Why 1+1=2 : Proving mathematical theorem
Seems like a silly thing to explain. You probably say, “Because it is.” Or maybe more formally, you’d say, “Because that’s how we’ve defined these concepts.” And the latter answer is basically correct, but I thought I’d go through the formal mathematics, giving rigorous definitions, and defining axioms. My goal is to explain these concept in a manner a non-math nerd could understand, with the goal of helping you understand how math “works”.
You can get information this all from the Wikipedia, but the Wikipedia doesn’t give you much hand holding, and it’s really hard to learn mathematical concepts from it. It focuses on giving mathematically formal definitions and constructions of systems. And mathematically correct formulations are not made to be easy to understand, just correct and rigorous. I’ll give you a mathematically correct treatment, but I’ll also break out of the strict math to give you examples of things that might refer to concepts we haven’t strictly defined yet. But I’ve mostly lifted this from the Wikipedia.
Anyway, so “1+1=2”. I have to define “1”,”+”,”=”, and “2”. I’ll define a few more things along the way and I’ll set out rules about the things I’m defining.
Along the way I’m going to define the whole numbers, (0,1,2,3…) using the Peano axioms. There’s actually other ways to define the whole numbers. (In other words the Peano system has isomorphisms. In math isomorphic systems are two systems that can be shown to actually be the same the thing.) Aside from the Peano axioms, there’s a lot of definitions for whole numbers that are based on set theory. I don’t like those ones as much because I feel like they basically just take sets and use them to develop examples of counting. While I can’t say they aren’t mathematically rigorous, they feel like cheating to me. Also, digging deep into set theory would be a lot to cover just to start defining numbers.
Basic outline of what we will do
Interestingly, laying out the basic terms was the longest part. But don’t worry it shouldn’t be too bad. The last section: “Bonus: More operations and bigger, better number systems!” is a bit long too. It also moves a little fast, but you can skip that part.
1 Define some basic mathematical terms a Mathematical Expression b Primitive notion i Definition ii Axiom c Theorem d Variable e Set f Binary relation g Function
2 Define equality a Reflexivity b Symmetry c Transitivity
3 Define whole numbers with the Peano axioms a Define zero b Define successor operation c A couple more successor axioms d Mathematical induction axiom
5 Define 1 and 2
6 Define Addition
7 Prove that 1+2=2
8 Bonus: More operations and bigger, better number systems!
Define some basic mathematical terms
Okay, we’ll need to get some definitions out of the way. The first ones are a little vague because we can’t really define them in terms of other things. You can look up some of these in the Wikipedia, but it tends to over-explain them. The big pointer for this first part is not to over think these terms. They all mean very basic and simple things.
My geometry teacher made a pretty good analogy, concerning starting off in a mathematical system. If you have a dictionary, it’s going to define words. But eventually it will have to come down to a set of words that it either doesn’t define, or it starts to define words in a circular manner. In math, we start out with a few basic concepts, that we can’t really define well. In practice you just have to intuitively “get” them. But we can circle around and put conditions on what those terms mean by making up axioms about them.
I’ll start with the term “Mathematical Expression”. By mathematical expression, I just thing we are using math to study. It could refer to a number or a shape or or an equation or whatever.
Now, let’s go to primitive notion. A primitive notion is an undefined concept or starting point of you mathematical system. I already gave you one, “mathematical expression”. An example from geometry is a “point”. A point just represents a location in space. It has zero size, etc. But it doesn’t really have a mathematical definition. You can define point and then make up rules about them, but you start with no definition. When you build your mathematical system, your goal is to have as few primitive notions as possible.
The main kind of primitive notion is an axiom. When you build your mathematical system, you just pull a couple of rules out of your butt and declare that that’s the way it is. A little later, I’m going to tell you that if a=b, then b=a. Why? Because I said so, that’s why. Making up this rule helps us create a mathematical structure that helps us understand various things.
