Hello. I’m Professor Von Schmohawk
and welcome to Why U.
For tens of thousands of years,
people invented different ways of counting things.
Things such as gazelles
or coconuts
or people
or days.
In mathematics, the counting numbers are called
“natural numbers”.
Natural numbers start with one.
There is no limit to the largest natural number.
Natural numbers do not include
the number zero.
When people started counting things
it probably seemed pointless
to invent a number for "no things".
Why would you say
"The number of bananas we have is zero."
when you could just say
"Yes, we have no bananas!"
However, once positional notation was invented
a symbol to represent zero was needed as a
place holder for columns containing no digits.
For instance, the number 2009 represents
two thousands
plus zero hundreds
plus zero tens
plus nine ones.
Without the zero symbol,
this number could get quite confusing.
At some point, people started including zero
along with the natural numbers.
The natural numbers plus zero
became known as the “whole numbers”.
Zero is a number with a unique property.
When you add zero to any number
the value of that number is unchanged.
In mathematics, an “identity element”
is a number that
leaves the value of something unchanged when
a particular mathematical operation is performed.
So zero is known as the “additive identity”.
One is also a number with a
unique identity property.
When any number is multiplied by one
its value is unchanged
so one is known as the “multiplicative identity”.
The existence of a number
which is an additive identity
and a number which is a multiplicative identity
is an important property for a number system.
Up until now, we have thought of numbers
as quantities.
But what if we visualize numbers as distances?
If we think of numbers as representing
distances from some point
then we can arrange the numbers on a line
like the numbers on a ruler.
The point from which the distances are measured
is called the “origin”.
It makes sense to place the number zero
at the origin
since it represents zero distance
from that point.
We must now choose some distance
for the number one.
This distance is called the “unit distance”.
Every whole number then corresponds to
a multiple of that unit distance.
This way of representing numbers
is called a “number line”.
Since there are an infinite number
of whole numbers
we place an arrow on the right end
of the number line
to show that it goes on forever in that direction.
The natural numbers and the whole numbers
both can be represented as points
on this number line.
Addition can be thought of as adding distances
on the number line.
For example, adding one unit distance
to one unit distance
gives us a distance of two units.
Adding a distance of three units
to a distance of four units
gives a distance of seven units.
Likewise, if we subtract a distance of
four units from a distance of seven units
we get a distance of three units.
When any two whole numbers are added
we always get another whole number.
Therefore, we say that whole numbers are
“closed” under the operation of addition.
A group being closed under some operation
means that the operation will
always create a result
which is also a member of that same group.
But are the whole numbers
closed under subtraction?
If you subtract a larger whole number
from a smaller whole number
there is no whole number which can
represent the result.
This is because we would need a negative number
to represent the result
and whole numbers do not include
negative numbers.
Therefore the whole numbers are
not closed under subtraction.
However, if we expand our collection of numbers
to include negative numbers
then we can always find a number
to represent the result
of any addition or subtraction operation.
These whole numbers which can be positive,
negative, or zero are called “integers”.
No matter how we add or subtract integers
the result can always be represented
by some integer.
Therefore integers are closed
under both addition and subtraction.
You may be wondering
what a negative number actually means.
As recently as the 18th century
negative numbers were not accepted as
legitimate numbers by many mathematicians.
It was thought that only positive numbers
represented things in the real world.
However, the idea that negative numbers
don't actually represent anything
in the real world is debatable
as anyone who has ever overdrawn their
bank account can tell you.
Someone who owes more money than they have
could be thought of as having
less than zero money
or having a negative net worth.
Death Valley is below sea level
so the altitude of Death Valley could be
thought of as a negative altitude.
A vacuum cleaner creates an air pressure
which is less than atmospheric pressure
so it can be thought of as creating
a negative pressure.
Integers can be represented on a number line
just like natural numbers and whole numbers
but now the number line must go off to infinity
in both directions.
With a positive or negative sign
a number can be thought of as representing
not only a distance, but also a direction.
Just as a positive number can be thought of
as representing a distance
to the right of the origin
a negative number can be thought of
as representing a distance to the left.
Adding a positive integer means
moving that number of units to the right.
For example, if we add a positive integer
to the number two
we start at two on the number line
and then move that number of units
to the right.
Adding a negative integer means
moving that number of units to the left.
In fact, adding a negative number is exactly
the same thing as subtracting a positive number.
So we can think of subtraction as just
the addition of a negative number.
For example, the problem
two
plus three
minus six
minus two
plus four
are instructions to start at two
on the number line
then move to the right three units
then move to the left six units
then left another two units
and finally to the right four units.
At the end of the journey
you will be at the one position.
So far we have seen that
adding a positive integer
means moving that number of units to the right.
Subtracting a positive integer
means moving that number of units,
but in the opposite direction.
We have also seen that adding a negative integer
means moving that number of units to the left.
So subtracting a negative integer must mean
to move that number of units to the right.
Subtracting a negative number is the same
as adding a positive number.
The distance from a number to the origin
is called its
“magnitude” or “absolute value”.
For instance, the numbers
positive three and negative three
have opposite signs but the same magnitude
since they are located
the same distance from the origin.
If you take any number, positive or negative
and add a number of the same magnitude
but the opposite sign
the result will be zero.
This number of equal magnitude and opposite
sign is called the number's “additive inverse”.
Any number plus its additive inverse is zero.
For example, the additive inverse of
positive three is negative three
and the additive inverse of
negative three is positive three.
With the invention of integers, we now have
a much more powerful number system.
Since the integers are closed under
addition and subtraction
we can represent the result of adding or subtracting
any numbers in our system.
However, as we shall soon see
there are still some operations which cannot
be represented using only integers.
