Pre-Algebra 6 – Commutative Property of Multiplication
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Pre-Algebra 6 – Commutative Property of Multiplication

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.


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