Another primitive notion can be a definition. However, a definition doesn’t really have to be a primitive notation. For example, I told you the term “point” doesn’t really have a definition. It’s a primitive concept. However, once you have some words and axioms, you can describe a mathematical object and then staple a word to it. So you can define parallel lines as lines that never intersect. So it’s not a totally primitive concept.
Okay, so math is just making up words and rules and then you go home? Of course not! Once you define some words and make some rules you can make logical deductions about how the system works. These deductions are called theorems. Your first theorems will just come directly from axioms and definitions, but once you write a few theorems, you can make theorems from your theorems.
If I tell you any number plus zero is that number, and then ask you what’s five plus zero, you can tell me five. That is a simple theorem, it just takes one rule and applies it to a specific case. Obviously if you want to write a theorem that will be published in a math journal, your going to have to put together a few axioms and come up with a more clever theorem.
For this article, we are going to create a theorem that 1+1=2.
A variable represents a mathematical expression that is known or unknown. So you’ve seen things like x=3+2. “x” is a variable. In this case, we can figure out what number x is. But sometimes we’ll say, ‘Let “a” and “b” be whole numbers, if a=b, then b=a.’ In this case, you can’t figure what “a” and “b” are. This is on purpose. I’m not saying “a” and “b” are mystery numbers to find, I’m saying they can be any number. They are a generic stand-in for all numbers.
I think this is a double usage of variable is a common point of confusion. In this article, variables are always going to be general stand-ins for any whole number. But when you were in school you probably got a question: “7=4+x”. What your teacher meant to say was ‘Given the axioms x is a number and “7=4+x”, create a theorem that proves that x is a specific number.’
A set is just a group of things. Maybe the set of things in your lunchbox is a sandwich, apple, and can of pop. My family is a set of people, my daughter, my wife, and me. Obviously there’s sets that are a little more math specific. The set of all whole numbers, the set of all odd numbers, the set of all sets of odd numbers. This is a good time to point out that a set might have an infinite number of things in it. Or it might have a just a finite number of things in it, it might have nothing in it.
A binary relation takes two things that are in the same set and defines a relationship between them. Remember what I said, don’t overthink this! So equality, which we’ll define later is a relation. You can make a statement that 5=5. Greater than is a relation too, 8<9. In geometry, if two triangles have the same angles, they have a congruence relationship.
A function is a relation in between a set of inputs and outputs, so that for every input, you get one output value. You normally define a function by saying f(x). So let’s define a function:
f(x)=x+4.
So for any number, you input the function gives you the number four higher. When you want to to apply the function to a certain input value, like 3, you write f(3). In our case, f(3) is 7. You can also use another letter like “S”. Later, we’ll define the successor function and that’s normal called S(x).
Define equality
Shit, that ground work was a little harder than I thought and I’m sorry about that. But now we can have some fun. Okay, equality can mean different things, but I’m going to use the equivalence relation definition. I suppose you can also say it’s a function, because any number is equal one number, itself.
It’s a binary relation and when we want to say the variables a and b are equal, we say:
a=b
Okay, so this is one of those primitive notions I was talking about. I just threw out equality and didn’t tell you anything about it. So in order to make it a useful concept, I’ll have to start writing the axioms of equality.
Our first axiom is called Reflexivity or Identity. This axiom states that anything is equal to itself.
a=a
We haven’t defined numbers yet, but when we will we’ll be able to say things like:
1=1
2=2
9=9
Our second axiom is Symmetry.
if a=b then:
b=a
When we develop a little more, we’ll be able to say, since 2+3=5, that means that 5=2+3.
The next one is Transitivity.
if a=b and b=c then:
a=c
So, since 5=2+3 and 5=4+1, that means 2+3=4+1. And any mathematical expression that evaluates to five, is part of this “five club” and they are all equal to each other, because they are all five.
Define whole numbers with the Peano axioms
Okay, we’re ready to define a system of numbers! Are you excited! We will define whole numbers with the Peano axioms. The whole numbers are 0, 1, 2, 3, … forever. We aren’t defining the integers, which also includes the negative numbers, we aren’t defining rational numbers, which include fractions, etc. We are just defining whole numbers.
Axiom 1: 0 is a whole number.
