5.1: Axioms of the Real Numbers
In this chapter we will take a deep dive into structure of the real numbers by building up the multitude of properties you are familiar with by starting with a collection of fundamental axioms. Recall that an axiom is a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved. It is worth pointing out that one can carefully construct the real numbers from the natural numbers. However, that will not be the approach we take. Instead, we will simply list the axioms that the real numbers satisfy
Our axioms for the real numbers fall into three categories:
- Field Axioms: These axioms provide the essential properties of arithmetic involving addition and subtraction.
- Order Axioms: These axioms provide the necessary properties of inequalities.
- Completeness Axiom: This axiom ensures that the familiar number line that we use to model the real numbers does not have any holes in it.
We begin with the Field Axioms.
Axioms 5.1. There exist operations \(+\) (addition) and \(\cdot\) (multiplication) on \(\mathbb{R}\) satisfying:
- (Associativity for Addition) For all \(a, b, c\in \mathbb{R}\) we have \((a+b)+c = a+(b+c)\) ;
- (Commutativity for Addition) For all \(a,b\in \mathbb{R}\) , we have \(a+b=b+a\) ;
- (Additive Identity) There exists \(0\in\mathbb{R}\) such that for all \(a\in\mathbb{R}\) , \(0+a=a\) ;
- (Additive Inverses) For all \(a\in\mathbb{R}\) there exists \(-a\in\mathbb{R}\) such that \(a+(-a)=0\) ;
- (Associativity for Multiplication) For all \(a, b, c\in \mathbb{R}\) we have \((ab)c = a(bc)\) ;
- (Commutativity for Multiplication) For all \(a,b\in \mathbb{R}\) , we have \(ab=ba\) ;
- (Multiplicative Identity) There exists \(1\in\mathbb{R}\) such that \(1\neq 0\) and for all \(a\in\mathbb{R}\) , \(1a=a\) ;
- (Multiplicative Inverses) For all \(a\in\mathbb{R}\setminus\{0\}\) there exists \(a^{-1}\in\mathbb{R}\) such that \(aa^{-1}=1\) .
- (Distributive Property) For all \(a,b,c\in \mathbb{R}\) , \(a(b+c)=ab+ac\) ;
In the language of abstract algebra, Axioms F1–F4 and F5–F8 make each of \(\mathbb{R}\) and \(\mathbb{R}\setminus\{0\}\) an abelian group under addition and multiplication, respectively. Axiom F9 provides a way for the operations of addition and multiplication to interact. Collectively, Axioms F1–F9 make the real numbers a field . It follows from the axioms that the elements \(0\) and \(1\) of \(\mathbb{R}\) are the unique additive and multiplicative identities in \(\mathbb{R}\) . To prove the following theorem, suppose \(0\) and \(0'\) are both additive identities in \(\mathbb{R}\) and then show that \(0=0'\) . This shows that there can only be one additive identity.
Theorem 5.2. The additive identity of \(\mathbb{R}\) is unique.
To prove the next theorem, mimic the approach you used to prove Theorem 5.2.
Theorem 5.3. The multiplicative identity of \(\mathbb{R}\) is unique.
For every \(a\in\mathbb{R}\) , the elements \(-a\) and \(a^{-1}\) (as long as \(a\neq 0\) ) are also the unique additive and multiplicative inverses , respectively.
Theorem 5.4. Every real number has a unique additive inverse.
Theorem 5.5. Every nonzero real number has a unique multiplicative inverse.
Since we are taking a formal axiomatic approach to the real numbers, we should make it clear how the natural numbers are embedded in \(\mathbb{R}\) .
Definition 5.6. We define the natural numbers , denoted by \(\mathbb{N}\) , to be the smallest subset of \(\mathbb{R}\) satisfying:
- \(1\in\mathbb{N}\) , and
- for all \(n\in\mathbb{N}\) , we have \(n+1\in\mathbb{N}\) .
Notice the similarity between the definition of the natural numbers presented above and the Axiom of Induction given in Section 4.1. Of course, we use the standard numeral system to represent the natural numbers, so that \(\mathbb{N}= \{1,2,3,4,5,6,7,8,9,10\ldots\}\) .
