Express the complex number 2 + i ( 1 + i ) ( 1 - 2 i ) , in the standard form a + i b .

Let z = 2 + i ( 1 + i ) ( 1 - 2 i ) = 2 + i 1 - 2 i + i - 2 i 2 = 2 + i 1 - 2 i + i + 2 ∵ i 2 = - 1 = 2 + i 3 - i = 2 + i 3 - i × 3 + i 3 + i (multiply and divide by conjugate of denominator) = 6 + 2 i + 3 i + i 2 9 - i 2 = 6 + 2 i + 3 i - 1 9 + 1 ∵ i 2 = - 1 = 5 + 5 i 10 So, the standard form is 1 2 + 1 2 i .

Express the complex number 2 + i ( 1 + i ) ( 1 - 2 i ) , in the standard form a + i b .

Let z = 2 + i ( 1 + i ) ( 1 - 2 i ) = 2 + i 1 - 2 i + i - 2 i 2 = 2 + i 1 - 2 i + i + 2 ∵ i 2 = - 1 = 2 + i 3 - i = 2 + i 3 - i × 3 + i 3 + i (multiply and divide by conjugate of denominator) = 6 + 2 i + 3 i + i 2 9 - i 2 = 6 + 2 i + 3 i - 1 9 + 1 ∵ i 2 = - 1 = 5 + 5 i 10 So, the standard form is 1 2 + 1 2 i .

Express the complex number ( 3 - 2 i ) ( 2 + 3 i ) ( 1 + 2 i ) ( 2 - i ) , in the standard form a + i b .

Let z = ( 3 - 2 i ) ( 2 + 3 i ) ( 1 + 2 i ) ( 2 - i ) = 6 + 9 i - 4 i - 6 i 2 2 - i + 4 i - 2 i 2 = 6 + 9 i - 4 i + 6 2 - i + 4 i + 2 ∵ i 2 = - 1 = 12 + 5 i 4 + 3 i = 12 + 5 i 4 + 3 i × 4 - 3 i 4 - 3 i (multiply and divide by conjugate of denominator) = 48 - 36 i + 20 i - 15 i 2 16 - 9 i 2 = 48 - 36 i + 20 i + 15 16 + 9 ∵ i 2 = - 1 = 63 - 16 i 25 So, the standard form is 63 25 - 16 25 i .

Express the complex number 3 + - 1 2 - - 1 , in the standard form a + i b . Also find the conjugate and modulus of each number.

Let z = 3 + - 1 2 - - 1 = 3 + i 2 2 - i 2 ∵ i 2 = - 1 = 3 + i 2 - i = 3 + i 2 - i × 2 + i 2 + i (multiply and divide by conjugate of denominator) = 6 + 3 i + 2 i + i 2 4 - i 2 = 6 + 3 i + 2 i - 1 4 + 1 ∵ i 2 = - 1 = 5 + 5 i 5 = 1 + i . So, the standard form is 1 + i . Therefore, the conjugate of 1 + i is 1 - i . We know that, the modulus of any complex number x + i y is x 2 + y 2 . Hence, the modulus of a complex number 1 + i is 1 2 + 1 2 . ⇒ 1 + i = 2 .

Express the square of i 1 + i in the form x + i y .

According to the question, x + i y = i 1 + i 2 ⇒ x + i y = i 2 1 + i 2 ⇒ x + i y = i 2 1 + i 2 + 2 i ⇒ x + i y = - 1 1 - 1 + 2 i ∵ i 2 = - 1 ⇒ x + i y = - 1 2 i ⇒ x + i y = - 1 2 i × i i ⇒ x + i y = - i 2 i 2 ⇒ x + i y = - i - 2 ∵ i 2 = - 1 ⇒ x + i y = 0 + 1 2 i .

Show that 7 + 3 i 7 - 3 i + 7 - 3 i 7 + 3 i is real.

Let z = 7 + 3 i 7 - 3 i + 7 - 3 i 7 + 3 i ⇒ z = 7 + 3 i 2 + 7 - 3 i 2 7 2 - 3 i 2 ⇒ z = 7 2 + 3 i 2 + i 2 . 7 . 3 + 7 2 + 3 i 2 - i 2 . 7 . 3 7 - 3 i 2 ⇒ z = 7 + 3 i 2 + 7 + 3 i 2 7 - 3 i 2 ⇒ z = 14 + 6 i 2 7 - 3 i 2 ⇒ z = 14 - 6 7 + 3 ∵ i 2 = - 1 ⇒ z = 8 10 ⇒ z = 4 5 Since the imaginary part is zero. Therefore, it is a real.