My Ten Minute Intro to the Development of the Real Number System. Draft
Development of the real number system.
This is an abbreviated account of the development of the real number system from the counting numbers through the real numbers (all ∞ repeating & non-repeating decimals. While we acknowledge the difference in context between the words Number and Numeral we choose to use the two terms interchangeably as is customary in most elementary school textbooks. A number is an abstract concept denoting a specific quantity of distinct elements in a set. A numeral is a name of a number. The numeral 5 stands for the number five. The number five is the concept we associate with any set of elements, real or just thought about, that can be put into a one-to-one correspondence with the quantity of fingers that you probably have on any one of your hands.
We also acknowledge that the concept of having a number depends on the concept of having a Unit. We must all agree on what is one of something before we can talk about having more than one of the same things. For example, if you go to the market to buy a standard carton of eggs, you most probably would go to the store and buy one carton labeled to be a dozen eggs. The unit is one group or set of a dozen eggs. If however, you are coloring eggs for Easter, each individual egg becomes a unit of which you will need twelve units (eggs) to give them one-to-one to a dozen of your students. The concept of number depends on the quantity that makes up the designated unit. For example, it is unlikely that if your mother sends you to the store to get three eggs that she really means that you are to buy and come home with three individual eggs. Most stores do not sell single eggs. It is more likely the case that your mother wants you to get three cartons of eggs where each carton encloses a dozen eggs totaling thirty-six individual eggs out of the cartons. Similarly, it you say that you have five cents, it is the case that you could have five pennies or a single nickel that is equivalent in monetary value to having five pennies. Your unit in the first case is a penny, of which you have five versus the second case in which the unit is a nickel of which you have just one.
Development of the number concept begins with the
N: Counting aka Natural numbers
The Natural or Counting number constitute the positive integers of the real system, the number one, two, etc. Excludes the concept of zero. Except for the Mayans who were advanced mathematically and needed the concept of zero to give meaning to their place value number system no other early civilization except the Babylonian had a need to come up with the abstract concept of having "something" that stood for "nothing." I took the Babylonian ages before they reconciled their "zero problem" with a symbol separating their base sixty place values. The abstractness of the concept of zero is confirmed by the fact that zero as we recognize it as a number today did not come into general use until the Hindus and Arabians got together circa 900 AD.
The system of Natural numbers, N includes
· the relationships of "less than" (fewer) and "greater than" (more) represented by the symbols "<" and ">" along with ≤ (less than or equal to) and ≥ (greater than or equal to).
·
Includes the Operations +, x, -, and 
(div)but
require some rather specific conditions needing to be met. A discussion
of these conditions is given below.
+, Addition:
The process of addition assumes that if you take two distinct groups, each containing specific quantities of individual unit things and dump them all into one pile, then the sum of those two numbers is the resulting count of all of the element units in the pile. This means that if you have two disjoint sets each of which contains an apple (not the same apple) then when you dump them together you have two apples even though both objects are apples. Today this idea is expressed as follows: Let A be a nonempty set with a=n(A) elements and B a nonempty set with b=n(B) elements, then the sum a+b=n(A È B) provided AÇB=Æ. See the discussion in the addition algorithms sections for the properties of addition: closure, commutativity and associativity and the existence of identity and inverse elements. All of the algorithms we use to simplify the addition process are based on the validity of these properties.
*, Multiplication:
Multiplication (repeated addition) is colloquially called fast addition.
The notation of multiplication, ab= a•b=a*b=(a)(b) is a shorthand way
of writing that the product of the numbers a and b means that we add b to
itself a–many times. The sum of these repeated additions is the
same fixed number as the number of ordered pairs in the Cartesian Product
(Cross Product) of two distinct (nonempty) sets. (When we extend the natural
numbers to the whole numbers including the number zero, then zero is defined
as the number of elements in the empty set, i.e. 0=n({})=n( 
)).
Today, if what we used to call a definition of an operation yields a visual representation of the definition, then we refer to it as a "Model" for the operation. So, take two disjoint nonempty sets A and B where a is the number of elements in set A, a=n(A) and b is the number of elements in set B, b=n(B), then the product ab is modeled (defined) by the number of elements in the Cross Product of the two sets A and B. That is, a*b=n(AxB) where AxB is the set, AxB={(a,b)| a is an element in set a and b is another distinct element in the disjoint set B}. We can further represent this cross product by a graph, that is a subset of the integral valued coordinate points in first quadrant of the Cartesian plane, geometrically by an axb rectangle with a-many rows and b-many columns or by the going verbiage today, any other axb array (a cow by any other name is still a cow). So whether the student counts the number of units in an a*b rectangle or the number of dots, spaces, stars or whatever else one uses to put or picture in a row by column array, s/he will get the same numerical answer. Other models such as a tree and number line are discussed with the multiplication algorithms that history and we use today to simplify the process of finding the answer to a multiplication problem.
