Imaginary Complex Numbers

 An imaginary number is the number in the form bi where b is a non-zero, real number and i, defined by i2 = − 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)

 

Examples for complex numbers:

 

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)`  = `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 − i`sqrt(3)`  (ii)`3/5` i

Solution:

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

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

 

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Example 3:

Find the complex conjugate of (i) 2 + 3, (ii) − 4 − i5 (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 − 3, (ii) − 4 + i5 and (iii) √7 (`+-` the conjugate of any real number is itself).

 

Examples for imaginary numbers:

 

Example 1:

Express the following in the standard form of ib

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

(iii) (2 − 3i) (4 + 2i)

Solution:

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

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

 (iii) (2 − 3i) (4 + 2i) =8 + 4− 12− 6i2 = 14 − 8i

 

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Example 2:

Find the real and imaginary parts of the complex number =4i20 − i19/ 2− 1

=4i20 − i19 / 2− 1

Solution:

(i) =4i20 − i19/ 2− 1

=4(i2)10− (i2)9i/2− 1

=4(− 1)10 − (− 1)9i / − 1 + 2i

=4 + i / − 1 + 2i

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

=[− 4 − 6− − 2i2 ] /  (− 1)2 + 22

=(− 2 − 7i )/ 5 =− 2 / 5 −7 / 5 i

Re(z) = −`2 / 5` and Im(z) =− `7 / 5`

 

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