Question

Floating Point Representation

Consider a computer that stores information using 10 bits words. The first bit is for the sign of the number, the next 5 for the sign and magnitude of the exponent and the last 4 for the magnitude of the mantissa. The mantissa is normalized as described in class and in the textbook.

a. Convert 1 00010 1001 to a base-10 system

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.

Answer 1

a)

Sign bit = 1 => this means that the number is negative

Exponent sign = 0 => this means that it is positive

Exponent magnitude = 0010 = 2

Normalized Mantissa = 1.1001 = 1.5625

Hence, the number is -1.5625 * 2² = -6.25

b)

For the highest number,

sign bit = 0

Exponent sign = 0

Exponent magnitude = 1111 = 15

Mantissa = 1111

The number is equivalent to 1.9375 * 2¹⁵ = 63588

c)

And, for the lowest positive number

sign bit = 0

Exponent sign = 1

Exponent magnitude = 1111 = 15

Mantissa = 0000

The number is equivalent to 1 * 2^-15 = 0.00003051757812

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