Teacher2Teacher 
Q&A #3962 
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From: George Zeliger To: Teacher2Teacher Public Discussion Date: 2000080218:14:15 Subject: What Algebra is Kelli, 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. Regards, George
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