand the product as usual and exploit the fact that i2=1i2=−1 . So the product of two complex number is

Complex Number Properties

It can be shown that the complex numbers satisfy many useful and familiar properties, which are similar to properties of the real numbers. If uu , ww , and zz , are complex numbers, then

1. w+z=z+ww+z=z+w
2. u+(w+z)=(u+w)+zu+(w+z)=(u+w)+z
3. The complex number 0=0+0i0=0+0i is an additive identity, that is z+0=zz+0=z .
4. If z=a+biz=a+bi , then the additive inverse of zz is z=(a)+(b)i−z=(−a)+(−b)i . That is, z+(z)=0z+(−z)=0 .
5. wz=zwwz=zw
6. u(wz)=(uw)zu(wz)=(uw)z
7. u(w+z)=uw+uzu(w+z)=uw+uz
8. If wz=0wz=0 , then w=0w=0 or z=0z=0 .