Lab 3 Problems: Floating Point Representation

These questions ask you to convert between decimal and
32-bit IEEE 754 floating point values.  You may use
a calculator for these problems.  For more information
about the steps and exactly what to write for each step,
see the notes on converting between decimal and
binary floating-point numbers, available in the week
2 materials on the course website.


---Conversion of Decimal to Floating Point---

0.) What is the binary floating-point representation
    of decimal -1.5?  Show all eight steps.

0.step1)
Sign: 1

0.step2)
1 = 1

0.step3)
0.5 * 2 = 1.0 | 1

0.step4)
1.1
Move decimal point 0 places: 0

0.step5)
127 + 0 = 127

0.step6)
127 = 0111 1111

0.step7)
Right of decimal point: 1000 0000 0000 0000 0000 000

0.step8)
Altogether: 1011 1111 1100 0000 0000 0000 0000 0000


1.) What is the binary floating-point representation
    of decimal 13.125?  Show all eight steps.

1.step1)
Sign: 0

1.step2)
13 = 1101

1.step3)
0. 125 * 2 = 0.25 | 0
0.25 * 2 = 0.5    | 0
0.5 * 2 = 1.0     | 1

1.step4)
1101.001
Move decimal point left 3 places: 3

1.step5)
127 + 3 = 130

1.step6)
130 = 1000 0010

1.step7)
Right of decimal point: 1010 0100 0000 0000 0000 000

1.step8)
Altogether: 0100 0001 0101 0010 0000 0000 0000 0000


2.) What is the binary floating-point representation of
    decimal 22.75?  Show all eight steps.

2.step1)
Sign:

2.step2)


2.step3)


2.step4)


2.step5)


2.step6)


2.step7)
Right of decimal point:

2.step8)
Altogether:


3.) What is the binary floating-point representation
    of decimal 16.1875?  Show all eight steps.

3.step1)
Sign:

3.step2)


3.step3)


3.step4)


3.step5)


3.step6)


3.step7)
Right of decimal point:

3.step8)
Altogether:


4.) What is the binary floating-point representation
    of decimal 16.1875?  Show all eight steps.

4.step1)
Sign:

4.step2)


4.step3)


4.step4)


4.step5)


4.step6)


4.step7)
Right of decimal point:

4.step8)
Altogether:


5.) What is the binary floating-point representation
    of decimal 66.71875?  Show all eight steps.

5.step1)
Sign:

5.step2)


5.step3)


5.step4)


5.step5)


5.step6)


5.step7)
Right of decimal point:

5.step8)
Altogether:


6.) What is the binary floating-point representation
    of decimal -128.1?  Show all eight steps.

6.step1)
Sign:

6.step2)


6.step3)


6.step4)


6.step5)


6.step6)


6.step7)
Right of decimal point:

6.step8)
Altogether:



---Conversion of Floating Point to Decimal---

7.) Find the decimal representation of the following
    floating point number.  Show all six steps.
    0100 0000 1011 0000 0000 0000 0000 0000

7.step1)
Sign: positive

7.step2)
Exponent bits: 1000 0001
Decimal representation: 129

7.step3)
129 - 127 = 2

7.step4)
Mantissa bits: 0110 0000 0000 0000 0000 000
(0 * 2^-1) + (1 * 2^-2) + (1 * 2^-3) = 0.375

7.step5)
(1 + 0.375) * 2^2 = 5.5

7.step6)
+5.5


8.) Find the decimal representation of the following
    floating point number.  Show all six steps.
    1100 0000 0010 0100 0000 0000 0000 0000 

8.step1)
Sign: negative

9.step2)
Exponent bits: 1000 0000
Decimal representation: 128

9.step3)
128 - 127 = 1

9.step4)
Mantissa bits: 0100 1000 0000 0000 0000 000
(0 * 2^-1) + (1 * 2-2) + (0 * 2-3) + (0 * 2-4) + (1 * 2-5) = 0.28125

9.step5)
(1 + 0.28125) * 2^1 = 2.5625

9.step6)
-2.5625


10.) Find the decimal representation of the following
     floating point number.  Show all six steps.
     0100 0001 0001 0000 0000 0000 0000 0000

10.step1)
Sign: 

10.step2)
Exponent bits: 
Decimal representation: 

10.step3)


10.step4)
Mantissa bits:


10.step5)


10.step6)


11.) Find the decimal representation of the following
     floating point number.  Show all six steps.
     1100 0001 0111 0100 0000 0000 0000 0000

11.step1)
Sign: 

11.step2)
Exponent bits: 
Decimal representation: 

11.step3)


11.step4)
Mantissa bits:


11.step5)


11.step6)


12.) Find the decimal representation of the following
     floating point number.  Show all six steps.
     1100 0010 1111 1111 1000 0000 0000 0000

12.step1)
Sign: 

12.step2)
Exponent bits: 
Decimal representation: 

12.step3)


12.step4)
Mantissa bits:


12.step5)


12.step6)