Integer is the mathematical name for a number that can be positive or negative (or zero).
So what is a negative number, anyway?
Well, first let’s start with zero.
Suppose I give you five dollars.
Then I take two dollars back.
Then I take back three dollars.
How many dollars have you got left?
Of course, you don’t have any dollars.
But we really would like a number to represent that state, so that our bank account can still
have a number instead of our having to say, “there is no money left in this account”.
So we say instead, “the balance is zero”.
Think about this carefully.
It’s actually a very abstract concept.
Zero becomes a new number, representing not-having-any.
I can’t actually point to zero of anything, because they don’t exist, by definition!
It’s not a whole number, because I don’t have any whole things.
But it’s definitely a very useful number.
Okay, so what about negative numbers?
Well, just like zero, it’s an abstract way of keeping track of things when—well, to
be honest, when there isn’t actually anything physical to keep track of!
Suppose again I give you five dollars.
Then let’s say I take seven dollars from you.
What’s the balance of your account now?
It’s two dollars, but now I owe you instead of you owing me!
It’s two dollars, but I can’t actually point to the two dollars in your hand.
(I have to point to my own hand instead, in this case.)
So there’s the idea of two dollars, but there aren’t two whole anything that you
actually have.
How are we going to represent this in our account balance?
What we do is we invent negative numbers.
We say the balance is negative two.
That means it’s two dollars, but “in the opposite direction”, so to speak.
(Some people call negative numbers “directed numbers”.
We’ll come back to this idea again when we look at performing operations on integers.)
Okay, so if we can never actually point to negative three of anything, do negative numbers
really exist?
And this is where Mathematics takes a turn towards the abstract.
A mathematician will say, yes, the number itself exists, as an abstract concept—even
though you will never actually have a negative number of anything in the physical world.
Mathematics is really a study of abstract ideas and abstract truths.
… And occasionally it may actually be a useful way to do things in the real world,
but that’s a bonus.
Remember when we talked about whole numbers, and we said that the number or meaning was
an abstract concept, represented by the numeral, which we write using the digits.
The number 13, the idea of having thirteen of something, was represented by the numeral
“13” (a “one” followed by a “three”), made up of the digits “one” and “three”.
Well now we have the idea, the concept of having negative thirteen—if you like, the
idea of a lack of thirteen things, or thirteen steps in the opposite direction, or whatever—represented
by the numeral “–13” (written “minus sign”, “one”, “three”), made up
of the digits “one” and “three” and now also the symbol “minus”.
It’s exactly the same as it was with whole numbers.
The number itself is abstract.
It’s just that now I can’t point to anything physical in the real world to show you what
the number represents.
Just like with whole numbers, one of the simplest ways to represent and work with integers is
with a number line—except our number line now goes backwards past zero into the negative
numbers as well as forwards from 1 through the positive whole numbers.
We’re still applying the convention that positive numbers increase to the right.
So what’s the first integer?
As with whole numbers, there’s some disagreement.
Most people would still start with one.
The first positive number is certainly one, and the first negative number is negative
one.
But others say zero, because it sits neatly in between the positive and negative sides—and
there are some other nice theoretical reasons too, that abstract mathematicians like because
of the way number theory and group theory work.
That’s well beyond us here.
You decide for yourself; personally I like zero.
The smallest integer then is zero, because zero means you have none at all… or is it?
Wait a minute, don’t I get smaller numbers by moving left on the number line?
So then the smallest number would actually be… well, negative something-very-big?
We have to be careful with our language here.
If the positive numbers increase to the right, does that mean the negative numbers increase
to the left?
Or, are our numbers decreasing as we go to the left?
This is a tricky question, and we’re going to solve it by establishing another convention.
There has to be difference between “bigger” and “greater than”, a difference between
“smaller” and “less than”.
With whole numbers they’re the same, but with integers we have to be more careful.
Most people would say three is bigger than two, so we’d like three to be bigger than
negative two.
But what about negative three and two?
And which is bigger, four or negative four?
By convention, we’re going to say that numbers to the left on the number line are “less”,
and numbers to the right are “greater”; and we’re going to try to avoid using words
like “bigger” and “smaller” when the answer is ambiguous.
That means a positive number is always “greater” than a negative number—regardless of the
size of the number.
Actually, mathematicians use the word “magnitude” instead of “size”, to avoid exactly this
confusion.
The magnitude of a number is its distance from zero along the number line, regardless
of direction.
So, three is greater than negative two.
Negative three is less than positive two.
And negative four is less than positive four.