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.