Axioms of the Real Numbers Explainer

Charlie Ahrendts

There are many different systems of axioms, but one that provides a foundation for many others is that of the real numbers. These axioms consist of statements about the real numbers and the relationships between them. Many of them are very intuitive to us, since we often learn them as a young child when learning basic arithmetic for the first time, we are just unaware of the fundamental role they play in mathematics. In this article we will go through each of the real number axioms one by one and explore what they mean.

a + b = b + a  and  a*b = b*a

We call this first axiom the commutative law, and it simply states that whenever I add (or multiply) two real numbers together, the order in which I add (or multiply) them doesn’t matter. This is the case for operations like addition or multiplication but not for subtraction, where 5-3 gives a different result to 3-5.

(x + y) + z = x + (y + z)  and  (x*y)*z = x*(y*z)

This law is called associativity and it is very closely related to commutativity above. It states that if we do one type of operation like addition or multiplication two or more times, the order in which we perform those operations doesn’t matter. Or in other words, adding or removing parentheses does not affect the result. Almost all operations that are commutative are also associative. Some that are only associative include matrix multiplication and nested functions, which means functions within other functions.

∃0 : a + 0 = 0 + a = a

This axiom is called the existence of the additive identity. The symbol ∃ is short for ‘there exists’ and the : can be read as ‘such that’. The axiom therefore states that there exists an element, in this case zero, such that adding it to any number will not change that number. 

∃1 ≠ 0 : a*1 = 1*a = a

This axiom states the existence of a multiplicative identity and is very similar to the previous one. We can read it as: there exists a real number (which we call one) that is not equal to zero, such that multiplying it by any other number doesn’t change that number. 

a + (−a) = (−a) + a = 0

There exists not only an additive identity, but also an additive inverse. This axiom basically introduces the notion of negative numbers who form a counterpart to the positive numbers, such that adding up a number and its inverse will always produce zero (the additive identity element).

a*a-1 = a-1*a = 1

Since there is an additive inverse, it makes sense for there to be a multiplicative inverse as well. We can rewrite a-1 as 1 divided by a, so that the a and 1/a will cancel out and only 1, the multiplicative identity, remains. 

a*(b + c) = a*b + a*c

This axiom is called the distributive law. It states that if we multiply with two or more numbers added together, we can either first add those numbers together and multiply, or multiply each one separately and then add them together. The solution will in both cases be the same. 

And that’s it – all 9 of the real number axioms. We can see that most, if not all, of these axioms are very intuitive for us. When we learn maths in school, these axioms are just assumed to be true, but if we want to be able to formally and correctly prove other statements, we need these axioms to be the foundations from which we build.

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