Given the natural numbers, Axiom F3/Theorem 5.2 and Axiom F4/Theorem 5.4 together with the operation of addition allow us to define the integers , denoted by \(\mathbb{Z}\) , in the obvious way. That is, the integers consist of the natural numbers together with the additive identity and all of the additive inverses of the natural numbers.
We now introduce some common notation that you are likely familiar with. Take a moment to think about why the following is a definition as opposed to an axiom or theorem.
Definition 5.7. For every \(a,b\in\mathbb{R}\) and \(n\in\mathbb{Z}\) , we define the following:
- \(a-b:= a+(-b)\)
- \(\displaystyle\frac{a}{b}:= ab^{-1}\) (for \(b\neq 0\) )
- \(\displaystyle a^n:= \begin{cases} \overbrace{aa\cdots a}^n, &\text{if }n\in \mathbb{N}\\ 1, & \text{if }n=0\text{ and }a\neq 0\\ \displaystyle\frac{1}{a^{-n}}, & \text{if }-n\in \mathbb{N}\text{ and }a\neq 0 \end{cases}\)
The set of rational numbers , denoted by \(\mathbb{Q}\) , is defined to be the collection of all real numbers having the form given in Part (b) of Definition 5.7 The irrational numbers are defined to be \(\mathbb{R}\setminus\mathbb{Q}\) .
Using the Field Axioms, we can prove each of the statements in the following theorem.
Theorem 5.8. For all \(a,b,c\in\mathbb{R}\) , we have the following:
- \(a=b\) if and only if \(a+c=b+c\) ;
- \(0a=0\) ;
- \(-a=(-1)a\) ;
- \((-1)^2 = 1\) ;
- \(-(-a)=a\) ;
- If \(a\neq 0\) , then \((a^{-1})^{-1}=a\) ;
- If \(a\neq 0\) and \(ab = ac\) , then \(b = c\) .
- If \(ab=0\) , then either \(a=0\) or \(b=0\) .
Carefully prove the next theorem by explicitly citing where you are utilizing the Field Axioms and Theorem 5.8.
Theorem 5.9. For all \(a,b\in\mathbb{R}\) , we have \((a+b)(a-b)=a^2-b^2\) .
We now introduce the Order Axioms of the real numbers.
Axioms 5.10. For \(a,b,c\in \mathbb{R}\) , there is a relation \(<\) on \(\mathbb{R}\) satisfying:
- (Trichotomy Law) If \(a\neq b\) , then either \(a<b\) or \(b<a\) but not both;
- (Transitivity) If \(a<b\) and \(b<c\) , then \(a<c\) ;
- If \(a<b\) , then \(a+c<b+c\) ;
- If \(a<b\) and \(0<c\) , then \(ac<bc\) ;
Given Axioms 1–4, we say that the real numbers are a linearly ordered field . We call numbers greater than zero positive and those greater than or equal to zero nonnegative . There are similar definitions for negative and nonpositive .
Notice that the Order Axioms are phrased in terms of “ \(<\) ". We would also like to be able to utilize “ \(>\) ", “ \(\leq\) ", and “ \(\geq\) ".
Definition 5.11. For \(a,b\in\mathbb{R}\) , we define:
- \(a>b\) if \(b<a\) ;
- \(a\leq b\) if \(a<b\) or \(a=b\) ;
- \(a\geq b\) if \(b\leq a\) .
Notice that we took the existence of the inequalities “ \(<\) ", “ \(>\) ", “ \(\leq\) ", and “ \(\geq\) " on the real numbers for granted when we defined intervals of real numbers in Definition 3.4.
Using the Order Axioms, we can prove many familiar facts.
Theorem 5.12. For all \(a,b\in\mathbb{R}\) , if \(a,b>0\) , then \(a+b>0\) ; and if \(a,b<0\) , then \(a+b<0\) .
The next result extends Axiom 3.
Theorem 5.13. For all \(a,b,c,d\in\mathbb{R}\) , if \(a<b\) and \(c<d\) , then \(a+c<b+d\) .