-, Subtraction :
Subtraction is defined in terms of addition, a-b=c if and only if b+c=a. But subtraction in the set of natural numbers, a–b=? has a natural number answer only if a<b. Subtraction is also said to be the "inverse of addition. " But this statement cannot be completely explained until the integers enter the system circa 1600 AD or seven centuries after civilization reached the point of accepting the number concept of zero circa 900 AD. Until then, we can just say that subtraction is the "inverse of addition" because it is the operation that "undoes" addition, whatever that means. We revisit this meaning of subtraction when we discuss the subtraction algorithms and Take Away, Comparison, Missing Addend, and Number Line models. We cover these various models to give you some suggestions as to how you can alleviate the frustration many students feel when confronted with all the different ways subtraction problems are written in word problems.
Division:
Division is defined in terms of multiplication,
a 
b=c
if and only if c*b=a=b*c. There is no zero in the natural
numbers, so we do not need to add the restriction that b≠0. Just as
we must wait until we have the integers to systematically explain why subtraction
is called the inverse of addition, we must wait until we introduce the rational
numbers into our developing number system to explain why division is called
the inverse operation of multiplication. For now, we'll just give the elementary
level explanation. Namely, division is the inverse of multiplication
because it "undoes or reverses" the multiplication process.
Also, just as multiplication is colloquially called" fast or repeated
addition", division is called "fast or repeated subtraction".
A division problem is said to ask the question, "How many times does
'b' go into 'a'" But what is meant by "goes into?" For
the natural numbers, it is understood to mean, "How many times can the
number b be subtracted from the number
a before one arrives at a state of have nothing left to subtract
from?" The division question is simplified in the whole numbers
to, "How many times can b be subtracted
from a before one reaches zero." So, until we grow up enough
to investigate division in the more inclusive system, for all natural numbers
a, b, and c, a 
b=c if and only
if c*b=a.
And it is not unusual to get a remainder when we repeatedly subtract one
natural from another. Then we are applying the Division Algorithm which tells
us that if we repeatedly subtract natural numbers 3 from 28, we wee that 28=9*3+1.
We write, 28 
3=9
w/remainder 1.
See the number theory discussion for a complete discussion of the mathematical differences among the terms, "divide", "divides", " division and divisibility". For a complete discussion of the four basic models we have today for division, Repeated Subtraction, Partition, Missing Factor and the Number Line, see the division algorithm discussion.
Closure: The desire to solve increasingly complex equations
and the property of closure for all four operations +, -, * and 
go hand
in hand in any discussion of the development of the real number system.
An operation is closed if sum, difference, product or quotient of two numbers
from any given system is again a member of the same system. This is
best understood by an example. Take the two natural numbers 5 and 3;
5–3=2 and 2 is a natural number. All three numbers belong to the
same numerical system. But the equation, 3–5=? has no solution
in the natural numbers. We don't get 
2 until we get
to the integers. So algebraically: we get a natural number difference for
any two distinct natural numbers in one direction but not for other. To get
a natural number solution to the equation a-b=? a and b must satisfy
the inequality 1≤b<a. Our example shows why a must be greater than
b to get to a natural number solution, but what if one chose to take a and
b to be the same number? There is no solution to the equation a-b=?
in the natural numbers. We must wait until we are allowed to play with
the whole numbers to validly withdraw all of our money in the bank leaving
us with a zero balance.
Consider the scenario. You live in natural-number land and your neighbor says that s/he now has enough grass to support seven sheep and s/he asks you to buy enough sheep for him/her to have seven sheep. You are now in the animal section of the market and realize that you did not ask your neighbor how many sheep it is that he wants you to buy. However, you know that your neighbor has five sheep and you know that in natural-number land you can subtract five from seven to see that the solution to how many sheep you should buy is two. So you solve the equation, a-b=c where a is the number of sheep wanted, b is the number of sheep already owned and c is the number of sheep that is yet to be obtained. But you have no numerical equation you can write to say that you are satisfied with exactly the number of sheep that you have and that you do not need nor want any more sheep. You have no solution to the equation a-a=? The equation has no meaning in the natural numbers. What you do is apply to be admitted as a member of whole number land. Their system contains a solution, namely zero for subtracting any natural (actually any whole) number from itself; for example, 5-5=0, 3-3=0 and 0-0=0 in the whole numbers.
