Floating point representation Consider a computer that stores information using

Question

Floating point representation Consider a computer that stores information using 7 bits words. The first bit is for the sign of the number, the next 3 for the sign and magnitude of the exponent and the last 3 for the magnitude of the mantissa. The mantissa is normalized as described in class and in the textbook. (a) Convert 1 110 110 to the base 10 (b) What is the highest number that can be stored on this computer? (c) What is the lowest positive number that can be stored on this computer? (d) Determine the machine epsilon.

Explanation / Answer

(a)As number of having 7 bits.

1 110 110

first bit is sign bit. It is 1 for negative sign.

Next three bit for sign and magnitude of exponenet.

sign -> 1 which is negative

value of exponenet -> 10 = 2

last three is mantissa value e.i. significant value

110 = 6

So number is -6*10^(-2) = -(.06)

(b)

Highest number is positive number

sign bit = 0

NExt three sign and magnitude of exponenet

sign = 0

magnitude = 11 = 3

last three for mantissa

111 = 7

0 011 111

so number is 7*10^3 = 7000

(c)

Lowest positive number is

sign bit = 0

NExt three sign and magnitude of exponenet

sign = 0

magnitude = 00 = 0

last three for mantissa

000 = 0

0 000 000

so number is 0*10^0 = 0

(d)

p is the number of bits in significant which is 3

machine epsilon = (base)^-(p-1)

(2)^(-2)

= 1/4

= 0.25

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