The Numbers

TagsMATH 115

What is analysis?

Analysis is the breakdown of calculus and the rigorous buildup of each subject. We start by the definitions of certain spaces and symbols (here), then we build up to sequences, continuity, differentiation, and finally integration. At times, it will feel backward, because we know everything that will be proven in this course. But that’s kinda fun!

PROOF TIPS πŸ”¨

Natural Numbers

The set of natural numbers N\mathbb{N}ο»Ώ is just the set of positive integers, with a special successor. For every nnο»Ώ, let succ(n)=n+1succ(n) = n + 1ο»Ώ.

Peano Axioms 🐧

  1. 1∈N1 \in N
  1. if n∈Nn \in N, then succ(n)∈Nsucc(n) \in N
  1. 11ο»Ώ is not the successor of any element in NNο»Ώ
  1. If succ(n)=succ(m)succ(n) = succ(m)ο»Ώ, then n=mn = mο»Ώ
  1. If SβŠ†NS \subseteq Nο»Ώ and
    1. 1∈S1 \in S
    1. if n∈S,succ(n)∈Sn \in S, succ(n) \in S
    1. then, S=NS = Nο»Ώ

Now, these axioms are showing a test about a set and a successor definition. It happens that N\mathbb{N}ο»Ώ and succ(n)=n+1succ(n) = n + 1ο»Ώ satisfy these axioms. But you can easily define other sets that work too. For example, let T={1,3,5,…}T = \{1, 3, 5, …\}ο»Ώ and let succ(n)=n+2succ(n) = n + 2ο»Ώ. This works!

Peano and Induction πŸš€

We can show that the fifth statement actually shows the basis of matematical induction.

Let P1,P2,P3,...P_1, P_2,P_3, ...ο»Ώ be statements that may or may not be true. Let SSο»Ώ be the set of all true statements. With induction, we need to show that

And by the fifth peano axiom, if we have shown this, then we have shown that all Pk,k∈NP_k, k \in \mathbb{N} is true.

Formal definition of addition 🐧

We want to define a+ba + bο»Ώ, and we do this inductively. We start with the base case of a+1a + 1ο»Ώ, which is just succ(a)succ(a)ο»Ώ.

Now, we move to the inductive case. Given a+ba + bο»Ώ, we want to see what a+succ(b)a + succ(b)ο»Ώ is.

a+succ(b)=a+(b+1)=(a+b)+1=succ(a+b)a + succ(b) = a + (b + 1) = (a + b) + 1 = succ(a + b)

which completes the induction

Here, we define the associative property, which allows us to inductively define addition. Without the associative property, there is no addition.

Formal definition of multiplication 🐧

We want to define ababο»Ώ inductively. The base case is aβˆ—1a * 1ο»Ώ, which is just aaο»Ώ. Inductively, we assume that we have ababο»Ώ. We want to see what a(succ(b))a(succ(b))ο»Ώ is.

aβˆ—succ(b)=a(b+1)=ab+aa*succ(b) = a(b + 1) = ab + a

which completes the induction, with help from our formal definition of addition.

Here, we define the distributive property. without the distribution property, there is no multiplication

Formal definition of order 🐧

Similarly, we want to define what it means to have a<ba < bο»Ώ and so on. We do this through two stages of induction. First, as the base case, we set a=1a = 1ο»Ώ. Now, we show in an inner induction that $$1 < b for all b>1b > 1ο»Ώ.

In this base case, we have 1<succ(1)1 < succ(1)ο»Ώ, which is true. For the inductive step, we are given 1<b1 < bο»Ώ and want to show that 1<succ(b)1 < succ(b)ο»Ώ. We can expand to get 1<b+11 < b + 1ο»Ώ, which gets us 0<b0 < bο»Ώ, and because we have established that 0<10 < 1ο»Ώ (I think) and we use the transitive property, we are done with this induction. Again, like addition and multiplication, we notice that transitivity is necessary for the definition of order.

