13.3 Floating-point numbers, representation and manipulation

Question 1

Convert this binary floating point number into denary

Mantissa: 01011010
Exponent: 00000100

Answer 1

11.25

Question 2

10 bit Mantissa
6 bit Exponent

Both two’s complement.
Calculate -7.25 in this system.

Answer 2

Mantissa: 1000110000
Exponent: 000011

Question 3

Two’s complement system. Calculate denary value of given binary floating-point number.

Mantissa: 1011000111
Exponent: 000111

Answer 3

• 78.25

Question 4

What does changing Mantissa vs Exponent do?

Answer 4

Mantissa = Precision
Exponent = Range

Question 5

Normalize the two’s complement floating-point number.

Mantissa: 0000000111
Exponent: 100111

Answer 5

Mantissa = 0111000000
Exponent = 100001

Question 6

Two’s complement form.

12 bit Mantissa. 4 bit Exponent.

Represent +2.5 in floating-point representation.

Answer 6

Mantissa = 010100000000
Exponent = 0010

Question 7

Two’s complement form.

12 bit Mantissa. 4 bit Exponent.

Represent -2.5 in floating-point representation.

Answer 7

Mantissa = 101100000000
Exponent = 0010

Question 8

What makes a floating-point system normalized?

Answer 8

The first two bits should be different.

Question 9

Write the largest possible number that can be represented with an 8 bit Mantissa and 4 bit Exponent.

Answer 9

Mantissa = 01111111
Exponent = 0111

Question 10

Write the smallest positive number that can be represented with an 8 bit Mantissa and 4 bit Exponent.

Answer 10

Mantissa = 01000000
Exponent = 1000

Question 11

A student enters the following code into an interpreter.
X = 0.1
Y = 0.2
Z = 0.3
OUTPUT (X + Y + Z)

The student is surprised to see the output:
0.6000000000000001
Explain why this is output.

Answer 11

• As these numbers cannot be represented exactly in binary as there are rounding errors
• Adding two or more of these inaccurate representations increases the probability of inaccuracy.
• Which may give an answer where the different added inaccuracies can be large enough to be seen.

Question 12

………………………………………… can occur in the exponent of a floating-point number, when the exponent has become too large to be represented using the number of bits available.

Answer 12

Overflow

Question 13

A calculation results in a number so small that it cannot be represented by the number of bits
available. This is called …………………………………………

Answer 13

Underflow