Theorem 5.14. For all \(a\in\mathbb{R}\) , \(a>0\) if and only if \(-a<0\) .
Theorem 5.15. If \(a\) , \(b\) , \(c\) , and \(d\) are positive real numbers such that \(a<b\) and \(c<d\) , then \(ac<bd\) .
Theorem 5.16. For all \(a,b\in\mathbb{R}\) , we have the following:
- \(ab>0\) if and only if either \(a,b>0\) or \(a,b<0\) ;
- \(ab<0\) if and only if \(a<0<b\) or \(b<0<a\) .
Theorem 5.17. For all positive real numbers \(a\) and \(b\) , \(a< b\) if and only if \(a^2< b^2\) .
Consider using three cases when approaching the proof of the following theorem.
Theorem 5.18. For all \(a\in\mathbb{R}\) , we have \(a^2\geq 0\) .
It might come as a surprise that the following result requires proof.
Theorem 5.19. We have \(0<1\) .
The previous theorem together with Theorem 5.14 implies that \(-1<0\) as you expect. It also follows from Axiom O3 that for all \(n\in\mathbb{Z}\) , we have \(n<n+1\) . We assume that there are no integers between \(n\) and \(n+1\) .
Theorem 5.20. For all \(a\in\mathbb{R}\) , if \(a>0\) , then \(a^{-1}>0\) , and if \(a<0\) , then \(a^{-1}<0\) .
Theorem 5.21. For all \(a,b\in \mathbb{R}\) , if \(a<b\) , then \(-b<-a\) .
The last few results allow us to take for granted our usual understanding of which real numbers are positive and which are negative. The next theorem yields a result that extends Theorem 5.21.
Theorem 5.22. For all \(a,b,c\in \mathbb{R}\) , if \(a<b\) and \(c<0\) , then \(bc<ac\) .
There is a special function that we can now introduce.
Definition 5.23. Given \(a\in\mathbb{R}\) , we define the absolute value of \(a\) , denoted \(|a|\) , via \[|a|:= \begin{cases} a, & \text{if }a\geq 0\\ -a, & \text{if }a<0. \end{cases}\]
Theorem 5.24. For all \(a\in\mathbb{R}\) , \(|a|\geq 0\) with equality only if \(a=0\) .
We can interpret \(|a|\) as the distance between \(a\) and 0 as depicted in Figure 5.1.
Theorem 5.25. For all \(a,b\in\mathbb{R}\) , we have \(|a-b|=|b-a|\) .
Given two points \(a\) and \(b\) , \(|a-b|\) , and hence \(|b-a|\) by the previous theorem, is the distance between \(a\) and \(b\) as shown in Figure 5.2.
Theorem 5.26. For all \(a,b\in\mathbb{R}\) , \(|ab|=|a||b|\) .
In the next theorem, writing \(\pm a\leq b\) is an abbreviation for \(a\leq b\) and \(-a\leq b\) .
Theorem 5.27. For all \(a,b\in\mathbb{R}\) , if \(\pm a\leq b\) , then \(|a|\leq b\) .
Theorem 5.28. For all \(a\in\mathbb{R}\) , \(|a|^2=a^2\) .
Theorem 5.29. For all \(a\in\mathbb{R}\) , \(\pm a\leq |a|\) .
Theorem 5.30. For all \(a,r\in\mathbb{R}\) with \(r\) nonnegative, \(|a|\leq r\) if and only if \(-r\leq a\leq r\) .
The letter \(r\) was used in the previous theorem because it is the first letter of the word “radius". If \(r\) is positive, we can think of the interval \((-r,r)\) as the interior of a one-dimensional circle with radius \(r\) centered at 0. Figure 5.3 provides a visual interpretation of Theorem 5.30.
Corollary 5.31. For all \(a,b,r\in\mathbb{R}\) with \(r\) nonnegative, \(|a-b|\leq r\) if and only if \(b-r\leq a\leq b+r\) .
Since \(|a-b|\) represents the distance between \(a\) and \(b\) , we can interpret \(|a-b|\leq r\) as saying that the distance between \(a\) and \(b\) is less than or equal to \(r\) . In other words, \(a\) is within \(r\) units of \(b\) . See Figure 5.4.
