Show that every nonzero complex number can be written in one

Show that every nonzero complex number can be written in one and only one way as the product of a positive real number and a number of absolute value one.

Solution

let that nonzero complex number be z = x+iy

it means x and y can not both take value of 0 at the same time.

now let \'r\' is a positive real number

and a number of absolute value \'1\' can be taken as cos + i sin

now product of these two will be r ( cos + i sin )

now if x+ i y = r ( cos + i sin )

then x = r cos , and y = r sin

since cos and sin can not both be zero at the same time for any value of

and r is positive real number

so, x = r cos , and y = r sin can not both be zero at the same time for any value of r and

hence, z = x+iy can be equivalently written as r( cos + i sin )

now on contrary to the statement, let us assume that another non zero complex number, say, u = a + i b is also represented by   r( cos + i sin )

then,  x + i y = r( cos + i sin ) = a + i b

so, x + i y = a + i b

which gives a = x and b = y

which proves that r( cos + i sin ) represents one and only one no zero complex number.