Now, for the rest of the induction, we just need to do the same inductive step as above, but with aaο»Ώ generically. The same steps hold.

Minimal Principle πŸš€

If SβŠ†NS \subseteq \mathbb{N}ο»Ώ and is not an empty set, then SSο»Ώ has a minimal element.

Integers

Moving from natural numbers 🐧

We’ve shown addition on the natural numbers. But we are left unsatisfied, because it is possible to have some x+a=bx + a = bο»Ώ such that xβˆ‰Nx \notin \mathbb{N}ο»Ώ. This motivates the need for Z\mathbb{Z}ο»Ώ. This general strategy is something we will use to develop the whole number system. We start at one place, and find that we can construct something else with what we have.

The symbol Z\mathbb{Z}ο»Ώ is the set of integers, which can also be expressed as

βˆ’Nβˆͺ{0}βˆͺN-\mathbb{N}\cup \{0\} \cup \mathbb{N}

With ZZο»Ώ, we have addition, multiplication, subtraction, and ordering. These can be proved in a similar way.

Modified minimal principle 🐧

Any finite set of SβŠ‚ZS \subset Zο»Ώ does have a minimal element, but we can’t say that this is the case for all sets. For infinite sets, we can say the following:

Let BBο»Ώ be a lower bound on the set SβŠ†ZS \subseteq \mathbb{Z}ο»Ώ. We can perform a bijection on this set to S’S’ by setting

Sβ€²=Sβˆ’B+1S' = S - B + 1

This brings us into the realm of natural numbers. Now, we can safely apply the minimal principle.

So the tl;dr is that the following are necessary and sufficient conditions to have a minimum

If the set is infinite and lower bounded, you end up with an infimum.

Rational Numbers

Moving from integers 🐧

Similarly to how we motivated ZZο»Ώ, we see that equations of xβˆ—a=bx * a = bο»Ώ where b,a∈Zb, a \in Zο»Ώ may yield an xβˆ‰Zx \notin Zο»Ώ. This requires the development of QQο»Ώ.

The symbol Q\mathbb{Q} represents the set of rational numbers m/nm/n such that m,n∈Zm, n\in \mathbb{Z}. There are a ton of rational numbers (in fact, infinitely many of the same cardinality of Z\mathbb{Z}), but there are still gaps.

Modified minimal principle 🐧

Like before, any finite set SβŠ‚QS \subset \mathbb{Q}ο»Ώ does have a minimal element. However, unlike ZZο»Ώ, having a lower bound on a set SβŠ†QS \subseteq \mathbb{Q}ο»Ώ does NOT guarentee that there is a smallest element. For example, we can define

S={n;n∈Q,n>0}S = \{n; n\in \mathbb{Q}, n >0\}

The lower bound here is 00ο»Ώ, but there are an infinite number of elements in the set SSο»Ώ, with none of them being the smallest. You can always propose something smaller.

Therefore, in QQο»Ώ, we encounter some weirdness. We can define lower bounds, but this doesn’t necessarily translate to having a smallest element in the set.

Infimum and supremum ⭐🐧

The infimum is the greatest lower bound. This is formally with two necessary and sufficient terms

  1. for all x∈Sx \in Sο»Ώ, xβ‰₯inf⁑(S)x \geq \inf(S)ο»Ώ. This establishes the lower bound part
  1. If there exists some x’β‰₯inf⁑(S)x’ \geq \inf(S)ο»Ώ, then there exists some x∈Sx\in Sο»Ώ such that x’>xx’ > xο»Ώ. This establishes the greatest lower bound part.

The supremum is the lowest upper bound. There are two very similar necessary and sufficient terms

  1. for all x∈Sx \in Sο»Ώ, x≀sup⁑(S)x \leq \sup(S)ο»Ώ. This establishes the upper bound part
  1. If there exists some x’≀sup⁑(S)x’ \leq \sup(S)ο»Ώ, then there exists some x∈Sx\in Sο»Ώ such that x>xβ€²x > x'ο»Ώ. This establishes the lowest upper bound part.

