Modern Mathematics: Axiomatic Foundations and Proofs

• Modern Mathematics began with Axiomatic Foundations.
• A theory in Mathematics requires that assumptions (axioms) be properly defined.
• All terminology must be clearly defined.
• Every statement (theorem) must be made "universally valid".
• Every statement has to be "consistent" with the rest of the theory.
• Proofs and clearly stated axioms and definitions are needed for Universal Validity.
• A statement must be paired with a convincing argument to be universally valid.
• A proof uses axioms and previously validated theorems to show the validity of a claimed argument.
• A proof begins with assumptions of the statement (theorem).
• Following rules of logic, one deduces a chain of arguments.
• Each piece of the chain is discussed and validated by mathematical tools.
• The chain of argument continues until the conclusion is obtained.
• Numerical observations alone are not sufficient for mathematical proof.
• Numerical observations can lead to incorrect conclusions.

Gauss Sum

• Carl Friedrich Gauss was one of the greatest mathematicians of all time.
• Gauss found the sum of the first 100 integers by writing the sequence forward and backwards: 1 + 2 + 3 + ... + 98 + 99 + 100
• Gauss recognized the the sum of each pair equaled 101, and realized he had 100 of these
• Thus, the sum equals (101x100) / 2 = 5050
• This generalizes to the formula: 1 + 2 + 3 + ... + n = n(n+1) / 2
• Gauss's Sum can be represented visually as the area of a shape.

Sum of Squares

• The sum of squares of positive integers up to n is given by: 1² + 2² + 3² + ... + n²
• The sum of squares can also be visually demonstrated
• A visual proof of the sum of squares is very limited in application and is not applicable in most mathematical statements.

Sum of Cubes

• The sum of cubes of positive integers up to n is given by: 1³ + 2³ + 3³ + ... + n³
• The sum of cubes can also be visually demonstrated
• A visual proof of the sum of cubes is very limited in application and is not applicable in most mathematical statements.

Sigma Notation

• Sigma Notation simplifies summation representations.
• ∑ represents the sum of values with respect to consecutive indices.
• General form: ∑ak from k=1 to n equals a1 + a2 + a3 + ··· + an−1 + an
• The sum of integers using sigma notation: ∑k from k=1 to n = n(n+1) / 2
• The sum of square using sigma notation: ∑k² from k=1 to n = n(n+1)(2n+1) / 6
• The sum of cubes using sigma notation: ∑k³ from k=1 to n = ((n(n+1) / 2) ²

Proof by Induction

• Proof by induction is a chain of arguments.
• In a proof by induction, each argument is evoked and proved by the pervious argument.
• The chain has a "first statement" called the base statement which is labeled P(1).
• P(n) represents the inductive step, where one shows "as long as one step is true, the next step is always true”.
• If both the "Base Statement" and "Inductive Step" are true, then P(1) is true.
• If P(1) is true, then P(2) is true by definition of algorithm.

Proof of Gauss Sum by Induction

• Claim: ∑k from k=1 to n = n(n+1) / 2
• The Base Step shown when n = 1: P(1) : 1 = 1*(1+1) / 2 = 1
• Inductive Step: Assume P(n) is true and prove that P(n+1) will be true:
  • P(n) : 1+...+n = n*(n+1) / 2
  • P(n+1) : 1+...+n+(n+1) = (n+1)*(n+2) / 2

Proof of Sum of Squares by Induction

• Claim: ∑k² from k=1 to n = n(n+1)*(2n+1) / 6
• The Base Step shown when n = 1: P(1) : 1² = 1*(1+1)*(2+1) / 6 = 1
• Inductive Step: Assume P(n) is true and prove that P(n+1) will be true:
  • P(n): 1² + ... + n² = (n * (n + 1) * (2n + 1)) / 6
  • P(n+1): 1² + ... + n² + (n + 1)² = ((n + 1) * (n + 2) * (2n + 3)) / 6

Proof of Sum of Cubes by Induction

• Claim: ∑k³ from k=1 to n = (n*(n+1) / 2)²
• The Base Step shown when n = 1: P(1) : 1³ = (1*(1+1) / 2 )² = 1
• Inductive Step: Assume P(n) is true and prove that P(n+1) will be true:
  • P(n): 1³ + ... + n³ = ((n * (n + 1)) / 2)²
  • P(n+1): 1³ + ... + n³ + (n + 1)³ = (((n + 1) * (n + 2)) / 2)²