In part one of two videos on the real
number system, we will explore the most
simple and basic sets of the real number
system being the natural numbers, the
whole numbers, and the integers. Before we
go over the subsets of the real numbers
let's think about what it means to be a
mathematical object. To begin with, let's
think about this: Is a phone number a
mathematical object? Would it make sense
to take your phone number and subtract
it from another person's phone number?
Could you put phone numbers in any type
of order?
Would you be able to find the average of
five different phone numbers? would any of
those things make sense to us in the
real world or have any value?
The answer is No. So let's put things
together. These symbols that we use in
mathematics are also used in other areas
as identifiers are not always used as
mathematical objects. In order for
mathematics to be occurring we need
these symbols or objects to go along
with operations and a set of properties.
So when we talk about objects or symbols
we have our number system that we
currently use but many other cultures
use different symbols. One that we may be
familiar with his roman numerals. I've
shown one here at the end of my list.
Operations of course are addition,
subtraction; we can talk about square
roots or fraction bars. Properties are
greater than, less than, and so forth. So
we need those things to be happening in
order to be considered mathematics.
Now let's use the history of mathematics
to go through and figure out how these
different sets of numbers came about. So
if you were asked to think about some of
the first examples of the use of
mathematics in different cultures, one
of the things you would come up with
quickly would be counting. You've heard
of counting sheep. These early cultures
didn't have formal writing system so
they used objects like stones or
pictures on walls to keep track of
things that were very important to them.
So it makes sense that the most basic
set of numbers is dealing with counting
or called the counting numbers. The more
formal name for the set is called the
natural numbers and we'll use a letter
N to indicate this set going forward.
So we will use this set not only
account 1, 2, 3, 4, and so forth; We will rank
things first place, second place, third
place, and ordering we can obviously make
sense of four races or different things
but we've also come up with our
calendar system using these ideas as well.
So as culture became more structured you
can see some of the earlier forms of
writing in mathematics delt with tallying. The
Babylonians, the Egyptians, and the Mayans
all had a certain tallying system to keep
track of quantities. Only the Greeks and
the ancient Chinese had symbols that
represent a larger quantities. Now an
interesting thing that a lot of people
don't realize is for these earlier
counting cultures we did not have a
symbol for 0. If you had cows or sheep, you
didn't have to say I have zero sheep. I
could just say I have no sheep. So for
many of these cultures they did not come
up with a symbol for 0 for thousands of
years after they came up with these
counting symbols; and that brings us to
our next set of numbers. This name did
not come about until much later in history,
but adding that number zero creates a
new set of numbers called the whole
numbers; indicated by the letter W. So we
have our natural numbers plus 0
representing the whole numbers. Now 0
obviously represents nothing but it also
is very important in our modern number
system because it acts as a placeholder.
It allows us to tell the difference
between  101, 110, and the number 11.
It allows us to have a difference
between the tens, the ones, and the
thousands place. Now as our number
system continued to grow, so too did
culture.
Well as larger cultures began to demand
more and more across the world, our
travel and transportation of goods and
services were very important. And that
brought about having to have a common
language and a common notation for a
lot of cultures in the world to work
together. So you think about where in the
world was the major trade hub? A
communications and brought a lot of
culture together; and that was in India.
Many people were trading goods. So it
makes sense that our number system that
most of our culture's use is from the
hindu-arabic background. Because they had
to communicate; they had to arrange goods
and work together. So you can see the
Hindu and Hindu-Arabic is derived
over time and people are trading goods
and communicating and they have a common
system that is developed into what we
have now.
Now you know how does that relate to our
next to the number? When we are
purchasing and selling goods you always
are going to have an equal trade. So that
brings into the concept of debt. When we think
of debt in mathematics, we think of
negative numbers. Now, just like the name
whole numbers the name integers is
something more modern and has evolved
over time into what it has become today.
This set is made up of zero, the positive
natural numbers, and their opposites. So
we can think of all the numbers on our
number line that we learned earlier in
arithmetic as our integers.
Please note that this symbol Z is used
for the integers because the letter I
will be used again for later set. This
concept of negative is not only applied
to money and debt, but we use it a lot in
science for different types of forces,
direction, and other concepts. In addition
to looking at our number line we can
also think of a Venn diagram is a way to
indicate the relationship between these
numbers. So if you think about a box
containing all integers that would be
the blue area shown on the screen here
With all the negatives. Zero and the
positives inside that box is the set of
whole numbers you can make that a red box.
And the only difference between the set
of whole numbers and the set of natural
numbers is the number zero. Within that
box for whole numbers we have another
white box in this case indicating are
counting numbers or natural numbers. So
this is a good way to display the
relationship between our first three
sets of numbers. Now if we only had to do
addition, subtraction, and multiplication,
these three sets of numbers would be
sufficient enough for our work. But
unfortunately we need to divide our
values into smaller parts and that
involve fractions and decimals and other
operations. So we will have to introduces
a few other sets of numbers as we go
forward here in part two of our study of
the real number system. That's it.
please look forward to part 2 of this
video set on the real number system
where I'll introduce the rational and
irrational numbers and finish off the
entire set as the real numbers.
Thank you so much for watching! Please
continue to visit my channel for more
ideas on number sense, arithmetic, algebra, and
another topics.