Here’s an important thing to note: all minimums are infimums, and all infimums are lower bounds. All maximums are supremums, and all supremums are upper bounds. We are going from tight to loose to looser.

In subsets of N and Z, either infimums/supremums don’t exist, or they are equal to maximum/minimum. In subsets of Q, it may not be another element from the set. It may not even be another element from Q itself. This fact is actually critical for establishing R, which we will do in a later section.

πŸ‘‰
Infimum and supreums are typically not defined for unbounded sets; the notion of inf⁑S=∞\inf S = \infty is not very common. Typically, you would say taht inf⁑S\inf S is not defined.

Proving irrationality in general πŸ”¨

Typically, if something is irrational, we can prove by contradiction. Let m/nm/nο»Ώ be equal to the number in question and solve for mmο»Ώ and nnο»Ώ.

Proving irrationality through rational zeros theorem πŸ”¨

We define a number an algebraic number if when you plug it into a polynomial cnxn+…+c0c_nx^n + … + c_0ο»Ώ, it yields 0, where all ccο»Ώ are integers.

Rational numbers are always algebraic numbers, because if r=m/nr = m/nο»Ώ, then we can set up nxβˆ’m=0nx - m = 0ο»Ώ with that solution. However, not all algebraic numbers are rational, because it includes irrational square roots, etc.

Given a polynomial

if r=c/dr = c/dο»Ώ, a rational number that satisfies the equation, then ccο»Ώ divides c0c_0ο»Ώ and ddο»Ώ divides cnc_nο»Ώ.

This means all rational roots must satisfy the above constraints. Therefore, you can actually enumerate out the rational roots and find all of them easily.

And therefore, we stumble upon another way of proving rationality. Every algebraic number can be expressed as a polynomial (set xxο»Ώ equal to the expression, and square/cube, etc stuff until it’s integers). Find all the rational solutions using the rational zeros theorem and enumerate. If none of them yield a zero, then you say that none of the roots are rational, and therefore the original number, which was a root, can’t possibly be rational.

Pro tips

Real Numbers

One thing we note is that we can define some set SβŠ‚QS \subset \mathbb{Q}ο»Ώ such that inf(S)βˆ‰Qinf(S) \notin \mathbb{Q}ο»Ώ. An easy example is

S={r∈Q,r>0,r2>2}S = \{r \in \mathbb{Q}, r > 0, r^2 >2\}

which has inf(S)=2inf(S) = \sqrt{2}ο»Ώ. This shows us that QQο»Ώ is incomplete, and it also gives us an idea for how to β€œfill in” those gaps through the infimum definition

Dedekind cuts 🐧

A dedekind cut is a pair of subsets S,LS, Lο»Ώ such that

  1. SβŠ‚Q,LβŠ‚QS\subset Q, L \subset Qο»Ώ
  1. SβˆͺL=QS \cup L = Qο»Ώ
  1. S≠Q,L≠QS \neq Q, L \neq Q
  1. for any x∈S,y∈L,x<yx \in S, y \in L, x < y
  1. SSο»Ώ has no maximal element, and LLο»Ώ has no minimal element (although you might have a minimal element if LLο»Ώ is defined to have a minimal element…)

Graphically this looks like a number line with a singular cut in it.

The key insight here is that we can use Dedekind cuts to establish numbers! Notationally, we usually say that r=(S,L)r = (S, L)ο»Ώ where the () represents the cut.

We observed in the previous section that infimums of rational element sets don’t necessarily belong to Q. The Dedekind cuts exploits this to make the jump from Q to R, as we alluded to before. In fact, we now define the real numbers R as the set of possible Dedekind cuts with constraints in Q.

