The Development of Numbers Arithmetic and Algebra

This article discusses the development of numbers, arithmetic, and algebra. It presents the content as a simplified pseudo-history. Real world events were much more complicated with gains and losses made rather than a simple progression forward.

The Origin of Numbers (Natural Numbers)

In the beginning, ancient people could only understand having something or not having it. Eventually this became more nuanced and people realized you can have a thing and then have another of the same thing. We gave names to these concepts. One was used to describe having a thing. Two was used to describe having one more than one thing. Three was used for one more than two things. The pattern continued eventually yielding the Natural Numbers. Different number systems were developed to organize these ever increasing quantities. The one currently used is a base ten system called the Arabic or Decimal Number System. Almost all of the math taught in school has this system in mind (there are notable exceptions in computer science, like binary and hexadecimal; and historical holdovers, like the Roman Numerals). What we discuss here is universal to all adequate number systems. The particulars of how to do there computations in a particular base are discussed elsewhere.

Binary Operations

A Binary Operation in defined as an operation that takes two elements of a set and combines them to produce a third element which may or may not be from the same set. Addition, subtraction, multiplication and division of numbers are good examples.

The Origin of Negative Numbers and Zero (Whole Numbers and Integers)

People were able to count forward and backward using the natural numbers, but eventually tired of recounting when they put groups of things together. They developed addition, and later subtraction, to minimize the need for recounting. Subtraction was also useful to allow the undoing of addition ie. $3+7=10$ and then $10-7=3$

Addition

Addition, on the Natural Numbers, has some very nice properties. If we consider two natural numbers, they always add to give another Natural Number. $2+3=5$, not $-7$ or ${\frac {5}{9}}$, and especially not something nonsensical like blue or tree. This property is called Closure. Closure means that when you perform a binary operation on two elements of the same set the result is also an element of the same set.

The order in which one adds multiple Natural Numbers also doesn't matter as long as the sequence is not changed. For example $1+(2+3)$ gives the same result as $(1+2)+3$. This property is called associativity. Associativity means that the order repeated instances of the same binary operation are performed in doesn't matter.

Addition on the Natural Numbers also allows us the add two numbers in either order. For example $1+2$ gives the same result as $2+1$. This property is called commutativity. Commutativity means that the elements a single binary operation is acting on can appear in either order without changing the result.

Associativity and Commutativity together give us the option to shift the elements in repeated instances of the same binary operation in any way we want.

Subtraction

Subtraction on the Natural Numbers is a mess. It isn't even closed. What happens when you subtract a number from itself? You get nothing. We must invent the number zero to account for this (giving the Whole Numbers).

Even worse what happens if you subtract even more than you started with? We need to invent symbols for this too! Thus we get the negative numbers (and therefore the Integers). Finally we find that the integers are closed under subtraction, but subtraction remains without the associative property $(1-2)-3=-4$ while $1-(2-3)=2$, or the commutative property $1-2=-1$ while $2-1=1$, but do not fear there is hope.

The Integers

We begin by noting that when we extend the Natural Numbers to the Integers addition remains closed, associative, and commutative. We now also note two more important properties that now exist for addition on the Integers. We no have an element called $0$ which has the lovely property that adding it to any other Integer (from the left or the right) gives that same integer back. ie. $3+0=3=0+3$. We call this element the Additive Identity. For general binary operations an element like this would be called the Identity Element.

Further we notice that for every Integer, there exists another Integer such that when we add them (in either order), they give zero. ie. $3+(-3)=0=(-3)+3$. Every Integer has a negative counterpart also called an Additive Inverse. For a general binary operation we would say that every element has an Inverse Element.

Next we notice that everything we wanted to do with subtraction can be achieved by adding additive inverse. We can thus decide to replace subtraction with addition to ensure we can use the nice properties inherent to addition. In practise the subtraction operation is kept around, because people like to think in terms of doing things (doing operations), but it is not required once one has the Integers to work with.

The Origin of Fractions (Rational Numbers)

Addition and subtraction were not powerful enough. People began to want to repeatedly add and subtract, so we developed multiplication and division.

Multiplication

Multiplcation on the Natural numbers has the same nice properties as addition does. It has closure $\forall a,b\in \mathbb {N} ;a\cdot b\in \mathbb {N}$, associativity $\forall a,b,c\in \mathbb {N} ;a\cdot (b\cdot c)=(a\cdot b)\cdot c$, and commutativity $\forall a,b\in \mathbb {N} ;a\cdot b=b\cdot a$. It also has the number $1$ which serves as the Multiplicative Identity. ie. $\forall a\in \mathbb {N} ;a\cdot 1=a=1\cdot a$. We note the absence of inverses and expect the need to extend the naturals in response.

