Problem 8. We wish to do formal addition and subtraction with a set of three stones. For this purpose we shall model the stones by three letters S1, S2 and S3. Our aim is to develop a for- mal algebra of stones that allows us to do addition and subtraction as we do with integers, so expressions like S1 + S2, or -5 × S3 should be possible. (a) What could be a sensible intended interpretation of S1 + S2? (b) What would be a sensible interpretation of -5 × S3? (c) Develop ideas how such a formal algebra could be defined and implemented. (d) Can you solve linear and quadratic equations in your formal algebra of stones? Discuss. (e) What relationship can you draw from your formal algebra of stones to working with numbers like the reals, say?
(a) A sensible intended interpretation of in this formal algebra of stones could be that combining and results in a new stone or symbol, say , which represents the concept of having both and together. So, could be interpreted as , which signifies the presence of both and .
(b) A sensible interpretation of could mean that we have five stones of type , but with a negative sign indicating the absence or removal of these five stones. In other words, could represent a state where has been subtracted five times.
(c) Developing a formal algebra of stones can be done as follows:
1. Define the basic stones or symbols, such as , , , and so on, to represent different types of stones or objects.
2. Define addition () and subtraction () operations on these stones. For example, could be defined as the creation of a new stone, , to represent the combination of and . Subtraction can be defined as the removal or cancellation of stones.
3. Define multiplication () as a way to repeat or replicate stones. For example, could represent having five instances of , and could represent removing five instances of .
4. Establish rules and properties of this algebra, such as the distributive property, commutative property, and associative property, if applicable.
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