An imaginary number is the number in the form bi where b is a non-zero, real number and i, defined by i^{2} = − 1, is known as the imaginary unit. An imaginary number bi
can be added to the real number a to form a complex number of the form a + bi, where a and b are called respectively, the "real part" and the "imaginary part" of the complex number a + bi.
Imaginary numbers can be thought of as complex numbers where the real part is zero. The square of an imaginary number is negative real number.(Source.Wikipdia)

*Example 1:*

** **Write the following as complex numbers are (i) − √36 (ii) 3 −√ − 8

**Solution:**

(i) `sqrt(-36)` = `sqrt((-1) * (36))` = `sqrt(-1)` * `sqrt(36)` = *i `sqrt(36)` *= i 6

(ii) 3 − `sqrt(-8)` = 3 − `sqrt((-1) * (8))` = 3 − `sqrt(-1)` `sqrt(8)` = 3 − *i`sqrt(8)` *

*Example 2:*

** **Write the real and imaginary parts of the following number is (i) 4 −

**Solution:**

(i) Let *z *= 4 − *i √*3 ; *Re*(*z*) = 4, *Im*(*z*) = −√ 3

(ii) Let *z *=`3/5` *i *; *Re*(*z*) = 0, *Im*(*z*) =`3 / 5`

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*Example 3:*

Find the complex conjugate of (i) 2 + *i *3, (ii) − 4 − *i*5 (iii)√ 7

**Solution:**

By definition, the complex conjugate is obtained by the reversing the sign of the imaginary part of the complex number. Hence the required conjugate is

(i) 2 − *i *3, (ii) − 4 + *i*5 and (iii) √7 (`+-` the conjugate of any real number is itself).

**Example 1:**

Express the following in the **standard form** of *a *+ *ib*

(i) (3 + 4*i*) + (− 7 − *i*) (ii) (8 − 6*i*) − (2*i *− 8)

(iii) (2 − 3*i*) (4 + 2*i*)

**Solution:**

(i) (3 + 4*i*) + (− 7 − *i*) = 3 + 4*i *− 7 − *i *= − 4 +3 *i*

(ii) (8 − 6*i*) − (2*i *− 8) = 8 − 6*i *− 2*i *+ 8 = 16 − 8*i*

(iii) (2 − 3*i*) (4 + 2*i*) =8 + 4*i *− 12*i *− 6*i*2 = 14 − 8*i*

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*Example 2:*

Find the real and imaginary parts of the complex number *z *=4*i*^{20 }− *i*^{19}/ 2*i *− 1

*z *=4*i*^{20} − *i*^{19} / 2*i *− 1

**Solution:**

(i) *z *=4i^{20} − *i*^{19}/ 2*i *− 1

=4(*i*^{2})^{10}− (*i*^{2})^{9}*i*/2*i *− 1

=4(− 1)^{10} − (− 1)^{9}*i* / − 1 + 2*i*

=4 + i / − 1 + 2*i*

=(4 + i/− 1 + 2*i* )×(− 1 − 2*i* / − 1 − 2*i)*

=[− 4 − 6*i *− *i *− 2*i*^{2} ] / (− 1)^{2} + 2^{2}

=(− 2 − 7*i* )/ 5 =− 2 / 5 −7 / 5 *i*

*Re*(*z*) = −`2 / 5` and *Im*(*z*) =− `7 / 5`

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