Okay, that’s a start. We have a number. I haven’t told you anything about 0 yet that makes it 0 and not ten, but we’ll make some more axioms.
Axiom 2: If a is a whole number, and a=b, then b is also a whole number.
Okay, now we’re ready to define the successor function and really get off the ground!
Axiom 3: Let S(n) be the successor function. For every whole number n, S(n) is a whole number.
I haven’t said it explicitly but basically the successor is the number that is one bigger. Now we have a device for having more numbers other than 0. (Unless S(0)=0, but that’s the next thing for us to take care of.)
Axiom 4: For every whole number n, S(n) is not 0.
Okay, we’ve swung back around to 0. We now know that the whole numbers start with 0. You don’t get any smaller in the system of whole numbers.
Aside: Interestingly, you can define a system of numbers without this rule. Actually, in math you get to define anything you want. You sort of have to find interesting properties to make people care about it though. You can think about “clock arithmetic” in which you have numbers 1 through 12 and you can add and even multiply, just going around and around the clock. There’s an interesting kind of thing called a finite field. I won’t dig to deeply, into them. But the most basic kinds have all numbers from 0 up to (but not including) some prime number. So you can make a finite field of 0..6, for example. 6+1 is 0. What’s interesting, is in this field, you can define your four main operations, addition, subtraction, multiplication, and division, and everything will work as expected.
Axiom 5: For all natural number m and n, m=n if and only S(m)=S(n).
“If and only if” is a common piece of mathematical jargon called a bi-conditional. Basically it means “If A is true then B is true and also If B is true then A is true”. So in our case, if m and n are both 3, S(m) and S(n) are both 4. Likewise, if S(m) and S(n) are both 3, then m and n are both 2.
Aside: Axiom 5 guarantees that every number has a unique successor. We already know that 0 is not the successor to anything, and every number is defined by being a successor, so for any given number there exists another new number; its successor. So that means that there are an infinite number of whole numbers.
Axiom 6: Axiom of Induction.
This one is a little tricky. It actually sounds like common sense, but we can’t just declare common sense to be formally true in math.
Basically, the axiom of induction says that if a statement is:
* True about 0
* Can be shown to be true for S(n) whenever it is n.
Then:
It is true for all whole numbers.
Axiom of Induction is pretty important for proving things about numbers. For example, the nth number is Triangle number sequence is defined as the summation of every whole number up to n. Let’s call the nth Triangle number T(n). It can be shown that T(n)= n*(n+1)/2. This is easy to prove. Prove it’s true for zero, and it is because 0 times anything is 0.
T(0)=0*(0+1)/2=0*1/2=0/2=0
It is also pretty easy to show that if it is true for n, it is true for n+1.
Let n+1=m
and
Let T(n)=n*(n+1)/2
Then:
T(m)=m+T(n)=m+n*(n+1)/2=m+(m-1)*m/2=2*m/2+(m-1)*m/2=
(2*m+(m-1)*m)/2=(2+(m-1))*m/2=(m+1)*m/2=m*(m+1)/2
Define 1 and 2
Okay, this one isn’t too hard:
1=S(0)
2=S(1)=S(S(0))
Again, this is true because I said so.
Define Addition
Okay, let’s define addition. It’s a little tricky actually, because you have to define it recursively, but it’s not too bad once think about it.
First we will define a+0:
a+0=a
Next, we just have to define it for everything that isn’t 0. Everything that isn’t 0, is a successor. And we’ll define it this way:
a+S(b)=S(a+b)
Okay, this is a little confusing, because I defined addition in terms of addition of a smaller number. But we can always work back to 0 and get a good definition. So what’s a+1?
a+1=a+S(0)=S(a+0)=S(a)
So as you probably expected. a+1 is the same as it’s successor.
We can do 2 next.
a+2=S(a+1)=S(S(a))
Prove 1+1=2!
Okay, so what’s 1+1?
Since, a+1=S(a):
1+1=S(1)
And we already gave S(1) a name:
1+1=2
Yay!
Bonus: More operations and bigger, better number systems!
