# Find the multiplicative inverse of the complex number 4 - 2i.

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find the multiplicative inverse of 4 - 2i.

Now the multiplicative inverse of any number X is defined as X^-1, so that X*X^-1 = 1.

For 4 - 2i, let the multiplicative inverse be M.

Now M* (4 - 2i) = 1

=> M = 1/ (4 -2i)

=> M = (4 +2i)/ ( 4- 2i)( 4 +2i)

=> M = (4 + 2i) / [4^2 - (2i)^2]

=> M = ( 4 + 2i) / [ 16 + 4]

=> M = (4 + 2i) / 20

=> M = 4/20 + 2i /20

=> M = 1/5 + i/10

Therefore the multiplicative inverse of 4 - 2i is 1/5 + i/10.

neela | High School Teacher | (Level 3) Valedictorian

Posted on

To find the multiplicative inverse of the complex number.

We know the multiplication identity element of the complex number (x+yi) = e(x+iy) .

Therefore xe = x and  and xi*e = xi. Therefore e = 1.

Therfore if the  multiplicative inverse of  4-2i is x+yi, then

(4-2i)(x+yi) = 1.

4x+4yi-2xi -2yi^2 = 1+0*i

4x +(4y-2x)i +2y = 1+0*i.

(4x+2y) +(4y-2x)i = 1+0*i.

Equate imaginary parts and equate real parts both sides:

Imaginary parts: 4y -2x = 0.....(1)

Real Parts: 4x+2y = 1.....(2)

From (1): 4y- 2x = 0.

x= 2y. Substitute x= 2y in eq (2):

4(2y) +2y = 1

10y = 1.

y = 1/10.

x =2y= 2/10.

Therefore (4-2i) has the multiplicative inverse  2/10+i/10.

4y + 2(2y) =

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The multiplicative inverse is get when we multiply the given complex number by the inverse number we'll get the result 1.

We'll note the inverse as x:

(4-2i)*x = 1

We'll divide by (4-2i) both sides:

x = 1/(4-2i)

We are not allowed to keep a complex number to denominator, so we'll multiply the ratio by the conjugate of the complex number:

x = (4+2i)/(4-2i)(4+2i)

We'll re-write the denominator as a difference of squares:

(4-2i)(4+2i) = 4^2 - (2i)^2

(4-2i)(4+2i) = 16 - 4i^2

But i^2 = -1:

(4-2i)(4+2i) = 16 + 4

(4-2i)(4+2i) = 20

We'll re-write x:

x = (4+2i)/20

x = 4/20 + 2i/20

We'll simplify and we'll get:

x =  1/5 + i/10

The multiplicative inverse of the complex number 4 - 2i is 1/5 + i/10.