Complex Numbers Worksheet Page 3
ADVERTISEMENT
1
2
31.5
Exponentials and Logs
The logarithm of a complex number is easy to compute if the number is in polar form:
iθ
iθ
ln(a + bi) = ln re
= ln(r) + ln e
= ln(r) + iθ
The logarithm of zero is left undefined (as in the real case). However, we can now compute the log of a
negative number:
π
π
iπ
ln( 1) = ln 1 e
= iπ
or the log of i :
ln(i) = ln(1) +
i =
i
2
2
Note that the usual rules of exponentiation and logarithms still apply.
To exponentiate a number, we convert it to multiplication (a trick we used in Calculus when dealing with
x
things like x
):
b
b ln(a)
a
= e
i
i ln(2)
Example, 2
= e
= cos(ln(2)) + i sin(ln(2))
i π/4 1/2
1/2
1/4
i π/8
Example:
1 + i = (1 + i)
=
2e
= (2
)e
i
i ln(i)
i(iπ/2)
π/2
Example: i
= e
= e
= e
2
Real Polynomials and Complex Numbers
2
If ax
+ bx + c = 0, then the solutions come from the quadratic formula:
2
b
b
4ac
x =
2a
2
In the past, we only took real roots. Now we can use complex roots. For example, the roots of x
+ 1 = 0 are
x = i and x =
i.
Check:
2
2
2
(x
i)(x + i) = x
+ xi
xi
i
= x
+ 1
Some facts about polynomials when we allow complex roots:
th
1. An n
degree polynomial can always be factored into n roots. (Unlike if we only have real roots!) This
is the Fundamental Theorem of Algebra.
2. If a + bi is a root to a real polynomial, then a
bi must also be a root. This is sometimes referred to as
“roots must come in conjugate pairs”.
3
Exercises
1. Suppose the roots to a cubic polynomial are a = 3, b = 1 2i and c = 1+2i. Compute (x a)(x b)(x c).
2
2. Find the roots to x
2x + 10. Write them in polar form.
3. Show that:
z + ¯ z
z
¯ z
Re(z) =
Im(z) =
2
2i
4. For the following, let z
=
3 + 2i, z
=
4i
1
2
(a) Compute z
¯ z
, z
/z
1
2
2
1
(b) Write z
and z
in polar form.
1
2
5. In each problem, rewrite each of the following in the form a + bi:
1+2i
(a) e
2 3i
(b) e
iπ
(c) e
1 i
(d) 2
2
i
(e) e
i
(f) π
3
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education