Consider a new computer system which stores data in \"quads\", Le. it bas four s

Question

Consider a new computer system which stores data in "quads", Le. it bas four states 0, 1. 2 and 3 (as opposed to binary which has only two, 0 and 1). Now imagine that numbers are stored in a 12 quad formal where there is one quad for the sign of the number. three quads for the exponent, another one quad for the sign of the exponent and seven quads for the normalized mantissa (lie. it is of the from O.nqqqqqq, where n >= l). Find: The largest positive (decimal) number that can be represented in this system. The smallest positive (decimal) number that can be represented in this system. The machine epsilon. .

Explanation / Answer

(a) The largest exponent possible is

3334 = 3*42 + 3*4 + 3 = 63

The largest mantissa possible is 3333333.

0.33333334 = 0.9999390

Thus, the largest positive decimal number that can be represented is 0.9999390 x 463 = 8.507 x 1037

(b) The smallest exponent possible is -3334 = -63.

The smallest mantissa possible is 1000000.

0.14 = 0.25

Thus, the smallest positive decimal number that can be represented is 0.25 x 4-63 = 2.939 x 10-39.

(c) The machine epsilon is b-p/2 = 4-7/2 = 3.052 x 10-5.

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