Division

Division with remainders is considered first. It gives outputs which are not ideal. They are not a neat number but a number with some extra attached ie. $7\div 2=3$ with remainder $1$. We invent fractions (the positive Rational Numbers ie. greater than zero) to deal with this ie. $7\div 2={\frac {7}{2}}=3{\frac {1}{2}}=3.5$. Division is now closed on the positive Rationals, but (much like with subtraction) we find that the associative property ($(8\div 2)\div 2=2$ while $8\div (2\div 2)=8$) and commutative properties ($4\div 2=2$ while $2\div 4={\frac {1}{2}}$) cannot be salvaged.

The Rationals

We now have the positive rationals (which we will denote $\mathbb {Q} ^{+}$) which give us the Multiplicative Inverses (often called reciprocals in the context of multiplication). We denote the inverse of an element $a$ by $a^{-1}$ or sometimes ${\frac {1}{a}}$. More formally he property of multipcative inverse can be stated as $\forall a\in \mathbb {Q} ^{+},\exists a^{-1}\in \mathbb {Q} ^{+};a\cdot a^{-1}=1=a^{-1}\cdot a$. For example $2$ and ${\frac {1}{2}}$ are multiplicative inverses of each other $2\cdot {\frac {1}{2}}=1={\frac {1}{2}}\cdot 2$. Multiplication on the rationals also retains the closure, associative, and commutative properties and the multiplicative identities found in multiplication on the naturals. We also note we can do everything we wanted to do with division with multiplication on the positive rationals, so we don't really need division.

It would be nice to be able to do multiplication and addition both on the same set though. It is not difficult to see that we can extend the positive rationals, by including zero and the negative fractions, much the same way we extended the Naturals to the Integers. This gives us the Rational Numbers. Addition and Multiplication on the Rationals retain all the nice properties we have discussed, with the exception that we don't consider the multiplicative inverse of the additive identity (ie. we don't have zero in the denominator ie. we don't divide by zero). What's more we have a new property which allows the two operation to interact. The Distributive Property allows us to distribute multiplication over addition ie. $2\cdot (3+4)=2\cdot 3+2\cdot 4$. Said more mathematically $\forall a,b,c\in \mathbb {Q} ,a\cdot (b+c)=a\cdot b+a\cdot c,(b+c)\cdot a=b\cdot a+c\cdot a$.

The Real Numbers

If we take a look at the decimal representation of the rationals we have found every number which terminates after a finite number of digits ($-2.5$) and all the numbers which end in an infinitely repeating pattern ($4.7777777777\dots$). What about numbers which do not have a nicely behaved decimal representation though? What about numbers which have decimal expansions that do not terminate and do not have a pattern? Numbers like $\pi$ and ${\sqrt {2}}$. These are the irrationals. We can include these with the Rational Numbers to create the Real Numbers. The Reals also have all the nice properties we have discussed. Some properties are obvious (the identities $0$ and $1$ are the same). Other properties like closure, associativity, and commutativity can be correctly guessed at based on the decimal representation.

Arithmetic

Arithmetic on numbers is about computing values. Usually in high school we deal with arithmetic of Real Numbers. To do this we follow the order of operations. If we note that subtraction and division are not really needed, and choose to represent them by using addition and multiplication of inverses, we can simplify the order of operations significantly. The order becomes Brackets, Exponents, Multiplication, then Addition. We no longer have some operation which have the same precedence. Even more usefully we can do he operations at a precedence level any way we want (thanks to associativity and commutativity). Finally we can simplify our interpretation of the order of operation some. This is because, in some sense, exponents can be thought of ass repeated multiplication, multiplication can e thought of as repeated addition, and addition can be thought of as repeated counting. So, in some sense, the order of operations is just brackets telling us what to do first and very complicated ways of counting.

To do arithmetic efficiently many rules and algorithms, specific to the decimal (or other) number system in use, are used to speed up computation. Regrouping, borrowing, memorizing times tables, etc. are all tricks of arithmetic designed to speed up the computation process. When doing algebra, arithmetic is part of the closure property.

Algebra

Algebra is about rearranging the values presented is an equation (know and variable) into a more advantageous form. We do this using the properties painstakingly developed in this article.

We now define three of the most widely used algebraic structures. We will repurpose the symbols $+$ and $\cdot$ to mean general binary operations for a moment. We choose to keep this symbols in our new more abstract discussion, because of convention and the subtle reminder they give of the rules and how the operations interact. When we use one of the symbols we will say that we are using the additive notation or multiplicative notation depending on which symbol is used.

It is also common to repurpose the symbols $0$ and $1$ to be the additive and multiplicative identities. To emphasize the difference between the abstract the notation and explicit addition and multiplication on numbers we will use a different notation here. We will use $e$ to represent the identity when only one operation is being used. When two operations are being used, we will use $e_{0}$ and $e_{1}$.

Groups

A Group is a pair, often written $(G,\cdot )$ (or $(G,+)$) where G is a set and $\cdot$ (or $+$) is a binary operation on the set. In general the multiplicative notation is preferred as we wish to simplify the notation further. Often when we write multiplication of variables we do not bother to write the operation in between (ie. $a\cdot b=ab$) and we will take advantage of this notation here. We will call the binary operation the Group Operation.

