Representing Real Data

Question 1

Why can’t a computer represent all real numbers exactly?

Answer 1

Because there are uncountably many real numbers but only finitely many bit patterns

Question 2

What do we want from an encoding of real numbers?

Answer 2

It should be precise (as accurate as possible) and unambiguous (each bit pattern corresponds to exactly one number in the system).

Question 3

General form of scientific notation in base 10?

Answer 3

Number = 𝑚 × 10 ^ x , where 𝑚 is the mantissa (just the number) and 𝑥 is the exponent.

Question 4

What are the three parts of scientific notation?

Answer 4

Mantissa (significand), base (radix), and exponent.

Question 5

In decimal scientific notation, what is the base?

Answer 5

The base (radix) is 10

Question 6

In normalized decimal scientific notation, what range must the mantissa satisfy?

Answer 6

1 ≤ 𝑚 < 10

Question 7

How is zero represented in normalized scientific notation?

Answer 7

m = 0
x = 0

Question 8

Convert 16.37 to scientific notation.

Answer 8

16.37 = 1.637 × 10^1

Question 9

List three virtues of scientific notation.

Answer 9

It is simple to generate, understand and manipulate;

It can represent numbers of widely varying magnitudes in a compact representation;

It is accurate to the precision of the representation

Question 10

What happens when the mantissa has a fixed number of digits?

Answer 10

Numbers with more significant digits than the mantissa can hold must be approximated (rounded).

Question 11

Give two disadvantages of scientific notation.

Answer 11

It approximates numbers that are mathematically exact (e.g. 1/3 or pi)

Rounding errors can accumulate and grow in complex calculations.

Question 12

General form of binary floating point representation?

Answer 12

Number = ± 𝑚 × 2^𝑥 , with 1 ≤ |m| < 2 for normalized values.

Question 13

What is special about the mantissa in normalized binary floating point?

Answer 13

It always has a leading 1 to the left of the binary point (except for zero).

Question 14

Normalize
101.1101 in binary scientific notation.
2

Answer 14

101.1101 (2) = 1.011101 (2) × 2^2

Question 15

What does the sign bit 𝑆 represent in floating point?

Answer 15

𝑆 = 0 means positive
𝑆 = 1 means negative

Question 16

How to convert IEEE 754 Single Precision (32 bits) to decimal

Answer 16

First separate into its 3 parts (sign, exponent, mantissa)

The most left bit represents the sign
Next 8 bits represent the exponent
Next 23 bits represent the mantissa

Convert the exponent from binary to decimal then subtract 127.

For the mantissa do 1 + 2^m + 2^n where m,n represents the positions of all 1’s in the mantissa.

Then multiply the above number by 2^x where x represents exponent - 127

Question 17

How to convert the integer 13.1875 to single precision floating point number.

Answer 17

The number is positive so sign = 0

Mantissa:

13 in binary = 1101

0.1875 - keep multiplying by 2 until you reach the 0 after the point, or notice repetition

0.1875 x 2 = 0.375 0.375 x 2 = 0.75 0.75 x 2 = 1.5 0.5 x 2 = 1

we ignore the 1 in 1.5, only care about decimal part

LOOK AT FIRST NUMBER EITHER 0/1

Join the 2 numbers together : 1101.0011

Normalise it to 1.1010011 by moving binary point 3 times to the left so it is:

1.1010011 x 2^3 -> First 1 is not stored, so: 1010011

Add the bias so: 3 + 127 = 130 = 10000010

Answer:

01000001010100110000000000000000

Question 18

How to do floating point addition using the example : (9.6 x 10^2) + (6.6 x 10^1)

Answer 18

1) De-normalise the smaller operand and adjust its exponent to equal that of the other operand: (9.60 x 10^2) + (0.66 x 10^2)

2) Add the mantissae and retain the common exponent: (10.26 x 10^2)

3) Re-normalise the mantissa and adjust the exponent if necessary: (1.03 x 10^3)

Question 19

How to do floating point multiplication using example: (5.2 x 10^-4) x (2.6 x 10^7)

Answer 19

1) Multiply the mantissae and add the exponents. This will yield: (13.56 x 10^3)

2) Renormalise the mantissa and adjust the exponent if necessary: (1.356 x 10^4)