Download Complex numbers in Cartesian form: in principle . . . and in practice

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Complex numbers in Cartesian form: in principle
1. What equation defines the complex number i, and what “twin” of i do we pick up in the process?
2. Cartesian form of complex numbers:
(a) What do we mean by Cartesian form of a complex number?
(b) What are the real part and imaginary part of a complex number a + bi?
(c) How do we denote the set of all complex numbers?
How can we use Cartesian form to visualize this set, and where do the real numbers fit in?
3. How do we define the conjugate of a complex number a + bi? How is it denoted?
What are the basic properties of this operation?
4. Arithmetic in Cartesian form:
(a) How can we add and subtract complex numbers in Cartesian form?
(b) How can we compute the product of two complex numbers in Cartesian form?
(c) What trick allows us to compute the quotient of two complex numbers in Cartesian form?
5. What is the only basic relation among real numbers that does not make sense for complex numbers?
Why?
. . . and in practice
6. Simplify each of the following expressions, and find the real and imaginary parts of the results:
2+i
(a) (2 + i) + (4 − 3i)
(b) (2 + i) − (4 − 3i)
(c) (2 + i)(4 − 3i)
(d)
4 − 3i
1
(e) (3 + 2i) + 3 + 2i
(f) (3 + 2i) − 3 + 2i
(g) (3 + 2i)(3 + 2i)
(h)
2i
7. Simplify and plot the following powers of complex numbers:
(a) 2n , for n = 0, 1, 2, 3, 4
(b) in , for n = 0, 1, 2, 3, 4
(c) (1 + i)n , for n = 0, 1, 2, 3, 4, 5, 6, 7, 8
8. Use Cartesian form of a complex number z to answer the following questions:
(a) What can be concluded if z = z̄?
(b) Show that for every z ∈ C, z z̄ is a nonnegative real number.
(c) Using only conjugation and arithmetic, how can we compute the real and imaginary parts
of a complex number z?
[ Hint: Solve for a and b in terms of z and z̄. ]
9. Suppose that P (z) = a0 + a1 z + a2 z 2 + · · · + an−1 z n−1 + an z n is a polynomial with real coefficients.
Show that if P (z) = 0, then P (z̄) = 0 as well—that is, complex roots of real polynomials come in
conjugate pairs.
[ Hint: Write out the hypotheses, conjugate both sides, and carefully simplify. ]
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Polar form, powers, and roots of complex numbers: in principle
1. Magnitude:
(a) How do we define the magnitude of a complex number z? How is it denoted?
(b) If we write z in Cartesian form, how can we quickly compute its magnitude?
(c) What is the geometric meaning of the magnitude of z?
2. Arguments:
(a) What do we mean by an argument of a nonzero complex number z?
(b) How many arguments does each nonzero z have, and how are they related to one another?
(c) How can we find a nonzero complex number’s arguments?
3. Polar form:
(a) What is polar form of a complex number?
(b) How can we convert a complex number from polar to Cartesian form, and vice-versa?
(*c) What must be true in order for two complex numbers r[cos θ + i sin θ] and s[cos φ + i sin φ]
in polar form to be equal?
4. Arithmetic in polar form: suppose that z = r[cos θ + i sin θ] and w = s[cos φ + i sin φ].
(a) How can we compute zw in polar form? How can we remember this formula?
z
(b) How can we thus compute
in polar form?
w
(c) If n is a whole number, how can we quickly compute z n in polar form?
5. If n is a whole number, what do we mean by an “nth root” of a complex number z?
How many nth roots does each nonzero complex number z have?
How can we find them? How are they arranged in the complex plane?
. . . and in practice
6. Find Cartesian form for each of the following complex numbers:
5π
(a) 2(cos 5π
(b) 1(cos π4 + i sin π4 )
6 + i sin 6 )
(c) 6(cos π + i sin π)
7. Find polar form for each of the following complex numbers:
√
(a) 1 − i
(b) 3 + 4i
(c) − 3 + i
(d) −1
(e) 4i
8. Let z = 2(cos π6 + i sin π6 ), w = 3(cos π2 + i sin π2 ), and u = 1(cos π4 + i sin π4 ). Compute the following:
w
z6
(a) wz
(b)
(c) u8
(d) z 6
(e) 8
z
u
9. Find all 4th roots of −4, then plot them, convert them into Cartesian form, and simplify.
10. Find all cube roots of 8, then plot them, convert them into Cartesian form, and simplify.
11. Find all 6th roots of −1, then plot them, convert them into Cartesian form, and simplify.