But, while you have incorporated zero into your system, you still have no solution to the equation a–b=? whenever b>a. For that, we need to incorporate the integers. Before we decide to apply for integer citizenship, let's look at some of ways in which we benefited by adding the number zero, "0" to our system of natural numbers.
Whole numbers: W
= N 
{0}
The whole numbers, W is the union of the
element 0 added to the system of natural numbers. Recall that
today; zero is defined to be the number of elements in the empty set, 0=n(
). It seems unbelievable
to me that mankind, except for the Mayans, had to pass through all of pre-history
and 900 years AD before widely accepting 0 as the solution of the equation
a–a=? With the advent of zero, we now
had a number we could add to any other number and not change its value. Adding
zero to the natural numbers enables every number to be identified with itself
under addition. That is, a+0=0+a=a, for all whole numbers a.
Zero is therefore, called the additive identity.
What additional properties do we pickup from incorporating the concept of
zero in our numerical system? The biggest advantage numerically is that
it gives meaning to the concept of place value, which allows for algorithms
that enables us to simplify all of our four function algorithms. See
the section on the four function algorithms, +, –, *,
Multiplication also has a "zero"
property that says that any
number times zero is zero. And multiplication is commutative, so a*0=0*a=0,
for all a 
W. Jokingly,
we can look at the equation 0*a=0 and
say that it doesn't matter how many cars a the dealer has on
his/her lot, if you have zero dollars to put down, then you leave the dealer's
lot with no (zero) new car. It is also true, that it doesn't matter
how much money a you have, you are
still without a new car if the dealer doesn't have any of them to sell to
you, or a*0=0 for any whole
number a, including 0.
Division in the whole number system reaches a snag, however, with zero
added to a system. Recall that division is defined in terms of
multiplication, a 
b=c
if and only if c*b=a, for all
numbers a, b, and
c in the system. Division has an algebraic solution for the equation,
0 
5=0, because
0*5=5*0=0. But there is no whole number solution to the equations 5
0=?
or 0 
0=?
See the division algorithm discussion for an explanation of why this is so.
The inclusion of the number zero to the natural numbers, means that
the subtraction operation now has a solution for the equation a–a=0
for all whole numbers a, but
there is still no whole number solution for 3-5=? Closure under
subtraction requires that we extend the Whole number system to include the
set of negative numbers, Neg.={ 
n | n+
n= 
n+ n=0, for all
natural numbers n}. Our hierarchy so far can be expressed as the set
of integers, I=W 
Neg.
Integers, I=W 
Neg=N
{0}
Neg,
where the set of Negative Integers, Neg ={-n |for all natural numbers n such that n+(-n)=0=(-n)+n }.
The Integers,
is the set that contains all of the counting numbers, zero and all of the
negative integers. Subtraction becomes closed in the Integers because the
subtraction equation, "a-b=?" now has a solution for all whole
numbers, a and b without one needing to be more than or less than the other,
for example, 3-5= 
2, because 
2+5=3. The negative
numbers in I adds direction, which enables us to model the number line for
a visual representation of our number system. From any fixed starting
point designated as zero, the positive numbers go in one linear direction
from zero and the negative numbers go in the same linear but opposite direction
from zero. It is standard in the Cartesian Plane XxY, to have a horizontal
, x-axis and a
vertical 
, y-axis in any
representation of the Cartesian Plan e. But there is no fixed law that says
these horizontal/vertical designations are fixed and irreversible. For
example, hot/cold thermostats often have positive numbers go in the up, 
and the negative
numbers go in the down, 
direction
from zero. It really doesn't matter which direction the positive axis
is pointed. One could even hold the Cartesian plane in a slanted direction.
That is, the axes could go / or \ as long as zero is the linear separation
point of the opposite directions.
Zero also enables us to incorporate place value into our system, thereby making any calculation Bases, (e.g., base two, three, five, etc.), are good to emphasize the meaning of the concept of place value and of the elementary operations of + .–. Multiplication and division in bases further enhances a students understanding of operations of addition and subtraction, but often the time involved to cover these items in depth negates their usefulness in the average classroom.
The integers
are closed with respect to +, *, and -, but not by division. Asking how many
sets of three elements can one find if one has only a set of two elements,
i.e. 2 
3 still has no
meaning. To gain closure for all nonzero numbers under division, we
must further build up our number system to being able to work with parts of
a whole. So, to close division by all non-zero integers requires us
to add all possible ratios of the form, { 
,| a,b≠0
I} to the integers
to get the rational numbers.
Rational Numbers or Quotients,
Q=I 
{ 
, a, b≠0
I}=N
{0}
Neg
{ 
, a, b≠0
I}:
The letter Q commonly represents the rational numbers, for Quotient. Later we use the letter R to represent the real numbers that consists of the union of the rational numbers and the irrational numbers.
