Realm of Algebra by Isaac Asimov

Author:Isaac Asimov [Asimov, Isaac]
Format: epub
Published: 2010-05-22T00:24:58.526000+00:00

The Final Operations

117

let's consider a simple expression involving numerical symbols only.

-What is the value of (2)(2)(5)? The answer, you

can see at once, is 20. It doesn't matter whether

you first evaluate (2) (2) to get 4 and then evaluate

(4)(5) to get 20; or do it in another order, (2)(5)

giving you 10 and (10) (2) giving you 20. The

answer is the same no matter in what order you

multiply.

Suppose, though, you had written not (2) (2) (5)

but (22)(5). It is the same expression, but now it

involves an involution. The value of 22 is 4, of

course, and if that is multiplied by 5, the answer

is 20. But can you multiply 2 by 5 first and then

perform the involution? Suppose, just to see what

would happen, you try to do this. Well, (2) (5) is

10 and 102 is 100, and now you have an inconsistency.

To avoid the inconsistency, involutions must be

performed first, before multiplications are performed; just as multiplications are performed before

additions.

As an example, let's look at the expression 3x2

and ask ourselves what it means. Does it mean

To answer that question, let's remember that I

have explained that since multiplications must be

performed before additions, a multiplication is

always treated as though it were in parentheses.

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