Complex Numbers Flashcards
List the hierarchy of number systems.
N ⊂ Z ⊂ Q ⊂ R ⊂ C
Define a complex number.
A complex number is a number written in the form
z = a + b i, where
1. a∈R is the real part of z (Re(z)),
2. b∈R is the imaginary part of z (Im(z)),
3. i is a formal symbol satisfying i^2 = −1.
Define the set of complex numbers.
The set of complex numbers is
C = {a + b i| a, b ∈ R}.
If
1. b = 0, then z = a ∈ R is a real number,
2. a = 0, then z = b i ∈ C is a purely imaginary number.
What are the 5 Arithmetic with Complex Numbers?
- Equality: a+bi = c+d i ⇔ a=c and b=d,
- Addition: We add complex numbers by adding their real and
imaginary parts:
(a+bi) + (c+di) = (a+c) + (b+d)i, - Multiplication: We multiply complex numbers by expanding
and using i^2 = −1:
(a + b i)(c + d i) = (ac − bd) + (ad + bc)i, - Subtraction: (a + b i) − (c + d i) = (a − c) + (b − d)i,
- Negatives: −(a + b i) = −a − b i.
Define Complex Conjugation.
The conjugate of z = a + b i,(a, b ∈ R) is z” (really it’s a z with a bar on top) = a−bi (we
negate the imaginary part).
What does every complex number correspond to?
Every complex number a+bi corresponds to a vector (or point) (a, b) in the plane R^2
Define absolute values (Modulus) of complex numbers
If z=a+bi∈C, then the absolute value (or modulus) of z is
|z| =√(zz”) = √((a + b i)(a − b i)) = √(a^2 + b^2) ≥ 0,
which is always non-negative.
If z = a + 0 i, then |z| =√a^2 = |a|, which is the usual absolute
value of real numbers.
How is division of complex numbers related to multiplicative inverses?
For instance, if a,b∈R and b≠0, then a/b=a(1/b) = ab^−1
Dividing a complex number by a real number is easy: If
z=a+bi∈C and c∈R, then
z/c =(a+bi)/c =
(1/c)(a + b i) = (a/c)+(b/c)i.
Describe multiplicative inverses of z
If z≠0, then |z| > 0 and
z(z”/|z|^2) = zz”/|z|^2 =
(|z|^2)/(|z|^2) = 1,
and so 1/z = z^−1 = z”/(|z|^2)
How do you divide two complex numbers?
Suppose z=a+bi and w=c+di are two complex numbers. Then, z/w= z(1/w) = z(w^−1) = z(w"/|w|^2) = 1/(c^2 + d^2)((a + b i)(c + d i)) = 1/(c^2+d^2)((ac+bd)+(bc−ad)i)∈C
What are the (12) Properties of Complex Numbers?
- (z+w)” = z”+w”,
- (az)” = a(z”),
- (zw)” = z” w”,
4.(z/w)”=z”/w”, - (z”)” = z,
- z” = z ⇔ z∈R,
- z” = −z ⇔ z is purely
imaginary, - |z| ∈ R and |z| ≥ 0,
- |z| = |z”|,
- |zw| = |z| |w|,
- |z/w| =|z|/|w|,
- |z + w| ≤ |z| + |w| (triangle
inequality).
Define polar form of complex numbers.
If z=a+bi∈C with a,b∈R, then we can define polar coordinates (r, θ) such that z=a+bi = rcos(θ) + rsin(θ)i = re^(θ i), cos(θ) = a/r, sin(θ) = b/r, r = |z| = √(a^2 + b^2.)
Give more information on the polar form of complex numbers. (3)
- e is a constant (e = 2.718) i.e. the base of natural logarithm,
- θ is called the argument of z (or θ = arg(z)) and is not
uniquely determined, since θ’= θ + 2nπ also works (n∈Z), - If in addition, we assume −π ≤ θ ≤ π, then θ is called the principal argument of z (or θ = Arg(z)).
Describe arithmetic with polar form (5)
1. re^(θ i) = r'e^(θ'i) ⇔ r = r' and θ − θ' = 2nπ, where n∈Z, 2. (re(^θ i))" = r e^(−θ i), 3. | e^(θi) | = 1, for any θ, 4. Suppose z1 = r1 e^(θ1i) and z2 = r2 e^(θ2i), then z1z2 = (r1r2) e^((θ1+θ2)i), 5. Suppose z1 = r1 e^(θ1i) and z2 = r2 e^(θ2i), then z1/z2 = (r1/r2)e^((θ1−θ2)i)
What is the fundamental theorem of algebra?
There are certain mathematical theorems that are only true if we
work with complex numbers, for instance: …
Every polynomial p(x) = a0 + a1x + a2x^2 + · · · + anx^n with complex coefficients a0, a1, · · · , an ∈ C factors completely into linear forms ci x + di, with ci, di ∈ C and 1 ≤ i ≤ n, i.e. p(x) = (c1x + d1)(c2x + d2)· · ·(cnx + dn).
This is not true if we only work with real numbers.
