Chapter 2: Floating Point Numbers and Rounding

Question 1

What is truncation?

Answer 1

Dropping unwanted bits, rounding toward zero, rounding down

Question 2

What is rounding toward +- infinity?

Answer 2

rounding up, essentially rounding to the nearest possible floating-point number while increasing the magnitude

Question 3

What is rounding to the nearest?

Answer 3

rounding to the closest possible floating-point number irrespective of direction

Question 4

What is normalization?

Answer 4

When a number is made such that there is only one non-zero digit leading the radix point and an exponent is made to compensate the shifting of the radix point

Question 5

Explain how to normalize a fractional binary value

Answer 5

A

Question 6

What is the Significand also called?

Answer 6

Mantissa

Question 7

What is the significand/mantissa?

Answer 7

The significant bits of a normalized number

Question 8

What is the significand/mantissa represented in?

Answer 8

sign and magnitude format (leading bit is the sign bit)

Question 9

How many unique zeros are there in IEEE-754 format?

Answer 9

2:

00000000000000000000000000000000) and (10000000000000000000000000000000

Question 10

What is the general bias of 32-bit IEEE-754 format?

Answer 10

127

Question 11

What does the leading bit of IEEE-754 format represent?

Answer 11

The sign

Question 12

What do the 8 bits after the first bit represent in IEEE-754 format?

Answer 12

The biased exponent

Question 13

What do the final 23 bits in IEEE-754 format represent?

Answer 13

The fractional bits

Question 14

When normalizing a binary number for IEEE-754 format, what is removed?

Answer 14

The leading 1 to the left of the radix point

Question 15

If every digit but possibly not the first is a 0 in IEEE-754 format, the number is:

Answer 15

+- zero

Question 16

If the exponent is all zeros, but there is at least one non-zero bit in the mantissa in IEEE-754 format:

Answer 16

The number is un-normalized and the bias is 126

Question 17

What is the one case where the bias of IEEE-754 format is 126 not 127?

Answer 17

If the exponent is all zeros, but there is at least one non-zero bit in the mantissa

Question 18

If the exponent is all 1’s and there is NO fractional bit in IEEE-754 format, the number is:

Answer 18

Either +- infinity

Question 19

If the exponent is all 1’s and there is at least one non-zero fractional bit in IEEE-754 format, the number is:

Answer 19

NaN (Not a Number)

Question 20

Explain floating-point arithmetic:

Answer 20

Make the exponents the same and then perform addition or subtraction

Question 21

For floating-point arithmetic, if the difference between the two exponents of the normalized two numbers is greater than the number of significant bits:

Answer 21

The addition result of these two numbers will be the larger of them

Question 22

For floating-point arithmetic, when will the addition result of these two numbers be the larger of them?

Answer 22

If the difference between the two exponents of the normalized two numbers is greater than the number of significant bits

Question 23

What is the smallest possible normalized value to be represented in IEEE-754 format?

Answer 23

+- E = 1, F = 0x000000

+- 1.17549 x 10^-38

Question 24

What is the largest possible normalized value to be represented in IEEE-754 format?

Answer 24

+- E = 254, F = 0x7FFFFF

+- 3.40282 x 10^38

Question 25

How many many possible NaN values can be represented in IEEE-754 format in TOTAL?

Answer 25

((2^22) - 1) * 2

possible NaN values

Question 26

How many many possible NaN NEGATIVE values can be represented in IEEE-754 format in?

Answer 26

((2^22) - 1)

possible NaN values

Question 27

How many many possible NaN POSITIVE values can be represented in IEEE-754 format in?

Answer 27

((2^22) - 1)

possible NaN values