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.