Question

Using Horner's scheme, represent (evaluate) the 2's complement number 10011101 in decimal.

CAUTION: You will earn NO CREDITS if you don't use Horner's scheme to more efficiently evaluate polynomial.

You MUST apply Horner's scheme directly on the polynomial expansion for the number as given (NOT on the polynomial expansion for the 2's complement of the given number).

(This means that you should NOT, for instance, evaluate 1011 as -<corresponding positive value of 1011 using Horner's scheme>.)

(It also means that you should NOT, for instance, evaluate 1011 as <unsigned value of 1011 using Horner's scheme> - 24.)

You MUST apply Horner's scheme to cover all contributing bits, i.e., all contributing bits (not just the positive contributing bits, for instance) must participate in the repeated factoring.

You will earn NO CREDITS if you simply show the final result and not clearly show working (i.e., intermediate steps).

Answer 1

For below 8 bits (abcd efgh) the Horner's scheme is

= ((((((((-a*2 + b)2 + c)2 + d)2 + e)2 + f)2 + g)2 + h

Substituting 1001 1101 in abcd efgh

= ((((((((-1*2 + 0)2 + 0)2 + 1)2 + 1)2 + 1)2 + 0)2 + 1
((((((-2)2 + 0)2 + 1)2 + 1)2 + 1)2 + 0)2 + 1
(((((-4)2 + 1)2 + 1)2 + 1)2 + 0)2 + 1
((((-8 + 1)2 + 1)2 + 1)2 + 0)2 + 1
((((-7)2 + 1)2 + 1)2 + 0)2 + 1
(((-14 + 1)2 + 1)2 + 0)2 + 1
(((-13)2 + 1)2 + 0)2 + 1
((-26 + 1)2 + 0)2 + 1
((-25)2 + 0)2 + 1
(-50 + 0)2 + 1
(-50)2 + 1
-100 + 1
= -99
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