21 Expansion of Numbers and Operations

• 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 ...
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• 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)
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• 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.
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• 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.
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• 211 Decimal expansion
The decimal expansion of any rational number is either a terminating or a recurring decimal.
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• 211 √2
There is no rational number whose square is 2.
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• 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.
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• 213 ∀a,b∈ℚ and a≠b, ∃c in (a,b) and c∉ℚ
Between any two distinct rationals there is an irrational.
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• 223
The density of Q and other consequences of the Axiom of Completeness.
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• 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
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• 223 Infimum and supremum
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• 223 The Supremum and Completeness of ℝ
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• 233 Mathematical Induction
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• 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.
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