2. The set of complex numbers can be defined as the set C = {a + bi | a, b = R, i²=-1}. For any complex number z = a +bi € C, • the real part of z is Re(z) = a and the imaginary part of z is Im(z) = b; the conjugate of z is z = a - bi; and • the modulus of z is |z| = √a² + b². With this notation, answer the following (each can be done in one line of computation). 2+z z-z (a) Show that Re(z) = 2 2i (b) Show that z-z = |z|². and Im(z) = (c) Show there is an expression for the multiplicative inverse of z, the number in terms of just the conjugate 7 and the modulus |z|. (Hint: Refer to (b).)
2. The set of complex numbers can be defined as the set C = {a + bi | a, b = R, i²=-1}. For any complex number z = a +bi € C, • the real part of z is Re(z) = a and the imaginary part of z is Im(z) = b; the conjugate of z is z = a - bi; and • the modulus of z is |z| = √a² + b². With this notation, answer the following (each can be done in one line of computation). 2+z z-z (a) Show that Re(z) = 2 2i (b) Show that z-z = |z|². and Im(z) = (c) Show there is an expression for the multiplicative inverse of z, the number in terms of just the conjugate 7 and the modulus |z|. (Hint: Refer to (b).)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
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Chapter1: Functions And Models
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![2. The set of complex numbers can be defined as the set C = {a + bi | a, b = R, i²=-1}.
For any complex number z = a + bi € C,
• the real part of z is Re(z) = a and the imaginary part of z is Im(z) = b;
the conjugate of z is z = a -bi; and
• the modulus of z is |z| = √a² + b².
With this notation, answer the following (each can be done in one line of computation).
2+z
z-z
(a) Show that Re(z) =
2
2i
(b) Show that · z = |z|².
and Im(z)
=
(c) Show there is an expression for the multiplicative inverse of z, the number in terms of
just the conjugate 7 and the modulus [z]. (Hint: Refer to (b).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94707e96-7ca2-4be8-88e5-b4b51e0075ac%2Ffa7320a8-b75d-47d5-8de4-012f86d3d6ff%2F09lzjkr_processed.png&w=3840&q=75)
Transcribed Image Text:2. The set of complex numbers can be defined as the set C = {a + bi | a, b = R, i²=-1}.
For any complex number z = a + bi € C,
• the real part of z is Re(z) = a and the imaginary part of z is Im(z) = b;
the conjugate of z is z = a -bi; and
• the modulus of z is |z| = √a² + b².
With this notation, answer the following (each can be done in one line of computation).
2+z
z-z
(a) Show that Re(z) =
2
2i
(b) Show that · z = |z|².
and Im(z)
=
(c) Show there is an expression for the multiplicative inverse of z, the number in terms of
just the conjugate 7 and the modulus [z]. (Hint: Refer to (b).)
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