By John Earman
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Additional info for A primer on determinism
After you’ve made a few changes, use Check Equality, Save your work, and repeat the process. That way, you’ll have a record of your work just as you would if you were using paper and pencil. As a simple example of an inconclusive test, consider the apparently obvious equation x = ln ex . Exponentiation and taking the natural logarithm of a number are inverse operations, so mathematically they “undo” each other. You might expect Check Equality to verify that the equation is true. x = ln ex is undecidable Since x can be real or complex, the right-hand side may be a multivalued function so this equation may or may not be true.
Sin (θ + φ) = cos θ sin φ + cos φ sin θ cos 3θ = cos3 θ − 3 cos θ sin2 θ You can use Simplify to reduce them too. cos θ sin φ + cos φ sin θ = sin (θ + φ) cos3 θ − 3 cos θ sin2 θ = cos 3θ Applying Expand to a product of polynomials has the effect of what is often called “multiplying it out”. (x + 2) (x − 1) (2y + 3) (y − 1) = 2x2 y 2 +x2 y−3x2 +2xy 2 +xy−3x−4y 2 −2y +6 7 (x + a) = a7 + 7a6 x + 21a5 x2 + 35a4 x3 + 35a3 x4 + 21a2 x5 + 7ax6 + x7 The result for (x + a)7 is an example of a binomial expansion, and that name can remind you of which command to use.
2. Enclose the expression in expanding square brackets. 3. Create a Subscript to the right of the brackets. 4. List the values in the subscript input box separated by commas. 5. Click Evaluate or Evaluate Numerically. If you want to Substitute into an expression that you have not yet created, you can create the expanding square brackets ﬁrst and then create the expression inside the brackets. The Basics 15 When there is only one variable in the expression, you need only to include its value in the subscript input box without assigning it to a variable.
A primer on determinism by John Earman