Topic 2: Real Numbers Flashcards
(14 cards)
What are the 12 properties of numbers (4, 4, 4)
-The associative law for addition
-The existance of an additive identity
-The existence of additive inverses
-The commutitative law for addition
-The associative law for multiplication
-The existence of a multiplicative identity
-The existence of multiplicative inverses
-The commutitative law for multiplication
-The distributative law
-The trichtonomy law
-Additative closure
-Multiplicative closure
What are the associative law for addition, the existence of an additive identity, the existence of an additive inverse, and the commutitative law for addition (4)
-(a+b) + c = a + (b+c)
-There exists 0 ∈ A such that a + 0 = a
-There exists -a ∈ A such that a-a = 0
-a + b = b + a
What are the associative law, existance of identity, existence of inverse and commutative laws for multiplication (4)
-(a x b) x c = a x (b x c)
-There exists 1 ∈ A such that a x 1 = a
-There exists a-1 ∈ A such that a(a-1) = 1
-AB = BA
What are the distributive laws, trichotomy law, additative closure and multiplicative closure (4)
-a(b+c) = ab + ac
-Either a < b, a = b or a > b
-If a,b ∈ A, a + b ∈ A
-If a, b ∈ A, ab ∈ A
What is a ring, commutitative ring and field (3)
-If A satisfies p1-6 (additative laws, associative and identity for multiplication), p9 (distributive law), p11-12 (additive/multiplicative closure) it is a ring
-If A satisfies a ring + p8 (commutative law for multiplication) it is a commutative ring
-If A also satisfies a CR + p7 (existence of multiplicative inverse) it is a field
What are the natural numbers, and what properties of numbers do they follow (3)
-ℕ = {1, 2, 3, …} (positive integers)
-ℕ satisfies p1, p4-6, p8-12
-ℕ0 = {0} u ℕ, satisfying p1-2, p4-6, p8-12
What are the integers, and what properties of numbers do they follow (2)
-ℤ = {-2, -1, 0, 1, 2, ….}
-ℤ satisfies everything except p7
What are the rational numbers, and the properties of numbers they follow (2)
-ℚ - {m/n: m, n ∈ ℤ, n ≠ 0}
-This satisfies all 12 properties
What does it mean that the rational numbers are dense (2)
-Between any 2 rational numbers is another one
-Lemma: a, b ∈ ℚ, with a < b -> c = 0.5(a+b), so a < c < b
How do we prove √2 is irrational (2,10)
-Theorem: √2 ∉ ℤ
-Lemma: let n ∈ ℤ, then n2 is even iff n is even
-Proof: suppose √2 is rational, there then exists p, q ∈ ℤ, with q not 0, and p and q with no common factors
-√2 = p/q
-2 = p2/q2, p2 = 2q2
-p2 is thus even, and by the lemma, p must be even
-We can then say p = 2r for some r ∈ ℤ
-(2r)2 = 2q2
-2r2 = q2
-Hence q2 is even, and by the lemma, q must be even
-However, p and q being even contradicts the original assumption that p and q are coprime
-Hence our original assumption that √2 ∈ ℤ is false, and therefore √2 is not rational
What are the real numbers, and the classification of real numbers (2, 3)
-ℝ is the set of real numbers (rational + irrational)
-This is defined as finite or infinite decimal expansions
Classification:
-Any finite decimal is rational
-ANy infinite repeating decimal is rational (do 100…x - x to convert to fraction)
-Any infinite non-repeating decimal is irrational
What are the 2 different classes of irrational numbers (2)
-Algebraic numbers can be expressed as roots of polynomials with integer/rational coefficients (√2)
-Transcendental numbers cant (e)
What is the triangle inequality + working (1. 4)
- |a+b| ≤ |A| + |B| (a, b ∈ ℝ)
-(|a+b|)2 = a2 + 2ab + b2 (since a ≤ |a|, b ≤ |b|)
-(|a+b|)2 ≤ |a|2 + 2|a|b| + |b|2
-(|a+b|)2 ≤ (|a| + |b|)2
-|a+b| ≤ |a| + |b|
What are different interval notations for subsets of ℝ (1, 4, 1)
-Suppose a, b ∈ ℝ within a < b
-(a,b) = {x:a<x<b} is the open interval
-[a,b] = {x:a≤x≤b} is the closed interval
-(a, b] = {x:a<x≤b} is the half open/closed interval
-(a,∞) = {x:x>a} is semi-infinite intervals
-note ∞ ∉ ℝ
