Fsc Part 1 Mathematics (Complete Solution)

Q2
verify the multiplication properties of the complex numbers

Solution:

Let z1= u + vι  ,   z2= w + xι   ,and   z3= y + zι

(i) z1z2=(u + vι)(w + xι)

=uw + uxι + wvι + vxι2

=(uw - vx) + (ux + wv)ι

thus z1z2 is a complex no, so the closure law of multiplication holds in the set of complex numbers C.

(ii) The multiplicative identity is (1,0) i.e,

(x,y).(1.0) =(a.1-b.0,b.1+a.0)=(a,b)

so the it holds in the set of complex numbers C.

(iii) Every non-zero complex number {i.e,number not equal to (0,0)}has a multiplicative inverse.

that multiplicative inverse of (x,y) is

(
x
x2+y2
,
-y
x2+y2
)

(x,y) multiply with multiplicative inverse of(x,y),so its equal to (1,0), the identity element

(iv) (u,v)[(w,x)+-(y,z)]=(u,v)(w,x)+-(u,v)(y,z)

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