Читать книгу Algebra I All-in-One For Dummies - Mary Jane Sterling, Mary Sterling Jane - Страница 65

Before discussing the commutative property, take a look at the word commute. You probably commute to work or school and know that whether you’re traveling from home to work or from work to home, the distance is the same: The distance doesn’t change because you change directions (although getting home during rush hour may make that distance seem longer).

The same principle is true of some algebraic operations: It doesn’t matter whether you add or , the answer is still 3. Likewise, multiplying or yields 6.

The commutative property means that you can change the order of the numbers in an operation without affecting the result. Addition and multiplication are commutative. Subtraction and division are not. So,

In general, subtraction and division are not commutative. The special cases occur when you choose the numbers carefully. For example, if a and b are the same number, then the subtraction appears to be commutative because switching the order doesn’t change the answer. In the case of division, if a and b are opposites, then you get –1 no matter which order you divide them in. By the way, this is why, in mathematics, big deals are made about proofs. A few special cases of something may work, but a real rule or theorem has to work all the time.

You can use this rule to your advantage when doing math computations. Sometimes, changing the order in addition or multiplication situations can make the work much easier.

Q.

A. You don’t really want to multiply fractions unless it’s necessary. Notice that the first and last factors are multiplicative inverses of one another:

Just switch the order of the last two numbers:

Q.

A. The second and last terms are reversed, and then the first two terms are grouped. .

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