Consider using Theorems 5.29 and 5.30 when attacking the next result, which is known as the Triangle Inequality . This result can be extremely useful in some contexts.
Theorem 5.32. For all \(a,b\in\mathbb{R}\) , \(|a+b|\leq |a|+|b|\) .
Figure 5.5 depicts two of the cases for the Triangle Inequality.
Problem 5.33. Under what conditions do we have equality for the Triangle Inequality?
Where did the Triangle Inequality get its name? Why “Triangle"? For any triangle (including degenerate triangles), the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. That is, if \(x\) , \(y\) , and \(z\) are the lengths of the sides of the triangle, then \(z\leq x+y\) , where we have equality only in the degenerate case of a triangle with no area. In linear algebra, the Triangle Inequality is a theorem about lengths of vectors. If \(\mathbf{a}\) and \(\mathbf{b}\) are vectors in \(\mathbb{R}^n\) , then the Triangle Inequality states that \(\lVert \mathbf{a}+\mathbf{b}\rVert \leq \lVert\mathbf{a}\rVert +\lVert\mathbf{b}\rVert\) . Note that \(\lVert \mathbf{a}\rVert\) denotes the length of vector \(\mathbf{a}\) . See Figure 5.6. The version of the Triangle Inequality that we presented in Theorem 5.32 is precisely the one-dimensional version of the Triangle Inequality in terms of vectors.
The next theorem is sometimes called the Reverse Triangle Inequality .
Theroem 5.34. For all \(a,b\in\mathbb{R}\) , \(|a-b|\geq \left||a|-|b| \right|\) .
Before we introduce the Completeness Axiom, we need some additional terminology.
Definition 5.35. Let \(A\subseteq \mathbb{R}\) . A point \(b\) is called an upper bound of \(A\) if for all \(a\in A\) , \(a\leq b\) . The set \(A\) is said to be bounded above if it has an upper bound.
Problem 5.36. The notion of a lower bound and the property of a set being bounded below are defined similarly. Try defining them.
Problem 5.37. Find all upper bounds and all lower bounds for each of the following sets when they exist.
- \(\{5,11,17,42,103\}\)
- \(\mathbb{N}\)
- \(\mathbb{Z}\)
- \((0,1]\)
- \((0,1]\cap \mathbb{Q}\)
- \((0,\infty)\)
- \(\{42\}\)
- \(\{\frac{1}{n}\mid n\in\mathbb{N}\}\)
- \(\{\frac{1}{n}\mid n\in\mathbb{N}\}\cup\{0\}\)
- \(\emptyset\)
Definition 5.38. A set \(A\subseteq \mathbb{R}\) is bounded if \(A\) is bounded above and below.
Notice that a set \(A\subseteq\mathbb{R}\) is bounded if and only if it is a subset of some bounded closed interval.
Definition 5.39. Let \(A\subseteq \mathbb{R}\) . A point \(p\) is a supremum (or least upper bound ) of \(A\) if \(p\) is an upper bound of \(A\) and \(p\leq b\) for every upper bound \(b\) of \(A\) . Analogously, a point \(p\) is an infimum (or greatest lower bound ) of \(A\) if \(p\) is a lower bound of \(A\) and \(p\geq b\) for every lower bound \(b\) of \(A\) .
Our next result tells us that a supremum of a set and an infimum of a set are unique when they exist.
Theorem 5.40. If \(A\subseteq \mathbb{R}\) such that a supremum (respectively, infimum) of \(A\) exists, then the supremum (respectively, infimum) of \(A\) is unique.
In light of the previous theorem, if the supremum of \(A\) exists, it is denoted by \(\sup(A)\) . Similarly, if the infimum of \(A\) exists, it is denoted by \(\inf(A)\) .
Problem 5.41. Find the supremum and the infimum of each of the sets in Problem 5.37 when they exist.
It is important to recognize that the supremum or infimum of a set may or may not be contained in the set. In particular, we have the following theorem concerning suprema and maximums. The analogous result holds for infima and minimums.
