Question

Use the following relations to solve problem 1 and 2 z = x + jy Rectangular form z = r/? Polar form (9.15) z = re^{j?} Exponential form The relationship between the rectangular form and the polar form is shown in Fig. 9.6, where the x axis represents the real part and the y axis represents the imaginary part of a complex number. Given x and y, we can get r and ? as r = ?(x² + y²), ? = tan?¹(y/x) (9.16a) On the other hand, if we know r and ?, we can obtain x and y as x = r cos ?, y = r sin ? (9.16b) Thus, z may be written as z = x + jy = r/? = r(cos ? + j sin ?) (9.17) Figure 9.6 Representation of a complex number z = x + jy = r/?. Addition and subtraction of complex numbers are better performed in rectangular form; multiplication and division are better done in polar form. Given the complex numbers z? = x? + jy? = r?/?? z? = x? + jy? = r?/?? the following operations are important. Addition: z? + z? = (x? + x?) + j(y? + y?) (9.18a) Subtraction: z? - z? = (x? - x?) + j(y? - y?) (9.18b) Multiplication: z?z? = r?r?/?? + ?? (9.18c) Division: z?/z? = r?/r?/?? - ?? (9.18d) Reciprocal: 1/z = 1/r/-? (9.18e) Square Root: ? z = ? r/?/2 (9.18f) Complex Conjugate: z* = x - jy = r/-? = re??? (9.18g) Note that from Eq. (9.18e), 1/j = -j (9.18h) Alexander, Charles K. Fundamentals of electric circuits / Charles K. Alexander, Matthew N. O. Sadiku. — 5th ed.p. cm 1. Solve and write the following complex numbers in rectangular standard form( In the form a ± jb) i. (4 - 5j)(12 + 11j) ii. (-3 - j) - (6 - 5j) iii. (1 + 4j) + (-16 + 9j) iv. 8j(10 + 2j) v. (-3 - 9j)(1 + 10j) vi. (2 + 7j)(8 + 3j) vii. 4 - j2 + 10j/(4-j) viii. 1 + 5j/(8+3j) ix. (6+7i)/(3+5i) x. i²? 2. Convert all the answers into polar form

          Use the following relations to solve problem 1 and 2
z = x + jy Rectangular form
z = r/? Polar form (9.15)
z = re^{j?} Exponential form

The relationship between the rectangular form and the polar form is shown in Fig. 9.6, where the x axis represents the real part and the y axis represents the imaginary part of a complex number. Given x and y, we can get r and ? as
r = ?(x² + y²), ? = tan?¹(y/x) (9.16a)

On the other hand, if we know r and ?, we can obtain x and y as
x = r cos ?, y = r sin ? (9.16b)

Thus, z may be written as
z = x + jy = r/? = r(cos ? + j sin ?) (9.17)

Figure 9.6
Representation of a complex number z = x + jy = r/?.

Addition and subtraction of complex numbers are better performed in rectangular form; multiplication and division are better done in polar form. Given the complex numbers
z? = x? + jy? = r?/??
z? = x? + jy? = r?/??

the following operations are important.
Addition:
z? + z? = (x? + x?) + j(y? + y?) (9.18a)

Subtraction:
z? - z? = (x? - x?) + j(y? - y?) (9.18b)

Multiplication:
z?z? = r?r?/?? + ?? (9.18c)

Division:
z?/z? = r?/r?/?? - ?? (9.18d)

Reciprocal:
1/z = 1/r/-? (9.18e)

Square Root:
? z = ? r/?/2 (9.18f)

Complex Conjugate:
z* = x - jy = r/-? = re??? (9.18g)

Note that from Eq. (9.18e),
1/j = -j (9.18h)

Alexander, Charles K. Fundamentals of electric circuits / Charles K. Alexander, Matthew N. O. Sadiku. — 5th ed.p. cm

1. Solve and write the following complex numbers in rectangular standard form( In the form a ± jb)
i. (4 - 5j)(12 + 11j)
ii. (-3 - j) - (6 - 5j)
iii. (1 + 4j) + (-16 + 9j)
iv. 8j(10 + 2j)
v. (-3 - 9j)(1 + 10j)
vi. (2 + 7j)(8 + 3j)
vii. 4 - j2 + 10j/(4-j)
viii. 1 + 5j/(8+3j)
ix. (6+7i)/(3+5i)
x. i²?

2. Convert all the answers into polar form

Use the following relations to solve problem 1 and 2
z = x + jy Rectangular form
z = r/? Polar form (9.15)
z = re^j? Exponential form

The relationship between the rectangular form and the polar form is shown in Fig. 9.6, where the x axis represents the real part and the y axis represents the imaginary part of a complex number. Given x and y, we can get r and ? as
r = ?(x² + y²), ? = tan?¹(y/x) (9.16a)

On the other hand, if we know r and ?, we can obtain x and y as
x = r cos ?, y = r sin ? (9.16b)

Thus, z may be written as
z = x + jy = r/? = r(cos ? + j sin ?) (9.17)

Figure 9.6
Representation of a complex number z = x + jy = r/?.

Addition and subtraction of complex numbers are better performed in rectangular form; multiplication and division are better done in polar form. Given the complex numbers
z? = x? + jy? = r?/??
z? = x? + jy? = r?/??

the following operations are important.
Addition:
z? + z? = (x? + x?) + j(y? + y?) (9.18a)

Subtraction:
z? - z? = (x? - x?) + j(y? - y?) (9.18b)

Multiplication:
z?z? = r?r?/?? + ?? (9.18c)

Division:
z?/z? = r?/r?/?? - ?? (9.18d)

Reciprocal:
1/z = 1/r/-? (9.18e)

Square Root:
? z = ? r/?/2 (9.18f)

Complex Conjugate:
z* = x - jy = r/-? = re??? (9.18g)

Note that from Eq. (9.18e),
1/j = -j (9.18h)

Alexander, Charles K. Fundamentals of electric circuits / Charles K. Alexander, Matthew N. O. Sadiku. — 5th ed.p. cm

1. Solve and write the following complex numbers in rectangular standard form( In the form a ± jb)
i. (4 - 5j)(12 + 11j)
ii. (-3 - j) - (6 - 5j)
iii. (1 + 4j) + (-16 + 9j)
iv. 8j(10 + 2j)
v. (-3 - 9j)(1 + 10j)
vi. (2 + 7j)(8 + 3j)
vii. 4 - j2 + 10j/(4-j)
viii. 1 + 5j/(8+3j)
ix. (6+7i)/(3+5i)
x. i²?

2. Convert all the answers into polar form

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