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PROOF 86

Prove.

What each extension gains

On each number system.

What you may use

A1a + 0 = a
additive identity
A2there exists −a with a + (−a) = 0
additive inverse (from ℤ on)
A3(a + b) + c = a + (b + c)
associativity of addition
A4a + b = b + a
commutativity of addition
A5a × 1 = a
multiplicative identity
A6a ≠ 0 ⟹ there exists b with ab = 1
multiplicative inverse (from ℚ on)
A7(ab)c = a(bc)
associativity of multiplication
A8ab = ba
commutativity of multiplication
A9a × 0 = 0
lemma of Proof 01
A10n = 2k or n = 2k + 1
parity of integers (never both)
A11n ∈ ℕ ⟹ 0 < n
natural numbers are positive
A120 < 1
1 is positive
A13a < b ⟹ a + c < b + c
order and addition
A14a<b, b<c ⟹ a<c
transitivity of order
A15exactly one of a<b, a=b, b<a
trichotomy of order
A16polynomial functions are continuous
continuity
A17f(a) < k < f(b) ⟹ f(c)=k
completeness — intermediate value theorem
A18no rational x with x² = 2
theorem of Proof 09
A19no real x with x² = −1
theorem of Proof 14
A20i × i = −1
definition of i (Proof 14)
Ea = b ⟹ apply the same operation to both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

What an extension gains is measured by a single yardstick. Does an equation unsolvable in the previous system become solvable in the new one?

Proof.