Theorem 5.42. Let \(A\subseteq \mathbb{R}\) . Then \(A\) has a maximum if and only if \(A\) has a supremum and \(\sup(A)\in A\) , in which case the \(\max(A)=\sup(A)\) .
Intuitively, a point is the supremum of a set \(A\) if and only if no point smaller than the supremum can be an upper bound of \(A\) . The next result makes this more precise.
Theorem 5.43. Let \(A\subseteq \mathbb{R}\) such that \(A\) is bounded above and let \(b\) be an upper bound of \(A\) . Then \(b\) is the supremum of \(A\) if and only if for every \(\varepsilon >0\) , there exists \(a\in A\) such that \(b-\varepsilon <a\) .
Problem 5.44. State and prove the analogous result to Theorem 5.43 involving infimum.
The following axiom states that every nonempty subset of the real numbers that has an upper bound has a least upper bound.
Axioms 5.45. If \(A\) is a nonempty subset of \(\mathbb{R}\) that is bounded above, then \(\sup(A)\) exists.
Given the Completeness Axiom, we say that the real numbers satisfy the least upper bound property . It is worth mentioning that we do not need the Completeness Axiom to conclude that every nonempty subset of the integers that is bounded above has a supremum, as this follows from Theorem 4.40 (a generalized version of the Well-Ordering Principle).
Certainly, the real numbers also satisfy the analogous result involving infimum.
Theorem 5.46. If \(A\) is a nonempty subset of \(\mathbb{R}\) that is bounded below, then \(\inf(A)\) exists.
Our next result, called the Archimedean Property , tells us that for every real number, we can always find a natural number that is larger. To prove this theorem, consider a proof by contradiction and then utilize the Completeness Axiom and Theorem 5.43.
Theorem 5.47. For every \(x\in\mathbb{R}\) , there exists \(n\in\mathbb{N}\) such that \(x<n\) .
More generally, we can “squeeze" every real number between a pair of integers. The next result is sometimes referred to at the Generalized Archimedean Property .
Theorem 5.48. For every \(x\in\mathbb{R}\) , there exists \(k,n\in\mathbb{Z}\) such that \(k<x<n\) .
Theorem 5.49. For any positive real number \(x\) , there exists \(N\in \mathbb{N}\) such that \(0<\frac{1}{N}<x\) .
The next theorem strengthens the Generalized Archimedean Property and says that every real number is either an integer or lies between a pair of consecutive integers. To prove this theorem, let \(x\in\mathbb{R}\) and define \(L=\{k\in\mathbb{Z}\mid k\leq x\}\) . Use the Generalized Archimedean Property to conclude that \(L\) is nonempty and then utilize Theorem 4.40.
Theorem 5.50. For every \(x\in\mathbb{R}\) , there exists \(n\in \mathbb{Z}\) such that \(n\leq x<n+1\) .
To prove the next theorem, let \(a<b\) , utilize Theorem 5.49 on \(b-a\) to obtain \(N\in\mathbb{N}\) such that \(\frac{1}{N}<b-a\) , and then apply Theorem 5.50 to \(Na\) to conclude that there exists \(n\in\mathbb{N}\) such that \(n\leq Na<n+1\) . Lastly, argue that \(\frac{n+1}{N}\) is the rational number you seek.
Theorem 5.51. If \((a,b)\) is an open interval, then there exists a rational number \(p\) such that \(p\in(a,b)\) .
Recall that the real numbers consist of rational and irrational numbers. Two examples of an irrational number that you are likely familiar with are \(\pi\) and \(\sqrt{2}\) . In Section 6.2, we will prove that \(\sqrt{2}\) is irrational, but for now we will take this fact for granted. It turns out that \(\sqrt{2}\approx 1.41421356237\in (1,2)\) . This provides an example of an irrational number occurring between a pair of distinct rational numbers. The following theorem is a good challenge to generalize this.
Theorem 5.52. If \((a,b)\) is an open interval, then there exists an irrational number \(p\) such that \(p\in(a,b)\) .
Repeated applications of the previous two theorems implies that every open interval contains infinitely many rational numbers and infinitely many irrational numbers. In light of these two theorems, we say that both the rationals and irrationals are dense in the real numbers.