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COMS21103: Summary of Lecture 2
Some of the points that we discussed in the lecture two:
Basic sets
The set of natural numbers N, integers Z, rationals Q, reals R, and complex numbers C are all sets that show up in this course one way or another.
Polynomials over a set R
The set R[X] of polynomials over R is defined by:
R[X] = {a 0 + a 1 X + a 2 X^2 +... + anXn^ | n ∈ N}
Addition, multiplication, and division are defined using the corresponding operations in R. We didn’t go into any details here; this is really basic stuff.
Complex numbers
We have fixed i =
−1 (in particular i^2 = −1) and defined the set of complex number C to be
C = {a + bi | a, b ∈ R}
We have defined addition and multiplication for complex numbers. We have seen that complex numbers can be identified with points in the two dimensional plane, and that one can use two possible sets of coorinates to represent the same complex number. Cartesian coordinates essentially identify a complex number with the pair of projections on the Ox andOy axes. That is, z = a + bi is identified with the pair (a, b). Polar coordinates identify a complex number with the pair (r, θ) where r is the length of the segment that links the origin of the plane with z and θ is the angle that this segment makes with the axes Ox.
Euler’s formula
Euler’s formula says that eiθ^ = cos θ + i sin θ.
We can use this formula to conclude that the complex number z ∈ C identified by polar coordinates (r, θ) is z = r cos θ + ir sin θ.
Roots of unity
We then looked at the roots of the equation Xn^ = z where z ∈ C and n ∈ N are arbitrary. If z = reiθ^ the roots of the equation Xn^ = z are:
zk = r
1 n (^) · e θ+2kπ n i^ (1)
with k ∈ { 0 , 1 , 2 ,... , n − 1 }. For example, the first root z 0 is z 0 = r (^1) n e iθn (we’ve just replaced k
with 0) and the second root z 1 = r
1 n (^) e θ+2π n i^ (we’ve replaced θ with 1) etc.
The proof that the nth roots of z are given by Equation (1) is done by simply verifying that indeed (∀k ∈ { 0 , 1 ,... , n − 1 }) it holds that zkn = z. This can be seen as follows:
znk =
r
1 n (^) e θ+2kπ n i
)n = reθi+2kπi^ = reiθe^2 kπi
Since, by Euler’s formula, e^2 kπi^ = cos 2kπ + i sin 2kπ = 1 it follows that zkn = reiθ^ = z.
As an important particular case, we have looked at the n’th roots of unity, that is, the complex solutions of the equation Xn^ = 1. Since 1 = 1 · e^0 i, we obtain (by simply pluging in r = 1 and θ = 0 in Equation (1)) that the n’th roots of unity are give by the formula:
ωk = e
2 kπi n (^) (2)
for k = 0, 1 , 2 ,... , n − 1.
Exercise Show that the product of any two n’th roots of unity, say ωi, ωj (for i, j ∈ { 0 , 1 , 2 ,... , n− 1), is another root of unity. Finally, we have looked at a geometric interpretation for the n’th roots of unity. All of these roots sit on the circle of radius 1 with center (0, 0) at equal distance. The first root has coordinates (1, 0). It follows that the angle between the Ox axis and the line that goes through the origin and the k’th root is (^2) nπ · k (for k = 0, 1 ,... , n − 1).
Quantifiers
We have looked at how to use quantifiers to express various mathematical statements. Here are some examples:
- The function f : D → C is injective
(∀x 1 , x 2 ∈ D) (x 1 6 = x 2 ⇒ f (x 1 ) 6 = f (x 2 ))
- The function f : D → C is surjective:
(∀y ∈ C)(∃x ∈ D)(f (x) = y)
- The graph of function f : R → R is always strictly above the graph of function g : R → R:
(∀x ∈ R)(f (x) > g(x))
