Question #138035

Prove that the set of complex numbers is field.


1
Expert's answer
2020-10-13T19:38:49-0400

SOLUTION

The set of complex numbers, CC with addition and multiplication is defined as a field comprising of additive and multiplicative credentials (0,0)(0,0) and (1,0).(1,0).

With the complex number i=(0,1)i = (0,1), it is possible to interchangeably list each complex number as

z=(x,y)=x+(i×y)=x+iyz=(x,y) = x+(i×y)=x+iy

Now, in order to prove that set of complex numbers is field, we need to find the field axioms for the definitions of addition and multiplication i.e.

z+w=(x,y)+(u,v)=(x+u,y+v)z + w = (x,y) + (u,v) = (x+u, y+v)

z×w=(x,y)×(u,v)=(x×uy×v,x×v+y×u)z × w = (x,y) × (u,v) = (x×u - y×v, x×v + y×u)

Now, for the first case; (when (+)(+) and (×)(×) are associative). It is simple for addition, but multiplication requires illustrations that actually a×(b×c)=(a×b)×ca×(b×c)=(a×b)×c

Now, let a=(x1,y1),b=(x2,y2),and,c=(x3,y3)a=(x_1,y_1), b=(x_2,y_2), and, c=(x_3,y_3) such that;

a.(b.c)=(x1,y1).((x2,y2).(x3,y3))a.(b.c) = (x_1, y_1).((x_2, y_2). (x_3, y_3)) =(x1,y1).(x2x3y2y3,x2x3+y2y3)= (x_1, y_1). (x_2x_3 - y_2y_3, x_2x_3 + y_2y_3)

=(x1(x2x3y2y3)y1(x2x3+y2y3),x1(x2x3+y2y3)+y1(x2x3y2y3))=(x_1(x_2x_3 - y_2y_3) - y_1(x_2x_3 + y_2y_3), x_1(x_2x_3 + y_2y_3) + y_1(x_2x_3 - y_2y_3))

=(x1x2x3x1y2y3y1x2x3y1y2y3,x1x2x3+x1y2y3+y1x2x3y1y2y3)=(x_1x_2x_3 - x_1y_2y_3 - y_1x_2x_3 - y_1y_2y_3, x_1x_2x_3 + x_1y_2y_3 + y_1x_2x_3 - y_1y_2y_3)

=(x3(x1x2y1y2)y3(x1y2+y1x2),x3(x1y2+y1x2)+y3(x1x2y1y2))= (x_3(x_1x_2 - y_1y_2) - y_3(x_1y_2 + y_1x_2), x_3(x_1y_2 + y_1x_2)+ y_3(x_1x_2 - y_1y_2))

=(x1x2y1y2,x1y2+y1x2).(x3,y3)= (x_1x_2 - y_1y_2, x_1y_2 + y_1x_2). (x_3,y_3)

=((x1,y1).(x2,y2)).(x3,y3)=((x_1,y_1).(x_2,y_2)) .(x_3,y_3)

=(a.b).c=(a.b).c

For commutative of both (+)(+) and (×)(×), we can prove that

a+b=b+aa+b = b+a and a×b=b×aa×b = b×a

(x1+y1)+(x2+y2)=(x1+x2)+(y1+y2)(x_1+y_1)+(x_2+y_2) =(x_1 +x_2) + (y_1 +y_2)

=(x2+x1)+(y2+y1)=(x_2+x_1) + (y_2+y_1)

=(x2+y2)+(x1+y1)=(x_2+ y_2) +(x_1+y_1)

=b+a=b+a

This also applies to the multiplication, (×)(×)


For the distributive law a×(b+c)=(a×b)+(a×c)a×(b+c)=(a×b)+(a×c) we have;

a×(b+c)=(x1+y1)×([x2+x3]+[y2+y3])a×(b+c)=(x_1+y_1) × ([x_2+x_3]+[y_2+y_3])

=x1[x2+x3]+x1[y2+y3]+y1[x2+x3]+y1[y2+y3]=x_1[x_2+x_3]+x_1[y_2+y_3]+y_1[x_2+x_3]+y_1[y_2+y_3]

=[(x1x2y1y2)+(x1y2+y1x2)]+[(x1x3y1y3)+(x1y3+y1x3)]=[(x_1x_2−y_1y_2)+(x_1y_2+y_1x_2)]+[(x_1x_3−y_1y_3)+(x_1y_3+y_1x_3)]

=(a×b)+(a×c)=(a×b)+(a×c)


For the additive identity, (0,0)(0,0) we have;

0=0+0i.0=0+0i.

That is z1+0=(x1+y1i)+(0+0i)z_1+0=(x_1+y_1i)+(0+0i)

=(x1+0)+(y1+0)i=(x_1+0)+(y_1+0)i

=x1+y1i=x_1+y_1i

=z1=z_1


For the multiplicative identity (1,0)(1,0) we have

1=1+0i1=1+0i

That is z1×1=(x1+y1)(1+0i)z_1×1=(x_1+y_1)(1+0i)

=(x1+x1×0i)+(y1+y1×0)i=(x_1+x_1×0i)+(y_1+y_1×0)i

=x1+y1i=x_1+y_1i

=z1=z_1

For the additive inverse of (x,y)(x,y) gives (x,y)(-x,-y) i.e. Considering;

z1z_1 which is z1=xyi−z1=−x−yi

That is z1+(z1)=(x1+y1i)+(x1y1i)z_1+(−z_1)=(x_1+y_1i)+(−x_1−y_1i)

=(x1x1)+(y1y1)i=(x_1−x_1)+(y_1−y_1)i

=0+0i=0+0i

=0=0


For the multiplicative inverse to (x,y)(x,y) is (x(x2+y2),y(x2+y2))(\frac{x}{(x2+y2)}, -\frac{y}{(x2+y2) }) i.e.

considering complex number z1z_1 where z1=/0z_1 {=}\mathllap{/\,}0

z11=x1y1ix12+y12z_1^{-1} = \frac{x_1−y_1i}{x_1^2+y_1^2}

Now z1×z11=(x1+y1i)×(x1y1ix12+y12)z_1×z_1^{−1} = (x_1+y_1i) × (\frac{x_1−y_1i}{x_1^2+y_1^2})


=(x1+y1i)(x1y1i)x12+y12=\frac{(x_1+y_1i)(x_1−y_1i)}{x_1^2+y_1^2}


=x12x1y1i+x1y1iy1i2x12+y12= \frac{x_1^2−x_1y_1i + x_1y_1i−y_1i^2}{x_1^2+y_1^2}


=x12+y12x12+y12= \frac{x_1^2+y_1^2}{x_1^2+y_1^2}


=1=1

Therefore, Since the set of complex numbers CC used in the proof above

 satisfies the axioms working with the addition and multiplication, it can therefore be concluded that CC is a field.


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