Professor Dave here, let’s talk about multiplication and division.
It’s easy to understand how humans developed addition and subtraction out of necessity,
due to the increasingly complex interactions that burgeoning civilizations required.
The next operations that followed soon after were multiplication and division.
Multiplication is similar to addition in that it involves the combination of two numbers
to get a single number, but with multiplication this number is not a sum, it is something
else, which we call a product.
We can view multiplication as an abbreviated form of repeated addition.
Let’s go back to our basket of apples and pretend that we start with none.
We begin to collect apples and toss them in the basket two at a time.
There’s two, then two more, again, and again, and then a fifth time.
So we have added two apples to the basket five times.
How many are in there now?
Well, apart from just counting the ten apples, we could use addition and say two plus two
plus two plus two plus two.
We add up all the twos, and get ten.
But it’s easy to see how this could get very cumbersome very fast, if the number of
successive additions becomes more substantial.
Instead, we can use multiplication.
Rather than adding two five times, we can just do two times five.
The word “times” means multiplication, and in this case multiplying the two by five
represents the five successive twos that would otherwise be added together.
Two and five are called factors, and ten is the product.
Early on, we will see an X as the symbol for multiplication, but later on in our study
of math we won’t see this as often.
Instead we will just see a dot, an asterisk, or even nothing at all, because when two algebraic
terms are adjacent, it is implied that the terms are being multiplied.
So if multiplication solves the problem of how to combine several identical groups of
items into a single larger number, or a product, then division, as the inverse operation of
multiplication, does just the opposite.
It can break a larger number into smaller ones.
Let’s say you want fifteen apples, and at the market there are baskets, each containing
five apples.
How many baskets should you take?
We simply divide fifteen by five per basket to get three, which means that three baskets
will give us all the apples we need.
Fifteen is the dividend, and five is the divisor, while three is the quotient.
So multiplication gives us a product and division gives us a quotient.
With division, early on we see this symbol, but later on this will almost never be used.
Instead, we will see a line, either at an angle, or horizontal.
With this kind of notation, the first number will be divided by the second.
We should note that multiplication, like addition, is commutative.
Two times five is the same as five times two, because whether we have five piles of two
apples or two piles of five apples, we have ten apples in total.
Division, however, like subtraction, is not commutative.
Ten divided by two is not the same as two divided by ten.
Also, multiplication is associative.
Two times three times four will be twenty four no matter which numbers are multiplied
first.
We could do six times four, or two times twelve; it doesn’t matter.
Division, however, is not associative.
Ten divided by five divided by two will provide a different result depending on which number
is divided first.
This is not arbitrary, we defined these operations, and then observed whether or not they abide
by commutativity and associativity.
So there we have the four basic arithmetic operations: addition, subtraction, multiplication,
and division, as well as some of the properties of these operations.
The more we practice them, the easier they become, and it may be of some use to memorize
the multiplication tables.
Remembering offhand that eight times seven is fifty-six is not only faster than using
a calculator, it should also be a mild source of pride.
Our society has become increasingly dependent on calculators for the most trivial operations,
which is an incredible shame, because it robs us of considerable mental acuity, and lessens
our appreciation for the patterns that are inherent in numbers.
So the next time you pull out your phone to do something that should be done in your head,
just picture me making this face.
Don’t worry, later we will learn some tips on how to do mental math.
For now, let’s check comprehension.