MTH633 Group Theory Set 2

Topic No. 2

Properties of Complex Numbers

•  ∀ a+bi, c+di, e+fi ∊ℂ, [(a+bi)+(c+di)]+(e+fi)=

[(a+c)+(b+d)i]+(e+fi)=[(a+c)+e]+[(b+d)+f]i =[a+(c+e)]+[b+(d+f)]i=(a+bi)+[(c+e)+(d+f)i]= (a+bi)+[(c+di)+(e+fi)]

Properties of Complex Numbers

 ∀ a+bi, c+di, e+fi ∊ℂ, [(a+bi).(c+di)].(e+fi)

=[(ac-bd)+(bc+ad)i].(e+fi) =[(ac-bd)e-(bc+ad)f]+[(bc+ad)e+(ac-bd)f]i =[a(ce-df)-b(de+cf)]+[a(de+cf)]+b(ce-df)]i =(a+bi).[(ce-df)+(de+cf)i]=(a+bi).[(c+di).(e+fi)]

(a+bi)+(0+0i)=(a+0)+(b+0)i=a+bi=0+(a+bi)

(a+bi).(1+0i)=(a.1-0b)+(b.1+0.a)i=a+bi=1.(a+bi)

Properties of Complex Numbers

(a+bi)+(-a-bi)=(a+(-a))+(b+(-b))i=0+0i=0=(-a-bi)+(a+bi)

•  For every a+bi ∊ℂ\{0} there exists 1/(a+bi)=a/(a2+b2)-(b/(a2+b2))i ∊ℂ\{0}

such that (a+bi).(a/(a2+b2)-(b/(a2+b2))i ) = (a2+b2)/(a2+b2)+((ab-ab)/(a2+b2))i=1+0i=1 =(a/(a2+b2)-(b/(a2+b2))i )(a+bi)

Properties of Complex Numbers

 ∀ a+bi, c+di ∊ℂ, (a+bi)+(c+di)=(a+c)+(b+d)i =(c+a)+(d+b)i=(c+di)+(a+bi)  ∀ a+bi, c+di ∊ℂ,

(a+bi).(c+di) =(ac-bd)+ (ad+bc)i =(ca-db)+(cb+da)i =(c+di).(a+bi)

Properties of Complex Numbers

Topic No. 3

A binary operation ∗ on a set S is a function mapping S x S into S. For each (a, b) ∈ S x S, we will denote the element ∗((a, b)) of S by a∗b.  Usual addition ‘+’ is a binary operation on the

ℤ+  Usual multiplication ‘.’ is a binary operation on

 Usual multiplication ‘.’ is a binary operation on

Let M(ℝ) be the set of all matrices with real entries. The usual matrix addition is not a binary operation on this set since A+B is not defined for an ordered pair (A, B) of matrices having different numbers of rows or of columns. Usual addition ‘+’ is not a binary operation on the

ℤ\{0} since Let ∗ be a binary operation on S and let H be a subset of S. The subset H is closed under ∗ if for all a, b ∈ H we also have a∗b ∈ H. In this case, the binary operation on H given by restricting ∗ to H is the induced operation of ∗ on H. Usual addition ‘+’ on the set ℝ of real numbers does not induce a binary operation on the set ℝ\{0} of nonzero real numbers because

but 2+(-2)=0 ∉ ℝ\{0}. Thus ℝ\{0} is not closed under +.

Usual multiplication ‘.’ on

a binary operation on the

ℚ+, respectively.

Let be a set and S a b S Let be a set and

on is a rule which assigns to any ordered pair an element S a b S

S

a b a b S