First of all we’re going to be ingrates to our old friend the successor function. Since addition is such an awesome operation, and we know successor is the same as adding one, we don’t really have to refer to it directly anymore. We’ll give it its watch and send it home, but its presence will always be buried in our axiomic system.
Anyway, after defining addition we can define multiplication:
a*0=0
a*(b+1)=a+(a*b)
How about subtraction? Subtract is the inverse of addition. So we can define it like this:
a+b=c then:
c-b=a
But! Not every subtraction problem has an answer in the set of whole numbers! What’s 1–2? We can’t say -1, because it isn’t part of our system. We need to expand our set of numbers to make subtraction complete. We need to define the integers to have subtraction as a complete operation that is defined for everything.
If you look up integers in the Wikipedia, it’ll tell you how to construct the set of the integers. Interestingly, they do something with pairs of whole numbers to define integers. So integer 1, isn’t actually the same thing as whole number 1! It’s a more complex construct that just acts an awful lot like whole number 1.
Division? It’s the inverse of multiplication. If you want to have an answer to almost any division problem, you’ll have to construct the set of rational numbers. It’s base off a pretty similar scheme as the integers. Something with ordered pairs of integers to make the rationals. You still don’t have everything because you still can’t divide by zero. If a*0=0, a*0 never equals 5, so 5/0 doesn’t have an answer. I don’t know of a system of numbers that let’s you divide by 0.
Exponents? Integer exponents easy. How about roots or arbitrary rational exponents? Oh, we have a problem with the number -1. It doesn’t have a square root. Well, we can define one. We’ll call it i. (Or if you’re an electrical engineer you’ll call it j.) We’ll call expressions of the form a+i*b complex numbers. For now, we’ll say a and b are rational numbers but we’ll move up to calling them real numbers soon.
I want you to notice that moving from rational numbers to complex numbers wasn’t any sillier than moving from whole numbers to integers. Imaginary numbers aren’t actually more imaginary than any other number and scientist and engineers actually use them for things. That’s the only reason heard of them, actually. There are lots of weird mathematical objects you haven’t heard of like surreal numbers, octonions, etc. The just aren’t as useful as the complex numbers with their “imaginary” part.
Actually, we haven’t even finished giving everything roots! What about the square root of two? It’s irrational! Okay, let’s say we have an arbitrary polynomial equation with integer exponents and rational coefficients. (A lot to explain so if you aren’t keeping up, I’m sorry, you can just let this go or look it up.) The answer to that equation is an algebraic number!
Aside: The simplest way to express the square root of 2 is to say square root of 2. (I don’t know how to do a square root sign on this interface). The decimal representation just goes on forever with no pattern. If you have a polynomial of root 4 or lower (x^4+x=0, or x to the fourth power plus x equals 0), you’ll be able to express your answer with roots of rational numbers. But if you go up to degree five, you might not even have a good way to express the number. For example there are the Bring radicals: the answer to an equation like x^5-x+1=0. The answers just are what they are. Their decimal notations go on forever and there’s no convenient symbol to that you can use to express numbers of that form.
Okay! We’ve solved lot’s of problems and we’ve designed some really rich sets of numbers. Awesome! I want to calculate the circumference of a circle! FUCK! FUCK! FUCK! Pi isn’t even an algebraic number. It’s god damn transcendental number! The other really famous transcendental number is e. Anyway, the set of the algebraic and transcendental numbers (excluding those complex numbers that have an imaginary part) is the real numbers. And then the complex numbers, in normal usage is numbers of the form a+ib, where i is the square root of -1, and a and b are real numbers.
In between our algebraic numbers we have transcendental numbers. These numbers just go on forever like pi. Some of them can be defined by infinite sequences; equations that can just keep adding terms to, and they don’t go infinity. I suppose some of them you can’t.
There’s lots of transcendental numbers. There’s an infinite number of whole numbers. There’s an infinite number of algebraic numbers. But get this: the number of transcendental numbers is a bigger infinity!
If people like this I’ll write an article about “bigger infinities”.
