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  • Hello. I'm Professor Von Schmohawk and welcome to Why U.

  • In the first lecture, we explored the origins of the first number systems.

  • We also saw how the people on my primitive island of Cocoloco

  • first learned about the decimal number system.

  • Once the Cocoloconians discovered decimal numbers

  • we could do much more than count coconuts.

  • We could do arithmetic calculations with coconuts!

  • The first arithmetic operations we invented were addition and subtraction

  • which came in very handy when dealing with coconuts.

  • For instance, if you have three coconuts

  • and then your neighbor gives you five more

  • you will have eight coconuts.

  • Interestingly, if you start out with five coconuts

  • and then your neighbor gives you three more

  • you will also have eight coconuts.

  • For some reason, five plus three gives you the same answer as three plus five.

  • Eventually we figured out

  • that it doesn't matter in which order you acquire your coconuts.

  • You still end up with the same number of coconuts.

  • Since the two numbers on either side of the addition symbol

  • can switch positions without changing the answer

  • we say that theycommutesince commute means to travel back and forth.

  • Mathematical operations in which the numbers operated on

  • can be switched without affecting the result are said to becommutative”.

  • This illustrates what the Cocoloconians call

  • The Commutative Property of Addition of Coconuts.

  • Apparently, this property applies to adding anything

  • so we will just call it The Commutative Property of Addition.

  • So addition is a commutative operation.

  • Instead of talking about specific numbers like three and five

  • if we call these two quantities A and B

  • then we can write this property in a more general way.

  • Although addition is a commutative operation, subtraction is not.

  • For example, four minus three is one.

  • However, if you switch the order of the three and the four, the result will not be the same.

  • Three minus four is negative one.

  • As we saw in the previous lecture

  • we can write four minus three as an addition of positive four plus negative three.

  • Now, since addition is commutative

  • we can switch the four and the negative three without changing the result.

  • This is a trick which can come in handy in algebra problems.

  • Addition is a “binary operation”.

  • Binary operations are mathematical calculations which involve two numbers.

  • These numbers are calledoperandsand in the case of addition

  • these operands are added together to produce a result called thesum”.

  • In addition operations, the operands are sometimes referred to as theaddends”.

  • Even though addition is defined as a binary operation

  • you may often see additions involving more than two operands.

  • This is possible because pairs of operands can be added one at a time

  • with each sum replacing the pair.

  • In this way, an unlimited number of operands can be added sequentially.

  • The commutative property can be applied to addition operations

  • involving more than two numbers.

  • By switching adjacent pairs of numbers, operands can be reordered in any way we please.

  • For instance, in this addition involving four operands

  • the two at the end could be moved up to the front.

  • Or the five could be moved to the back.

  • In addition to the commutative property

  • here is another interesting property of addition that we discovered.

  • Let's say that you have five coconuts

  • and your neighbor on the left has three.

  • Both of you get together and pool your coconuts into one group of eight.

  • Then your neighbor on the right gives you four more.

  • Now you will have your group of eight plus four more, for a total of twelve coconuts.

  • On the other hand, let's say you started out

  • by pooling your five coconuts with your neighbor on the right who had four coconuts.

  • So you start out with a group of nine coconuts.

  • Then your neighbor on the left gives you his three

  • so you end up with three plus your group of nine

  • or a total of twelve coconuts.

  • You still end up with the same number of coconuts.

  • In these two scenarios the coconuts were grouped in different ways before they were added.

  • However, we ended up with the same number.

  • This illustrates what is called The Associative Property of Addition

  • because it doesn't matter in which way the coconuts are grouped

  • or associated with each other before they are added.

  • In the end, they all add up to the same number.

  • If we call these quantities A, B, and C

  • then we can write this property in a more general way.

  • The commutative property of addition involves moving around the numbers to be added

  • whereas the associative property of addition involves grouping them differently.

  • Using the associative and commutative properties, we can rearrange groups of numbers.

  • Let's see what this looks like on a number line.

  • As an example, we will take an addition problem involving positive and negative numbers.

  • Let's start at the origin and add positive two

  • plus positive three

  • plus negative six

  • plus negative two

  • plus positive four.

  • This all totals up to one.

  • However, because of the commutative property

  • we are free to rearrange this sequence of numbers in any order we like.

  • For instance, we could add all the negative numbers first

  • and we will still get the same result.

  • We could also use the associative property to group some of the numbers to be added.

  • For instance, the negative two and the positive two could be grouped.

  • Since this group adds up to zero, we could replace it with a zero

  • or eliminate it altogether.

  • Having a familiarity with the properties of addition

  • allows us to start building a tool chest of mathematical tricks

  • which we can use later to simplify complicated problems.

  • In the next few lectures, we will explore the properties of more arithmetic operations

  • such as multiplication and division.

Hello. I'm Professor Von Schmohawk and welcome to Why U.

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B1 中級

代数5 - 加法の可換的性質と連想的性質 (Pre-Algebra 5 - Commutative & Associative Properties of Addition)

  • 45 10
    g2 に公開 2021 年 01 月 14 日
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