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Problem 10 (Calculator Problem)

A calculator can find the reciprocal of a number as well as addition and subtraction. Describe how you would go about finding:

• the square of any number
• the product of two numbers

Solution:
(a) To find the square of any real number a different from zero, we use our calculator, which can add, subtract, and find reciprocals, to compute

where we can choose n as any positive integer as long as it is different from a. Using our calculator again, we compute the difference

and then add this last quantity to itself n times, getting the square of the original number a.

(b) To find the product of a and b, we observe the algebraic identity

If you look carefully, every term on the right-hand side of the equation can be calculated using our addition, subtraction, and reciprocal keys since

Note: This is one of those problems that seems to have more entertainment value than practical value, until you consider the fact that it might be easier to make electronic circuitry for adding, subtracting, and taking reciprocals than for computing products. The same was true for the Radon transform earlier in this century until a medical researcher at Tuft's University discovered that it could be used for making medical images, such as CAT scans and MRIs.

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Last modified on Tuesday, January 12, 1999