Squares of nonzero elements are positive

Depends on

• Ordered field
• Sign rules for products: $(-a)b = -(ab)$ and $(-a)(-b) = ab$

Used by

• The exponential is positive and satisfies exp(-x)=1/exp(x) Corollary
• The multiplicative identity is positive Corollary
• The first quadrant of ℝ² contains 0 and is closed under addition and is not a linear subspace, since it is not closed under multiplication by -1 Counterexample
• x/(1+(k+1)²x²) converges uniformly to zero on ℝ while every derivative at zero equals one Counterexample
• The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ Definition
• √· on [0,∞) is uniformly continuous and exactly 1/2-Hölder, and is not Lipschitz Example
• A quarter-turn of ℝ² has characteristic polynomial x²+1 and no real eigenvalue Example
• Finite Jensen for x↦ x² gives that the square of a weighted mean is at most the weighted mean of the squares Example
• ℚ(√2) carries exactly two distinct field orders, exchanged by the conjugation √2 ↦ -√2 Example
• The Babylonian sequence x₁ = 2, xₖ₊₁ = (xₖ + 2/xₖ)/2 decreases to √2 Example
• The map x ↦ (x + 2/x)/2 is a contraction of [1,2] with fixed point √2, and the a priori bound gives the error after n steps Example
• The rational function field ℝ(t) ordered by the eventual sign is an ordered field, worked out Example
• The real 2-dimensional irreducible representation of C₃ has endomorphism ring ℂ Example
• FALSE: a^m/n := (a^1/n)ᵐ extends to negative bases False statement
• FALSE: every real number has a real square root False statement
• FALSE: the complex field admits an order making it an ordered field False statement
• Bernoulli's inequality (1+x)ⁿ ≥ 1 + nx Lemma
• Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive Lemma
• Every ordered field is an ordered ring, and its order is the one its positive cone induces Lemma
• For n ≥ 1 the product topology on n copies of the usual topology of ℝ is the metric topology of d_∞ on ℝⁿ, and hence also of d₁ and d₂, so ℝⁿ as a product and ℝⁿ as a metric space are one space Lemma
• Homomorphisms out of a complete ordered field are order-preserving Lemma
• ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it Lemma
• Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ Lemma
• x²+1 is irreducible over ℝ Lemma
• ℂ=ℝ[x]/(x²+1) is a field, every element is uniquely a+bi, and every nonzero element has inverse (a-bi)/(a²+b²) Theorem
• ℍ is a division ring that is not commutative, hence not a field: q⁻¹ = q̄ / N(q) for q ≠ 0, while ij = k and ji = -k Theorem
• Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0} Theorem
• The arithmetic mean, geometric mean inequality Theorem
• The Cauchy-Schwarz inequality for finite sums Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
• M. Spivak, Calculus, 4th ed., Ch. 1 (standard reference, not scraped)
• University of Innsbruck notes: Ordered fields (standard reference, not scraped)