MATH1131 Lecture Notes - Lecture 10: Complex Plane, Ternary Numeral System, Verb
3 Jun 2018
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t
.3 Arithmetic with complex rrumbers
Before we go oa to learn how to do arithmetic with com-
plex numtrers, we aeed to lealn a bit of terminolory.
Definition 2.9. Let # urfh o,6 e R. The
realpart offieimaginary
part of
Exarnple 2.1O. Find
r Re(-3+i5): -J g
r Im(-S + rs) :5 *
.=+i5: -i-i5 u
Now, we will learn how to add, subtract and divide com-
plex numbers. Luckily, this is not hard: everythlng be-
I
haves just like you rvould erryect, with only one exception:
every time you get an i2, remember that this equals -1.
Let z: a*rb and w: c*i,elwrth o,&,c,d € IR.
Addition: z t w : (a+Lb) + Cc+zd)
=(n+c)+ilbtd) a
tr o[rol fi61 Ard ua;r4tt.lut! f4t&)
Subtraction: z - 11) -- (a+ vb) - (c*Ld)
= La_() +(e-l)L +
Multiplication: zu): (a+ ab) (c+fA)
. - Ac + qdL+ bcl *{Zt
-- ac + ^AL+ u'i -bA.
- ( ec**ba) '* t*d t adt +
Example 2.11. I.et z: L - ,i,w :3 + i2. Calculate
t 2z: z(t - i) = 2-21
,* : (r-t)( 3 +2 t) =,rr-,:;,r, -','
=t-ZB
z_ (l-it {3-i})
d- ifr' x-t?+i
- -5:a
- 1+*
ts r-
S 3-&r
rr ra- S
(2 + z)(i, - w) :(r+ t r- it)(i-{ 3 +,1))
: (a-1)t-i-;)
= - (r -i)( 3+ i)
' *d1+ '1
_*f0+
10
Division: 1: #^# , rrratlply by w
(A,
acclrli * cLi- -bdL'i
ea -.di * cdj-- Sz""t
tLar+l,l) + Ccb -a4) i
e*+ d'
{i#,.+
Note that &bove, we used the,fa[ordrg usefui fa*:
rdfr: ( c+aL) G*d>) = C'+ l' *
t1
ffiffi&'*
12
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