Calculus/Real numbers

Fields

You are probably already familiar with many different sets of numbers from your past experience. Some of the commonly used sets of numbers are

• Natural numbers, usually denoted with an $\mathbb {N}$ , are the numbers $1,2,3,\ldots$
• Integers, usually denoted with a $\mathbb {Z}$ , are the positive and negative natural numbers: $\ldots ,-3,-2,-1,0,1,2,3,\ldots$
• Rational numbers, denoted with a $\mathbb {Q}$ , are fractions of integers (excluding division by zero): $\left\{{\frac {p}{q}}:p,q\in \mathbb {Z} ,q\neq 0\right\}$
• Real numbers, denoted with a $\mathbb {R}$ , are constructed and discussed below.

Note that different sets of numbers have different properties. In the set integers for example, any number always has an additive inverse: for any integer $x$ , there is another integer $t$ such that $x+t=0$ This should not be terribly surprising: from basic arithmetic we know that $t=-x$ . Try to prove to yourself that not all natural numbers have an additive inverse.

In mathematics, it is useful to note the important properties of each of these sets of numbers. The rational numbers, which will be of primary concern in constructing the real numbers, have the following properties:

There exists a number 0 such that for any other number $a$ , $0+a=a+0=a$

For any two numbers $a,b$ , $a+b$ is another number

For any three numbers a,b, and c, a+(b+c)=(a+b)+c

For any number a there is another number -a such that a+(-a)=0

For any two numbers a and b, a+b=b+a

For any two numbers a and b, a*b is another number

There is a number 1 such that for any number a, a*1=1*a=a

For any two numbers a and b, a*b=b*a

For any three numbers a,b and c, a(bc)=(ab)c

For any three numbers a,b and c, a(b+c)=ab+ac

For every number a there is another number a^-1 such that aa^-1=1

As presented above, these may seem quite intimidating. However, these properties are nothing more than basic facts from arithmetic. Any collection of numbers (and operations + and * on those numbers) which satisfies the above properties is called a field. The properties above are usually called field axioms. As an exercise, determine if the integers form a field, and if not, which field axiom(s) they violate.

Even though the list of field axioms is quite extensive, it does not fully explore the properties of rational numbers. Rational numbers also have an ordering.' A total ordering must satisfy several properties: for any numbers a, b, and c

if a ≤ b and b ≤ a then a = b (antisymmetry)

if a ≤ b and b ≤ c then a ≤ c (transitivity)

a ≤ b or b ≤ a (totality)

To familiarize yourself with these properties, try to show that (a) natural numbers, integers and rational numbers are all totally ordered and more generally (b) convince yourself that any collection of rational numbers are totally ordered (note that the integers and natural numbers are both collections of rational numbers).

Finally, it is useful to recognize one more characterization of the rational numbers: every rational number has a decimal expansion which is either repeating or terminating. The proof of this fact is omitted, however it follows from the definition of each rational number as a fraction. When performing long division, the remainder at any stage can only take on positive integer values smaller than the denominator, of which there are finitely many.

Constructing the Real Numbers

There are two additional tools which are needed for the construction of the real numbers: the upper bound and the least upper bound. Definition A collection of numbers E is bounded above if there exists a number m such that for all x in E x≤m. Any number m which satisfies this condition is called an upper bound of the set E.

Definition If a collection of numbers E is bounded above with m as an upper bound of E, and all other upper bounds of E are bigger than m, we call m the least upper bound or supremum of E, denoted by sup E.

Many collections of rational numbers do not have a least upper bound which is also rational, although some do. Suppose the numbers 5 and 10/3 are, together, taken to be E. The number 5 is not only an upper bound of E, it is a least upper bound. In general, there are many upper bounds (12, for instance, is an upper bound of the collection above), but there can be at most one least upper bound.

Consider the collection of numbers $\{3,3.1,3.14,3.141,3.1415,\dots \}$: You may recognize these decimals as the first few digits of pi. Since each decimal terminates, each number in this collection is a rational number. This collection has infinitely many upper bounds. The number 4, for instance, is an upper bound. There is no least upper bound, at least not in the rational numbers. Try to convince yourself of this fact by attempting to construct such a least upper bound: (a) why does pi not work as a least upper bound (hint: pi does not have a repeating or terminating decimal expansion), (b) what happens if the proposed supremum is equal to pi up to some decimal place, and zeros after (c) if the proposed supremum is bigger than pi, can you find a smaller upper bound which will work?

In fact, there are infinitely many collections of rational numbers which do not have a rational least upper bound. We define a real number to be any number that is the least upper bound of some collection of rational numbers.

Properties of Real Numbers

The reals are totally ordered.

For all reals; a, b, c

Either b>a, b=a, or b<a.

If a<b and b<c then a<c

Also

b>a implies b+c>a+c

b>a and c>0 implies bc>ac

b>a implies -a>-b

Upper bound axiom

Every non-empty set of real numbers which is bounded above has a supremum.

The upper bound axiom is necessary for calculus. It is not true for rational numbers.

We can also define lower bounds in the same way.

Definition A set E is bounded below if there exists a real M such that for all x∈E x≥M Any M which satisfies this condition is called an lower bound of the set E

Definition If a set, E, is bounded below, M is an lower bound of E, and all other lower bounds of E are less than M, we call M the greatest lower bound or inifimum of E, denoted by inf E

The supremum and infimum of finite sets are the same as their maximum and minimum.

Theorem

Every non-empty set of real numbers which is bounded below has an infimum.

Proof:

Let E be a non-empty set of real numbers, bounded below

Let L be the set of all lower bounds of E

L is not empty, by definition of bounded below

Every element of E is an upper bound to the set L, by definition

Therefore, L is a non empty set which is bounded above

L has a supremum, by the upper bound axiom

1/ Every lower bound of E is ≤sup L, by definition of supremum

Suppose there were an e∈E such that e<sup L

Every element of L is ≤e, by definition

Therefore e is an upper bound of L and e<sup L

This contradicts the definition of supremum, so there can be no such e.

If e∈E then e≥sup L, proved by contradiction

2/ Therefore, sup L is a lower bound of E

inf E exists, and is equal to sup L, on comparing definition of infinum to lines 1 & 2

Bounds and inequalities, theorems: $A\subseteq B\Rightarrow \sup A\leq \sup B$ $A\subseteq B\Rightarrow \inf A\geq \inf B$ $\sup A\cup B=\max(\sup A,\sup B)$ $\inf A\cup B=\min(\inf A,\inf B)$

Theorem: (The triangle inequality)

$\forall a,b,c\in \mathbb {R} \quad |a-b|\leq |a-c|+|c-b|$

Proof by considering cases

If a≤b≤c then |a-c|+|c-b| = (c-a)+(c-b) = 2(c-b)+(b-a)≥b-a = |b-a|

Exercise: Prove the other five cases.

This theorem is a special case of the triangle inequality theorem from geometry: The sum of two sides of a triangle is greater than or equal to the third side. It is useful whenever we need to manipulate inequalities and absolute values.