A field is a mathematical structure in which you can add, subtract, multiply, and divide — that is, do all the operations that we call "algebra." Here are the rules that make our operations possible.

A binary operation is an operation that takes two inputs and produces one output. There exist two binary operations on $\mathbb{R}$, which we call addition ($+$) and multiplication ($\cdot$). We assume that these binary operations have the following properties:

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  • (a) Commutative Property: $+$ and $\cdot$ are commutative operations (so, $a+b = b+a$ and $a\cdot b = b\cdot a$).
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  • (b) Associative Property: $+$ and $\cdot$ are associative operations (so, $(a+b)+c = a+(b+c)$ and $(a\cdot b)\cdot c = a\cdot(b \cdot c)$ — which just means that we can unambiguously write $a+b+c$ without worrying about it).
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  • (c) Additive and Multiplicative Identities:
    • There exists a real number, which we call $0$ (zero), such that for all real numbers $x$, $x+ 0 =x$. This number $0$ is also called the "additive identity."
    • There exists a real number distinct from $0$, which we call $1$ (one), such that for all real numbers $x$, $x \cdot 1 =x$. This number $1$ is also called the "multiplicative identity."
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  • (d) Additive and Multiplicative Inverses:
    • For each real number $x$, there is a real number, which we call $−x$ ("minus $x$"), such that $x+ (−x) = 0$. This number $-x$ is also called the "additive inverse of $x$."
    • For each non-zero real number $x$, there exists a real number, which we call $\frac1x$, such that $x \cdot \frac1x= 1$. This number $\frac1x$ is also called the "multiplicative inverse of $x$", and we also denote it by $x^{−1}$.
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  • (e) Distributive Property: Multiplication distributes over addition, so that for all real numbers $x$, $y$, and $z$, $x \cdot (y+z) =x \cdot y+x \cdot z$.

We can thus define subtraction and division:

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  • (f) Subtraction: By $x-y$ we mean $x+(-y)$.
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  • (g) Division: By $\frac{x}{y}$ we mean $x\cdot\frac1y$.

It is honestly kind of amazing how much we can prove from these few basic rules!