Question

Using Hamilton’s definition of complex numbers, show that complex addition is
both associative and commutative. Show that complex multiplication is both
associative and commutative.

Answer 1

@terenzreignz

Answer 2

@Hunus

Answer 3

What in the world is Hamilton's definition?....
Stand by... googling...

Answer 4

Can't find it... you better tell me what Hamilton's definition is...
All I can find are quaternions...

Answer 5

Or... better call the big guns in :D
@satellite73 ?

Answer 6

@terenzreignz ^here are my notes on Hamilton from my History of Math course. Quaternions are mentioned in the notes also.

Answer 7

okay... it seems, and correct me if I'm wrong (and if I'm wrong you should really be able to spot it... assuming you read your notes... I just skimmed through it)
The definition for complex numbers is
\[\huge \mathbb{C} = \left\{a+bi \left| \quad a,b\in \mathbb{R} \quad , \quad i = \sqrt{-1}\right.\right\}\]

Answer 8

haha well yeah I know that in general from understanding complex numbers @terenzreignz

Answer 9

No need to keep tagging me... just call me TJ, and we'll get along ;)

So, let's so associativity, and let
\[\LARGE \color{green}{z_1 , z_2, z_3} \in \mathbb{C}\]
\[\Large z_1 =a_1+b_1i\]\[\Large z_2 = a_2+b_2i\]\[\Large z_3 = a_3+b_3i\]

Answer 10

*show* associativity

Answer 11

\[\Large (z_1+z_2)+z_3=[(a_1+b_1i)+(a_2+b^2i)]+a_3+b_3i\]

Answer 12

\[\Large = [(a_1+a_2)+(b_1+b_2)i]+a_3+b_3i\]\[\Large =(a_1+a_2+a_3)+(b_1+b_2+b_3)i\]
\[\Large = a_1+b_1i + [(a_2+a_3)+(b_2+b_3)i]\]\[\Large = a_1+b_1i+[(a_2+b_2i)+(a_3+b_3i)]\]

\[\Large \color{blue}{=z_1+(z_2+z_3)}\]

Thus showing associativity

Answer 13

Maybe the commutativity bit, you can do yourself? :)

Answer 14

Okay. Thank you. So for commutativity, do I just show some combination of z1 + z2 + z3 equaling another combination of z1 + z2 + z3?

Answer 15

Just z1 and z2 should be enough.
Just need to show that z1 + z2 = z2+ z1 and that's enough :)

Answer 16

Okay. Cool. And for the multiplication .. should they reflect the addition associativity and commutativity?

Answer 17

@terenzreignz

Answer 18

Yes, they should :)

Answer 19

Okay. Thank you