
Explore real mathematics, including the real number system, algebra, trigonometry, and geometry. Learn proofs of irrational numbers and concepts like upper and lower bounds, infimum, supremum, and absolute value.
The method uses contradiction to prove that sqrt(2) is irrational, showing that if sqrt(2)=a/b in lowest terms, both a and b are even, contradicting their coprimality.
This lecture proves that sqrt(2) is irrational using a contradiction: assume sqrt(2) = B/Q in lowest terms, and derive a contradiction with gcd(B, Q) = 1.
prove that sqrt(2) is irrational by contradiction. assume sqrt(2)=p/q with p and q integers and relatively prime; show both p and q are even, contradicting their relative primality; conclude irrational.
This lecture presents a proof by contradiction that a certain square root is irrational, analyzing composite numbers and integer factors to show no rational solution exists.
Prove the banali inequality using mathematical induction, establishing the base case at one and proving the step from k to k+1.
Show that the sum of a rational and an irrational number is irrational, using a contradiction by assuming the sum is rational and deriving a contradiction.
This exercise presents a counterexample where the sum and product of two irrational numbers are rational, using 2 plus 3 square root 2 and 2 minus 3 square root 2.
The lecture demonstrates that the sum and product of irrational numbers can be rational, using p1/q1 and p2/q2 to derive a rational sum and a rational product.
Define a field as a non-empty set with additive group structure, a multiplicative group on nonzero elements, and left and right distributive laws; examples include real, complex, and rational numbers.
Define an ordered field by the law of dichotomy and transitivity for non-empty real numbers. Show that multiplying by a positive preserves inequality; a negative reverses it for rational numbers.
Apply field properties to prove core propositions: if x+y = x+z, then y = z; if x+y = 0, then y = -x, using additive identity, additive inverses, and associativity.
Outline the multiplication axioms in a field, proving Y = 1 or 1/X when XY = X with X ≠ 0, using identity and inverse; introduce supremum, infimum, and uniqueness.
Define a complete ordered field and illustrate with real versus rational numbers. Show rationals aren’t complete via sqrt(2), and relate supremum and infimum properties.
Demonstrates how infimum and supremum relate via negation, and proves archimedean property, including existence of integers above any real and rationals between any two reals.
Explore the extended real number system with plus and minus infinity, and define the modulus (absolute value) and norm, including existence and uniqueness proofs and modulus properties.
Learn Schwarz's inequality for real vectors, proving the bound |<x,y>| <= ||x|| ||y|| using the norm and inner product, with multiple proofs and related norm properties.
Explains differentiability by evaluating left and right derivatives at zero using a function example, showing how limits determine whether f is differentiable at zero.
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