Fractional Numbers and Blackboard Notation.

Question

Infinite binary expansions of rational numbers are either pure recurring or

mixed recurring depending on whether the cycle starts immediately after the

point (pure recurring), or somewhat later (mixed recurring).

a) [math] Show the infinite binary expansion of 37.8 without normalization.

b) [math] Show the infinite hexadecimal expansion of 37.8 without normalization.

c) [math] Show the infinite binary expansion of 37.8 with normalization.

Include the scale factor, with a decimal exponent. (This is a _scaled_

binary expansion).

d) [math] Show the infinite hexadecimal expansion of 37.8 with normalization.

Include the scale factor, with a decimal exponent. Write the scale factor as

a power of 2. Note: normalization = 1.xxx in both cases (bin and hex).

e) Show, in normalized binary blackboard notation, the approximation to 37.8

for a fixed-size fractional field that is 20 bits long. Show these 20 bits.

Now, show these same 20 bits in hexadecimal (5 hexits).

Answer 1

0.8 = 0.5 + 0.25 + 0.05

= 1/2 + 1/4 + 0.05

= 2^(-1) + 2^(-2) + 0.05

is the binary expansion thus far....

The 0.05 MUST be written as the some

of the binary expansion

SUM (1/2)^N = 5/100 = 1/20

This is the sum of the geometric series

with common ratio 1/2

the sum of which is 2[R^(n+1) - 1]

Then, with R=1/2

2[R^(n+1) - 1]= 0.05

R^(n+1) - 1 = 0.025

(1/2)^(n+1) = 1.025

Takes LOG_2 of both sides

-(n+1) = log(1.025)

n+1 = -log(1.025)

n = -1-log(1.025)

change of base to log_10 = LOG

n = -1 - LOG(1.025)/LOG(2)

which is Highly irrational and NOT an integer

Base 16 will be no better