EngliSea > M > math > 55 Analysis > 2 Real Numbers

Between any two rationals there is an irrational.
 
21 Expansion of Numbers and  Operations
21 Expansion of Numbers and Operations
9545
 
Songs in Easy English
 
21 Expansion of Numbers and Operations
ᐥoperators + − ×(·) / ˄ ˅ ⍻ numbers ℕ ℤ ℚ ℝᐥ The whole numbers are closed under addition. Subtraction is the inverse operation of addition. l + m = n n - m = l Define negative integers then integers are closed under subtraction. Multiplication is repea ...
9540 COMMENT
 
21 Fundamental theorem of arithmetic
Every natural number either is a prime number itself or can be represented as the product of prime numbers and this representation is unique, except for the order of the factors. (unique factorization theorem, unique prime factorization theorem)
9706
 
21 Rational number
A number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.
9708
 
211 Decimal
The decimal numeral system (base-ten positional numeral system, denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system.
9712
 
211 Decimal expansion
The decimal expansion of any rational number is either a terminating or a recurring decimal.
9715
 
211 √2
There is no rational number whose square is 2.
9710
 
212 a+b√2(a,b∈ℚ) ∉ ℚ
If m/n and p/q are rationals, p≠0, then m/n+√2(p/q) is irrational.
9709
 
213 ∀a,b∈ℚ and a≠b, ∃c in (a,b) and c∉ℚ
Between any two distinct rationals there is an irrational.
9711
 
223
The density of Q and other consequences of the Axiom of Completeness.
9761
 
223 Completeness Axiom
ᐥEvery non-empty set of real numbers that is bounded above has a least upper bound. Every non-empty set of real numbers that is bounded below has a greatest lower bound.ᐥ
9746
 
223 Infimum and supremum
9748
 
223 The Supremum and Completeness of ℝ
9760
 
233 Mathematical Induction
9717
 
233 Mathematical Induction
Let P(n) be a statement for each natural number n. If (a) P(1) is true, and (b) P(k) true ⇒ P(k+1) true for every natural number k∈ℕ then P(n) is true for all n∈ℕ.
9749
 
◌◌◌ Equinumerosity
9762
 
Listionary
cosmologydictumreverenceritualtransgression

-