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.