Q&A #3962

Teachers' Lounge Discussion: Algebra

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From: George Zeliger <zeliger.g@asqnet.org>
To: Teacher2Teacher Public Discussion
Date: 2000080218:14:15
Subject: What Algebra is


You asked a very good question.
While Arithmetic is a science of properties of individual _numbers_
(its higher part is known as Number Theory), Algebra is a science of
properties of _operations_ we perform on numbers in Arithmetic.

For example, that 3+5=5+3 is a property of the numbers 3 and 5, and
can be verified experimentally.
However, that a+b=b+a, where letters a and b represent _any_ numbers,
is not a property of numbers, it is a property of the _operation_.

In Algebra we usually talk about the same operations that are
introduced and used in Arithmetic, and to which we got accustomed so
much.  It is possible, however, to define some other operations on
numbers, not so commonly known.  For example, the operation of
concatenation, which I will denote a@b, and which means that I write
the number b right behind the number a.  In this case 3@5, which is
35, is not equal to 5@3, which is 53.  Algebra would study general
features of this new operation asking questions like whether 
(a@b)@c=a@(b@c) for any numbers a,b, and c, and if not, then for which
sets of numbers it is true -- thus, properties of the _operation_
would define some structure in the set of _numbers_.

When I write (a+b)**2=a**2+2*a*b+b**2, this is a statement about the
operations of addition, multiplication, and raising to a degree (as
the matter of fact, raising to a whole degree is a special case of
multiplication, while multiplication of whole numbers is a special
case of addition, so we may say that everything is about addition
only).  Therefore, this is an algebraic statement.



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