Question

Throughout, the usual rules of arithmetic are assumed for the real numbers; in particular, a^2 > 0 for any nonzero real number a. 1. Show directly from rule (2) for multiplication that z^2 = (-z)^2. 2. Suppose that z = (x, y) and z^2 = (-1, 0). Show that z = i or z = -i. 3. Solve the equation z^2 = (0, 1). 4. Suppose that z^2 is real and negative; that is, z^2 = (a, 0), a < 0. Show that z = (0, b) and find b in terms of a. 5. Show by computation that addition of complex numbers is associative: (z1 + z2) + z3 = z1 + (z2 + z3); and commutative: z1 + z2 = z2 + z1. 6. Show by computation that multiplication of complex number is associative: (z1z2)z3 = z1(z2z3); and commutative: z1z2 = z2z1. 7. Define the absolute value, |z|, of z = (x, y) by |z| = ?(x^2 + y^2). Show directly that |z1z2| = |z1||z2|. 8. Define the complex conjugate, ?, of z = (x, y) by ? = (x, -y). Show that z? = (|z|^2, 0). 9. Show that z1z2 = 0 implies that either z1 or z2 is zero.

          Throughout, the usual rules of arithmetic are assumed for the real numbers; in particular, a^2 > 0 for any nonzero real number a.
1. Show directly from rule (2) for multiplication that z^2 = (-z)^2.
2. Suppose that z = (x, y) and z^2 = (-1, 0). Show that z = i or z = -i.
3. Solve the equation z^2 = (0, 1).
4. Suppose that z^2 is real and negative; that is, z^2 = (a, 0), a < 0. Show that z = (0, b) and find b in terms of a.
5. Show by computation that addition of complex numbers is associative:
(z1 + z2) + z3 = z1 + (z2 + z3); and commutative: z1 + z2 = z2 + z1.
6. Show by computation that multiplication of complex number is associative:
(z1z2)z3 = z1(z2z3); and commutative: z1z2 = z2z1.
7. Define the absolute value, |z|, of z = (x, y) by |z| = ?(x^2 + y^2). Show directly that |z1z2| = |z1||z2|.
8. Define the complex conjugate, ?, of z = (x, y) by ? = (x, -y). Show that z? = (|z|^2, 0).
9. Show that z1z2 = 0 implies that either z1 or z2 is zero.

Throughout, the usual rules of arithmetic are assumed for the real numbers; in particular, a^2 > 0 for any nonzero real number a.
1. Show directly from rule (2) for multiplication that z^2 = (-z)^2.
2. Suppose that z = (x, y) and z^2 = (-1, 0). Show that z = i or z = -i.
3. Solve the equation z^2 = (0, 1).
4. Suppose that z^2 is real and negative; that is, z^2 = (a, 0), a < 0. Show that z = (0, b) and find b in terms of a.
5. Show by computation that addition of complex numbers is associative:
(z1 + z2) + z3 = z1 + (z2 + z3); and commutative: z1 + z2 = z2 + z1.
6. Show by computation that multiplication of complex number is associative:
(z1z2)z3 = z1(z2z3); and commutative: z1z2 = z2z1.
7. Define the absolute value, |z|, of z = (x, y) by |z| = ?(x^2 + y^2). Show directly that |z1z2| = |z1||z2|.
8. Define the complex conjugate, ?, of z = (x, y) by ? = (x, -y). Show that z? = (|z|^2, 0).
9. Show that z1z2 = 0 implies that either z1 or z2 is zero.

Added by Paula A.

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