Hey it’s Professor Dave; let’s talk about addition and subtraction.
As we have just come to understand, math never started out as a bunch of seemingly arbitrary
operations meant to confuse and frustrate students.
It is simply that our understanding of math has become extremely sophisticated over the
past few hundred years, and the fact that all of this existing and sometimes difficult
math must be learned before anyone can contribute to the field is what is responsible for the
way that so many people despise this subject.
But let’s remember that math began very simply as part of our language, a set of models
and symbols that helped us to communicate ideas and describe our surroundings.
Mathematical innovations arose by necessity, and they still do, it’s simply that the
frontier of today’s math lies in an abstract place that very few can understand.
By the end of this series, maybe we can all get there, but for now, let’s start at the
very beginning.
What was the first kind of math that was developed by the human race?
That would undoubtedly be arithmetic.
Shortly after humans were able to count, we needed symbols to represent those counting
numbers, as well as methods to manipulate those numbers in ways that represent real-life
concepts.
How many people are in the tribe?
Two kids were just born, how many are there now?
How many years has the tribe’s wisest elder been alive?
Once civilizations formed and we began to trade goods with one another, we needed to
be able to keep track of inventory, price items appropriately, and so forth.
How many apples are in the basket?
How many do you want, and how many are now left?
Being that we like to count on our fingers, and we have ten of them, many counting systems
were based on a system of ten.
There were others based on twenty, or even sixty, but the one we use today is based on
ten, so rather than diving into the anthropology of arithmetic, let’s keep our study focused
on what can be readily applied.
Presently, our conceptualization of the number ten as the basis for our numerical system
is so ingrained in our collective logic that we sometimes forget that this number is completely
arbitrary.
If we had only eight fingers, things would be totally different.
But ten is what we went with, and it works just fine.
Getting back to the apples, the basic operations we will learn first are addition and subtraction.
Addition is the most basic arithmetic operation, and it represents the combination of two numbers
to become a single number, or a sum.
If you get two apples from one vendor, and then three from another, how many apples did
you get?
Of course we can easily count the resulting pile and see that there are five.
But how do we represent this mathematically?
Using the symbols that are common of today, we would put the number two, then the addition
or plus symbol, and then the number three, followed by an equals sign and then the number
five.
This is an equation, which is a statement of equality.
The expression on the left is numerically equivalent with the expression on the right.
This particular equation reads, “two plus three equals five”, with the word plus essentially
meaning “and”, and the equals sign meaning “is”.
Two and three is five, so five is the sum of this additive operation.
The next operation that became necessary was subtraction, which is the inverse, or opposite
of addition, in that it doesn’t find the sum of two numbers, it finds their difference.
You bring all five of your apples home and you eat one of them.
If one apple has been subtracted, how many are left?
Again, it is easy to count and see that there are four left.
But this result can also be calculated, which is much different from counting.
We can write another equation with a five, then the minus symbol, followed by a one,
then the equals sign, and the number four.
This reads, “five minus one equals four”, which essentially means five less one is four,
so four is the difference between one and five.
On a number line, this is the distance between the two numbers, and this is an excellent
way to visualize subtraction.
Fourteen minus eleven is three, because three is the difference between the two numbers.
It takes three to get from eleven to fourteen.
Now that we have become familiar with the symbolic representation of these simple operations,
we should discuss some applicable properties of numbers.
Addition is commutative, in that the order in which numbers are added does not matter.
Two plus three equals five, and three plus two also equals five.
Subtraction is not commutative.
It does indeed matter which number is being subtracted from the other.
Three minus two is not the same as two minus three.
Addition is also associative.
This means that if performing two successive additions, the order in which they are performed
does not matter.
Two plus three plus four will be nine no matter which numbers we add first.
We can add the first two to get five, and then add that to four, or we can add the latter
two to get seven, and then add that to two.
The result is the same.
Subtraction is not associative.
If we write down five minus three minus two, we could do five minus three first.
That gives us two, and subtracting the other two, we get zero.
If instead we do the three minus two first, we get one, and five minus one is four.
So we can see that subtraction is not associative.
We will learn all about the order of operations later, as well as other ways in which these
kinds of properties become less obvious yet very important.
No matter how far we choose to go with math, and no matter how complicated equations appear
to be, with symbols like square roots and logarithms and integrals, we must always remember
that these symbols mean something concrete.
They are rooted in the physical world, even if in a way that is more difficult to immediately
conceptualize than the plus symbol.
One goal of this series will be to make all the other mathematical operations as intelligible
and relatable as addition and subtraction.
Once you intimately understand what mathematical constructs represent, they no longer seem
arbitrary and infuriating, but instead powerful, as their utility becomes apparent.
So let’s move forward and learn some more arithmetic, but first, let’s check comprehension.