- Operands and operators. The “operands” of arithmetic are 1, 0, etc., and the “operators” are +, -, *, /, etc. Algebra, then, which builds on top of arithmetic, introduces the “operands” x, y, etc.
- The operands as distinguished into constants and variables. For example: 1 and 0 are “constants,” and x and y are “variables.” In arithmetic and algebra, the constants are numbers (e.g., 1, 0). The variables, then, are like blanks to be filled in with those constants (which, again, are numbers in the present context). x + 1 = 2 is like _ + 1 = 2. What constant/number, if put into that blank, would make a true proposition? The answer is of course 1. Thus, x = 1. That is: _ = 1. But why aren’t blanks actually used? The reason is that x is like a _ that must be filled in with the same constant/number everywhere—well, everywhere in the circumscribed sphere of that x’s usage. For example, consider: 2x + 3x = 10. That’s like 2_ + 3_ = 10 with the constraint that both _ must be filled in with the same constant/number. By contrast: 2x + 3y = 10 is like 2_ + 3_ without that constraint.
- Relations. The “relations” of arithmetic and algebra are =, >, <, etc. Without the relations, no proposition can be made (a proposition being anything that’s either true or false). For example: 1 + 1 isn’t a proposition, for it can neither be true nor false. But both 1 + 1 = 2 and 1 + 1 = 3 are propositions (with the former happening to be true and the latter happening to be false).
- x^2 + 6 = 5x is such that x = 2 or 3. Consider, though, that x + y = y + x is such that x can be any number and so can y. Thus: For some number(s) x, x^2 + 6 = 5x. And for all numbers x and y, x + y = y + x.
- A pair of propositions in English analogous to the foregoing: (1) “For some American people x, x’s parents are from America.” (2) “For all Japanese people x, x’s parents are from Japan.”
- Another pair: (1) “For some species of birds x, the prototypical x can fly.” (2) “For all species of fish x, the prototypical x can swim.”
- I’ll also need to define the vocabulary 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, along with the grammatical system inherent in putting, say, 1 (⚫︎) before 2 (⚫︎⚫︎), and getting 12 (⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎). That should come before bringing up the operands, the operators, and the relations, along with bringing up quantifiers etc.
The vocabulary and grammar of arithmetic and algebra
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