Real Analysis – Proofs of Basic Properties of Real Numbers

proof-writingreal-analysissoft-question

I have just begun my study of complex numbers and I learned where imaginary numbers came from and their importance. However there's one thing that I need to clarify and that is the properties of real numbers and their proofs.

  1. Closure Laws
    For all $a,b \in \mathbb{R}$, $a+b$, $a-b$, $ab$, $a/b$ are real numbers. Thus $\mathbb{R}$ is closed under four fundamental operations.

  2. Commutative Laws
    For all $a,b \in \mathbb{R}$ $a+b = b+a$ and $ab = ba$.

  3. Associative Laws
    For all $a,b,c \in \mathbb{R}$ $a+(b+c) = (a+b)+c$ and $a(bc) = (ab)c$.

  4. Additive Identity
    For all $a \in \mathbb{R}$ there exists $0\in \mathbb{R}$ such that $a+0 = 0+a = a$.

  5. Additive inverse
    For all $a \in \mathbb{R}$ there exists a $b \in \mathbb{R}$ such that $a+b = b+a = 0$, the additive identity $b = -a$ is called the additive inverse or the negative of $a$.

and similarly Multiplicative Identity, Multiplicative inverse, Distributive Law, Trichotomy Law, Transitivity of order, Monotone Law of Addition, Monotone law of multiplication.

I understand that the above laws hold good throughout mathematics. Should these laws be accepted as being true "on faith" or are there proofs?
If yes, I am curious to know the proofs. As per my understanding no textbook has ever talked about proofs for these.

Best Answer

One accessible account is in Michael Spivak's textbook Calculus. Here a real number is defined to be a subset $\alpha$ of the rational numbers that is non-empty, bounded above, and satisfies $x\in\alpha\mathrm{\ and\ }y\leq x\Rightarrow y\in\alpha$. (There's a natural way to identify these real numbers with real numbers as we usually think of them containing the rationals as a subset: for instance $\sqrt{2}$ in this sense is the following set $\{x\in\mathbb{Q}:x<\sqrt{2}\}$.) You can then define the field operations and the order, and patiently show that this gives you a complete ordered field (so proofs of the properties you ask about are given). The detailed proofs take several pages.

As far as I recall, Spivak gives a pretty full account of this, including the uniqueness of a complete ordered field.

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