Complex Numbers + Induction (1 Viewer)

[Riviet]

Need help with the proof of these two rules:
_ _ _ ___________
1) Prove by mathematical induction that z1 + z2 +... + zn = z1+ z2 +...+zn
_ _ _ __________
2) Prove by mathematical induction that z1 x z2... x zn = z1 x z2... x zn

I have had a go at them, i let Z= z1 + z2 +....+ zn and tried to prove it but i dont think the proof was solid enough. Thx for your time

 

[KFunk]

que? ...........

 

[Riviet]

EDIT- those lines were supposed to be on top of the z's
Ill correct it here:

(RANDOM LETTERS RANDOM LETTERS22) _(xx)_(xxxxxx)_ (xx)__________
1) Prove by mathematical induction that z1 + z2 +... + zn = z1+ z2 +...+zn

(RANDOM LETTERS RANDOM LETTERS22) _(xx)_(xxx) _ (x)__________
2) Prove by mathematical induction that z1 x z2... x zn = z1 x z2... x zn

the numbers next to the z's are subscripts (small numbers at bottom right of z)
P.S ignore all the xxxxx, that was just to align the underscores

 

[Antwan23q]

these are some gay questions, what text book are u using?

 

[robbo_145]

agreed, first one doesnt need induction.
second one let
z₁.z₂......z_k = x + iy and
z_k+1= a +bi to solve

 

[Riviet]

I'm using Advanced Mathematics by Terry Lee. Half the text book is like worked solutions lol

 

[Antwan23q]

get the arnolds 4 unit one, its top stuff

 

[Riviet]

Lol, my teacher actually setted us some random questions from the textbook the other day for homework and those 2 induction questions happened to be in the textbook!

 

[noah]

1) assume true for n = k

consider n = k + 1
we must prove:
__ __ _____ ________________
z1 + z2 + ... + z(k+1) = z1 + z2 +... + z(k+1)
__ __ __ ______________
(nB: z1 + z2 + ... + zk = z1 + z2 + ... + z k)
______________ _____
therefore LHS = z1 + z2 + ... + z k + z(k+1)

let Re(z1 + z2 + ... + z k) = a
Im(z1 + z2 + ... + z k) = b
Re(z(k+1)) = c
Im(z(k+1)) = d

therefore:
______________
z1 + z2 + ... + z k = a - ib and
_____
z(k+1) = c - id

therefore

LHS = (a+c) - i(b+d)
___________
= (a+c) + i(b+d)
_____________________
= z1 + z2 + ... + z k + z(k+1)

= RHS


the next question is pretty much the same

Edit: sorry the conjugate lines didn't come out quit right but im sure you get the picture

 

[Riviet]

Aah, i see. Thanks for that, i admit it was a stupid question because it's asking you to prove a simple rule which we already know.