The numerals or names of a rational numbers are called fractions because they represent whole units that are "fractured" into pieces. We will adopt the practice of not differentiating between a fraction (numeral) that technically stands for the name of a rational number or the ratio (proportional relationship) between two different numbers or quantities. We will the terms: fractions, rational numbers and quotients interchangeably.
All four number operations that are defined are closed in the rational number system. Division by zero is not defined in any number system invented by mankind to date. Because of this, we will understand that for any rational number a/b we write, the divisor or denominator b is not zero.
The concepts
rational number system is best understood when one has a good background in
modeling the different aspects and uses the four operations, +, -, *, 
take on in Q.
Besides writing our fractions as a/b, b≠0, we also have developed a
decimal system way of writing them. For example, 1/2=0.5 and 1/3=.3333333…
and during the course of our study, we find that all rational numbers can
be written as infinite repeating decimals. The Pythagorean Society had
a doctrine that all phenomena in the world could be expressed as ratios of
two integers (b≠0). But their whole society, as brilliant as they
were mathematically, fell apart when it was proven that there is no rational
number solution to the equation x*x=2 and they were put back to same stage
most civilizations were when they were not able to solve the equation, a-a=?
There was no solution to the problem that the square of any two numbers was
again a known number. It was easy to say that any two numbers when multiplied
had an answer but not every number could be the square or product of any number
by itself. It is interesting to me that the early Greeks pondered the
irrational problem a*a=2 but passed on the issue of finding a solution to
"2–2=?" or "3–5=?" by accepting the
rule that the subtrahend (B) had to be greater than the minuend (A)
to solve any subtraction problem b–a=? .
Once the use
of fractions and decimals became interchangeable and it was known that all
rational numbers could be written as infinite repeating decimals, it was reasonable
to ask what kind of number was an infinite non-repeating decimal. It
was certainly possible to infinitely input random whole numbers into the decimal
portion of a decimal number. Logically, if all rational numbers
(ratios a/b, b≠0) could be written as infinite repeating decimals,
then these infinite non-repeating decimals were clearly not ratios, therefore
they could not be rational numbers, ipso facto, they were called the set of
irrational numbers. The solution to the a squared equal to two problem
fits in here somewhere, but the result we want out of all of this is to define
the Real Numbers as the set of both infinite repeating and infinite non-repeating
decimals and to see what arithmetic and algebraic properties hold for the
basic four operations, +, -, *, 
.
Real Numbers,
R = Q 
Irrationals
= N 
{0}
Negatives
{
, a,
b 
I} 
Irrationals:
So the real number system is the set of all infinite
decimal numbers, infinite repeating that terminate like 1/2=0.5 or non-terminating
like. 
.
It is customary to put a bar above any set of numbers that continuously repeats
in the same manner to indicate the EXACT decimal value of the fraction rather
than approximating the number by rounding or adding ellipses, …
While we discuss square root and cube root numbers, our investigation centers
around what set the sum, difference, product or quotient of any two real numbers
belongs to. For example, you can get the product of two irrational numbers
to be a non-negative integer or another irrational, √2*√2=2 and
√2*√3=√6. However the sum, irrational ± irrational=irrational
as is the sum, rational ±irrational=irrational.
The irrational
numbers comprise only a part of the set of non-repeating decimals. There
are other transcendental numbers such as π, e. sin (π/3), cos (2+e
), etc. All four
basic number operations, including exponentiation, nth roots, etc. are defined
and closed in the nonzero real numbers excluding division by zero is not defined.
We stop our study of number systems at the real numbers in elementary school,
students. However, college bound high school mathematics students extend the
real number system to what we call the Complex Number System by adding the
solution, "i = 
" to the
equation a 
= (-1).
And what do mathematics majors in college do? They just keep extending
these existing number systems to include defined number solutions to other
previously undefined equations just as we included a=√2 to solve the
equation a 
=2 and the imaginary
number, "i= 
" to solve
the equation i 
=–1. The
object here is to make you aware that "Mathematics" as a system
extends far beyond the real numbers, to infinity? (to include infinity yes!).
Also we aim to show you that algebra is nothing more than writing equations using variables. Elementary algebra is nothing more than applying the general operations and their properties of the real number systems using non-specified parameters, such as a*(b+c)=a*b+a*c. These include all four basic operations and exponentiation and all associative, commutative, distributive, etc., properties. We solve these equations by find the real number values, which the variables can take to make the sentence numerically true. Now that calculators have resolved the tedious and time consuming calculations of real life problems, we have more time to algebraically look into the structure of our number system to better understand it and use it to help us get along in this twenty-first century world.