Complex Numbers
- Prove the following cancellation laws.
(a) If Z1 + Zz = Z1 + Z3, then Zz = Z3.
(b) If Z1Z2 = Z1Z3 and Z1 -=!= o, then Z2 :=:: Z3.
- Prove that the set of 2 x 2 matrices of the form
a, b real,
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with the usual definitions of equality, addition, and multiplication of
matrices, is isomorphic to <C.
- Prove that the set of polynomials in x with real coefficients, ~odulus
x^2 + 1, is isomorphic to <C.
9. Show that any three different complex numbers z 1 ,z 2 ,z 3 are linearly
dependent over the reals, i.e., there are :real numbers r 1 , r 2 , r 3 , not all
zeros, such that
riz1 + rzz 2 + raz 3 = 0
- Find the complex conjugate of each of the following,
(a) 2 - 3i (b) 4i(f + i)
(c) ia (d) l:i
- Show that:
(a) z1z2za = z1z2za (b). (lo1Z~) ~ = Z1Z2 :::---=--, Z3Z4 -1-r 0
Z3Z4 Z3Z4
- If z^2 = z^2 , show that z is either real or pure imaginary.
*13. Prove that
1 1 1
D = i z 1 z2 za
21 22 za
is real. Hint: Show that D = D. (* notl'Lt1on explained in Section 0.3.)
- Express each of the following in complex. conjugat~ coordinates.
(a) 2x + 3y + 5 = 0 (b) x^2 + y^2 - 2:i: - 8 = 0
(c) y^2 = 4x (cl) 9x^2 + 4y^2 = 36
- Express zz + (3 - 4i)z + (3 + 4i)2 + 24 = 0 in Cartesian coordinates.
1.4 Ordering of the Complex Numbers
We have extended the field of real mllnbers to th~ field of complex numbers,
so both systems of numbers satisfy the same formal laws (those of a field).
However, there is one important property of the reals that does not carry
over to the complex numbers. The field of the reals is a linearly ordered
field while that of the complex numbers is not.
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