Hello. I'm Professor Von Schmohawk and welcome to Why U.
In our last lecture, we saw how the people on my primitive island of Cocoloco
discovered addition and subtraction.
Once we had invented addition and subtraction
the Cocoloconians could calculate very complicated coconut transactions with great precision.
But we soon found out that with only addition and subtraction
some calculations could take a very long time.
For instance, once a year, everyone on Cocoloco
must donate three coconuts for the annual feast of Mombozo.
So if all 87 inhabitants of Cocoloco each donate three coconuts
then how many coconuts will we have for the feast?
Before we discovered multiplication, we had to add three, 87 times
to answer that question.
However, with multiplication, the answer can be found with a single calculation.
Multiplication is just a tricky way to do repeated addition.
For instance, when the king of Cocoloco wanted to tile the floor of his rectangular hut
with very expensive imported Bongoponganian tiles
we needed to know exactly how many square tiles to buy from the Bongoponganians.
We knew how big each tile was so we could have marked the floor into little squares
and counted all the squares.
But with multiplication, it was much easier.
All we had to do was to figure out how many rows of tiles we would need
and how many tiles were in each row
and then multiply the two numbers.
Since we figured it would take six rows of ten tiles
we knew that we would need six times ten, or sixty tiles.
But then someone suggested that it would be better to lay the tiles down in vertical rows
instead of horizontal rows.
We would then need ten rows of six tiles.
At first we thought that this might require a different number of tiles.
Then we realized that ten times six is also sixty
so you will still have to buy sixty very expensive imported Bongoponganian tiles.
It doesn't make any difference if you multiply six times ten, or ten times six.
You get the same number.
We originally called this
The Commutative Property of Multiplication of Very Expensive Imported Bongoponganian Tiles.
After a while we decided to shorten the name to The Commutative Property of Multiplication.
We can write this property as A times B equals B times A.
In Algebra, a dot is often used as a multiplication symbol
to avoid confusion with the letter X.
Just like addition, multiplication is a binary operation
which, as you may recall from our previous lecture
is a mathematical calculation involving two numbers.
These numbers are called "operands"
and in the case of multiplication
these operands are multiplied together to produce a result called the "product".
In multiplication operations, the operands are sometimes referred to as "factors".
Even though multiplication is defined as a binary operation
you may often see multiplications involving more than two operands.
Just as in addition, this is possible
because pairs of operands can be multiplied one at a time
with each product replacing the pair.
In this way an unlimited number of operands can be multiplied sequentially.
On Cocoloco, we soon discovered that the commutative property also applied to situations
where more than two numbers were multiplied together.
For example, let's say that you had 24 boxes.
You can stack these boxes in several different ways.
For instance, you could arrange them in three rows of four boxes
and stack them two levels high.
Or you could arrange them in four rows of two boxes
and stack them three levels high.
Or you could arrange them in two rows of three boxes
and stack them four levels high.
It doesn't matter in which order you multiply the dimensions of the stack.
It will always add up to the same number of boxes.
We can apply the commutative property to multiplication operations involving any number of operands.
By swapping adjacent pairs of numbers, the operands can be reordered in any way we please.
For instance, in this multiplication involving four operands
the two at the end could be moved up to the front.
Or the five could be moved to the back.
So two or more numbers which are multiplied can be reordered in any way
without affecting the result.
As we saw in the previous lecture, the same holds true for numbers which are added.
Addition and multiplication are both commutative.
Commutative properties are important algebraic tools
that allow us to rearrange groups of numbers which are added or multiplied.
In the next chapter, we will discover several more properties
which we will add to our tool chest of mathematical tricks.