To be a group the group operation must have four (familiar) properties.

• $\forall a,b\in G;ab\in G$ (Closure)
• $\forall a,b,c\in G;a(bc)=(ab)c$ (Associativity)
• $\exists e\in G;\forall a\in G,ae=a=ea$ (Identity Element)
• $\forall a\in G,\exists a^{-1}\in G;aa^{-1}=e=a^{-1}a$ (Inverse Elements)

Often a fifth property is present

• $\forall a,b\in G;ab=ba$ (Commutativity)

When this fifth property is also present we call the algebraic structure a Commutative Group or equally an Abelian Group.

Rings

A Ring is a triple often written ($R$,$+$,$\cdot$), where $R$ is a set and $+$ and $\cdot$ are binary operations on that set. A ring has the following properties.

It is an abelian group with respect to the additive binary operation.

• $\forall a,b\in R;a+b\in R$ (Closure)
• $\forall a,b,c\in R;a+(b+c)=(a+b)+c$ (Associativity)
• $\exists e_{0}\in R;\forall a\in R,a+e_{0}=a=e_{0}+a$ (Identity Element)
• $\forall a\in R,\exists -a\in R;a+(-a)=e_{0}=-a+a$ (Inverse Elements)
• $\forall a,b\in R;a+b=b+a$ (Commutativity)

The multiplicative binary operation is closed, associative, and has identity.

• $\forall a,b\in R;ab\in R$ (Closure)
• $\forall a,b,c\in R;a(bc)=(ab)c$ (Associativity)
• $\exists e_{1}\in R;\forall a\in R,ae_{1}=a=e_{1}a$ (Identity Element)

The multiplicative binary operation distributes over the additive binary operation

• $\forall a,b,c\in R,a\cdot (b+c)=a\cdot b+a\cdot c,(b+c)\cdot a=b\cdot a+c\cdot a$ (Distributivity)

Fields

A Field is a triple often written ($F$,$+$,$\cdot$), where $F$ is a set and $+$ and $\cdot$ are binary operations on that set. A Field has the following properties.

It is an abelian group with respect to the additive binary operation.

• $\forall a,b\in F;a+b\in F$ (Closure)
• $\forall a,b,c\in F;a+(b+c)=(a+b)+c$ (Associativity)
• $\exists e_{0}\in F;\forall a\in F,a+e_{0}=a=e_{0}+a$ (Identity Element)
• $\forall a\in F,\exists -a\in F;a+(-a)=e_{0}=-a+a$ (Inverse Elements)
• $\forall a,b\in F;a+b=b+a$ (Commutativity)

It almost is an abelian group with respect to the multiplicative binary operation. We must exclude the inverse of the additive identity (ie. division by zero is not allowed), but otherwise it it an Abelian Group.

• $\forall a,b\in F;ab\in F$ (Closure)
• $\forall a,b,c\in F;a(bc)=(ab)c$ (Associativity)
• $\exists e_{1}\in F;\forall a\in F,ae_{1}=a=e_{1}a$ (Identity Element)
• $\forall a\in F\setminus \{0\},\exists a^{-1}\in F;aa^{-1}=e_{1}=a^{-1}a$ (Inverse Elements)
• $\forall a,b\in F;ab=ba$ (Commutativity)

The multiplicative binary operation distributes over the additive binary operation

• $\forall a,b,c\in F,a\cdot (b+c)=a\cdot b+a\cdot c,(b+c)\cdot a=b\cdot a+c\cdot a$ (Distributivity)

All other tricks of algebra used are derived from these rules.

Exercises

Read through the article again and take the time to translate the notation. Becoming fluent in the language of math will be very helpful as time goes on. While you are reading consider different sets of numbers and binary operation on them (Naturals, Whole Numbers, Integers, Rationals, Positive Rationals, Reals, etc. under addition, subtraction, multiplication, and division). Which ones form groups? Rings? Fields? No formal proof is needed, just think about a variety of cases and see if the properties hold.

Consider what goes wrong if one attempts to find a subtractive or divisive identity. Is it unique? Does it work in both the left and right positions around the binary operation in question? If it only had to work on the right can you find one? How about the left?

Extension

Cyclic Groups

Be sure you have covered Modular Arithmetic and Relations and Functions before proceeding.

We begin by considering $\mathbb {Z} _{4}$ (the integers modulo $4$). We wish to show that this set along with addition and subtraction as defined form a ring. As there are a finite number of element is this ring we call it a finite ring. It also is a finite group if we only include the addition operation.

We begin by writing out the Cayley Table for each operation on the set. A Cayley Table is a table that shows the result of using the binary operation to combine every combination of elements in the group in every order. Technically the multiplication operation doesn't form a group so the table we get with multiplication will not strictly speaking be a Cayley Table, but it will be useful.