## Example 1.1.7 (a): Simple Complex Numbers |

Find the additive and multiplicative identities in |

The additive identity is easily found to be *(0, 0)*. As for
the multiplicative identity: we are looking for the complex number
*(u,v)* that does not change anything under multiplication, i.e.
for which:

(x,y)*(u,v)=(x,y)

for all *(x,y)*. According to the definition of multiplication:

(x,y)*(u,v) = (xu - yv, xv+yu) = (x,y)

According to the definition of equality this results in a system of two equations:

(1) xu - yv = x

(2) xv + yu = y

Multiplying the first equation by *y*, the second by *x*
gives:

xyu - y^{2}v = xy

x^{2}v + xyu = xy

Now we subtract the equations to get:

v(x^{2}+y^{2}) = 0

Since this has to be true for all real *x,y* we have that
*v=0*. Substituting that in equation (1) gives *u=1*.
Therefore, the multiplicative identity is:

(1, 0)