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

 

(a+bi)+(c+di)=ac+(ad)i+(bc)i+(bd)i2=(ac−bd)+(ad+bc)i(5.1.4)(5.1.4)(a+bi)+(c+di)=ac+(ad)i+(bc)i+(bd)i2=(ac−bd)+(ad+bc)i

 

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 .