By S. A. Amitsur, D. J. Saltman, George B. Seligman
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Extra info for Algebraists' Homage: Papers in Ring Theory and Related Topics
X(x − 5)(x 2 + 10) Solution: We look for trouble spots. First we consider the numerator. To make the quantity under the square root non-negative, we need x ≥ 3. The denominator has three factors: x, x − 5, and x 2 + 10. All of these are potential trouble spots. The factor x is 0 at x = 0. But this is already ruled out by our earlier restriction that x ≥ 3. The factor x − 5 is 0 at x = 5, and we must exclude this value. Now we focus on the factor x 2 + 10. Because x 2 ≥ 0, it follows that x 2 + 10 is positive for all x.
Surprisingly, it can be shown that the integers have the same cardinality as the rational numbers (fractions), but not the same cardinality as the real numbers. This has the intriguing consequence that “most” real numbers are not rational numbers. 1 SKILL BUILDING EXERCISES Evaluating formulas: In Exercises S-1 through S-24, evaluate the given function as required. √ x+1 S-1. f (x) = 2 at x = 2 x +1 S-2. 2 S-3. 6 x2 + y2 S-4. 1 − e4t . S-5. 1) if f (s) = s2 + 1 . s2 − 1 S-6. Calculate f (2, 5, 7) if f (r, s, t) = r+ s+ √ S-7.
95 . 95 represent? 9. Present value: Present value is the amount of money that must be invested now at a given rate of interest to produce a given future value. For a 1-year investment, the present value can be calculated using Future value , 1+r where r is the yearly interest rate expressed as a decimal. ) If an investment yielding a yearly interest rate of 12% is available, what is the present value of an investment that will be worth $5000 at the end of 1 year? That is, how much must be invested today at 12% in order for the investment to have a value of $5000 at the end of a year?
Algebraists' Homage: Papers in Ring Theory and Related Topics by S. A. Amitsur, D. J. Saltman, George B. Seligman