complex numbers

complex numbers §

for any imaginary numbers $bi$, $(bi{)}^2=-b^2$ now allow addition and subtraction of imaginary numbers $bi$ and real numbers $a$:

e.g. 2+3i$$$$-6 + 1.5i $2-\frac{9}{7}i$ each of these expressions are unable to be written more simply, so they are called complex numbers. ($a+bi$ where and and b are both real)

the word ‘complex’ here means ‘having more than one part’

for a complex number $z=a+bi$

a = Re(z), the real part of z (Re is getting the real part of the number z) b = Im(z), the imaginary part of z (Im is getting the imaginary part of the z)

THE IMAGINARY PART OF A COMPLEX NUMBER IS A REAL NUMBER????????!!!!!!!!!!! SHOCKING there are mathematical reasons for this definition, but its unfortunate it is just defined this way :(

e.g. if $z=\frac{1}{2}-6i$ then $Re(z)=\frac{1}{2}$ and $Re(z)=-6$

this is applicant in the solutions of quadratic equations

arithmetic with complex numbers §

for +, -: treat $i$ just like a variable. (2+i)+(6-4i)$$$$=2+6+i-4i=8-3i conjugate complex numbers and multiplication and division argand diagrams