Properties of Dedekind cuts πŸš€

Laws of addition & multiplication 🐧

Like before, we must establish simple laws in our new system of Dedekind cuts. Addition is pretty straightforward. If we have r1∈R,r2∈Rr_1 \in R, r_2 \in R, we have Dedekind cuts Sr1,Lr1,Sr2,Lr2S_{r1}, L_{r1}, S_{r2}, L_{r2}. We want to formulate a dedekind cut Sr1+r2S_{r_1 +r_2}, Lr1+r2L_{r_1 +r_2} that corresponds to r1+r2r_1 + r_2.

As it turns out, we can just do this

Sr1+r2={a+b,a∈Sr1,b∈Sr2}S_{r_1 +r_2} = \{a + b, a \in S_{r_1}, b \in S_{r_2}\}

which means that

Lr1+r2=Q\Sr1+r2L_{r_1 + r_2} = \mathbb{Q} \backslash S_{r_1 + r_2}

Because we were able to form a Dedekind cut that represents r1+r2r_1 + r_2ο»Ώ, we are done. Now, to show that this is a valid cut, you would need to show that the Dedekind cut properties actually hold, but let’s just assume that we can do that.

You can also define multiplication in a similar pairwise way.

Laws of ordering 🐧

We can also define ordering in terms of Dedekind cuts in actually a really elegant way

  1. r1≀r2r_1 \leq r_2ο»Ώ iff Sr1βŠ†Sr2S_{r_1} \subseteq S_{r_2}ο»Ώ and vice versa
  1. r1=r2r_1 = r_2ο»Ώ iff Sr1=Sr2S_{r_1} = S_{r_2}ο»Ώ.

This should make sense pictorially.

Density of rational numbers in R πŸš€β­

Theorem: for all r1,r2∈Rr_1, r_2 \in \mathbb{R}, there exists some q∈Qq \in \mathbb{Q} such that r1<Q<r2r_1 < Q < r_2.

Upshot: even though there are holes in Q, the set of rational numbers is dense in R. This is where it becomes funky, just because the concept of infinity is weird to us.

We can also show something stronger: there exists an infinite number of q∈Qq \in Q between any pair of real numbers. You can do this by finding two qq in [r1,r2][r_1, r_2] (just split the inteval in half and use the previous theorem to get the two qq, and from these two qq, you can make an infinite number of rational numbers

The completeness Axiom β­πŸš€

This is the most important result here. We show that R\mathbb{R}ο»Ώ doesn’t lack anything (i.e. no holes), unlike all other number system. We do this by showing that all subsets of R\mathbb{R}ο»Ώ with a lower bound have an infimum (naturally in R\mathbb{R}ο»Ώ).

Why does this show completeness? Well, suppose that we constructed a Dedekind cut, but this time, it’s across R\mathbb{R}ο»Ώ, i.e. SβˆͺL=RS\cup L = \mathbb{R}ο»Ώ. Well, in this case, the cut is exactly at inf⁑(L)\inf(L)ο»Ώ, so if we show that this always exists for any set with a lower bound, then we have shown taht the Dedekind cut is closed on R\mathbb{R}ο»Ώ, which shows completeness.

Archimedean property πŸš€

If a>0,b>0a > 0, b > 0ο»Ώ, then for some positive integer nnο»Ώ, we have na>bna > bο»Ώ.

What this is saying is that no b∈Rb \in R is infinitesimal with respect to another a∈Ra \in R.

Fields (additional topic)

Rational numbers (not natural numbers) and real numbers are fields. This means that they satisfy associativity, commutatitivy, and distribution.

Now, fields are mathematical objects; they are not derived, as they are, from Peano axioms. They are just a label we slap onto certain mathematical spaces and it is helpful.

The theory of rational numbers and real numbers rests on these (often obvious) properties. There are some natural consequences to field properties

Ordered fields

What’s more, rational and real numbers are ordered fields, which means that there are a few additional properties that hold

These yield certain results

Practical Max and Min Advice

Unfortunately, you can’t say much about max⁑(anβˆ’bn)\max(a_n - b_n)ο»Ώ unless all an>bna_n > b_nο»Ώ.