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**Week 2 Complex Numbers system**

Definition: Complex Numbers

A* complex number* is an object of the form

*a*+

*b*

*i*(5.1.1)(5.1.1)a+bi

where *a*a and *b*b are real numbers and *i*2=−1i2=−1 .

The form *a*+*b**i*a+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*+*b**i*z=a+bi , the number *a*a is called the **real part** of the complex number *z*z and the number *b*b is called the **imaginary part** of *z*z . Since i is not a real number, two complex numbers *a*+*b**i*a+bi and *c*+*d**i*c+di are equal if and only if *a*=*c*a=c and *b*=*d*b=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*+

*b*

*i*)+(

*c*+

*d*

*i*)=(

*a*+

*c*)+(

*b*+

*d*)

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

*a*+

*b*

*i*)+(

*c*+

*d*

*i*)=(

*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 *i*2=−1i2=−1 . So the product of two complex number is

*a*+

*b*

*i*)+(

*c*+

*d*

*i*)=

*a*

*c*+(

*a*

*d*)

*i*+(

*b*

*c*)

*i*+(

*b*

*d*)

*i*2=(

*a*

*c*−

*b*

*d*)+(

*a*

*d*+

*b*

*c*)

*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 *u*u , *w*w , and *z*z , are complex numbers, then

*w*+*z*=*z*+*w*w+z=z+w*u*+(*w*+*z*)=(*u*+*w*)+*z*u+(w+z)=(u+w)+z- The complex number 0=0+0
*i*0=0+0i is an additive identity, that is*z*+0=*z*z+0=z . - If
*z*=*a*+*b**i*z=a+bi , then the additive inverse of*z*z is −*z*=(−*a*)+(−*b*)*i*−z=(−a)+(−b)i . That is,*z*+(−*z*)=0z+(−z)=0 . *w**z*=*z**w*wz=zw*u*(*w**z*)=(*u**w*)*z*u(wz)=(uw)z*u*(*w*+*z*)=*u**w*+*u**z*u(w+z)=uw+uz- If
*w**z*=0wz=0 , then*w*=0w=0 or*z*=0z=0 .