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25 - Floating Point Representation

 1. Convert the number 0100100100 000100 to denary

  9.125

  7.5

  9.5

  8.125

 2. Represent the decimal value –57 as an 8-bit two’s complement binary integer

  10100000

  11000111

  1110011

  11011111

 3. The floating point representation is used to represent integers that are very large and also for the representation of real(fractional) numbers.

  TRUE

  FALSE

 4. The number 2.25 (Fixed point decimal) with an equal number of parts for the integer part and fractional part would be represented as:

  Answer:1010.0001

  Answer:0010.0100

  Answer:1111.0100

  Answer:0010.1111

 5. Using eight bits in fixed-point binary the number 11.6875 would be written as:

  Answer:1111.1111

  Answer:0011.1010

  Answer:0011.1000

  Answer:1011.1011

 6. Consider a really large integer - 1,234567. This number requires seven places to represent the value. If the no. of places was just four, then ________________________________

  None of the above

  certain digits could be stored (this is called doubling up) in the first four values

  certain digits would be added on to the fourth bit

  certain digits would need to be 'dropped'

 7. If more bits are assigned to the fractional part in fixed-point binary ___________________

  greater precision would be possible, but it would reduce the magnitude range

  None of the above

  a greater magnitude of number as well as greater precision would be possible

  a greater magnitude of number would be possible but it would reduce the precision

 8. Representing a large number like 1,234567 using only four places would make it: 1,234,000 and this would mean_____________
Note: In the number 1234567, the loss of the "most significant" digits (i.e. 123) would mean a greater loss! There are thus rules for determining significance.

  a loss of floating point recognition

  None of the above

  an increase in precision

  a loss of precision

 9. Read through the following rules for determining significance and answer the question. What are the significant digits?
Rules for determining significance (Integers)
=============================================
1. A nonzero digit is always significant
2. The digit '0' is significant if it lies between other significant digits
3. The digit '0' is never significant if it precedes all the nonzero digits

What then, are the "significant digits" in 00012340?

  All three of the zeros are significant

  The significant digits are: 01234

  The significant digits are: 1234

  The significant digits are: 12340

 10. So, how does all this apply to Binary? Review the modified significance rules for significance and answer the question. What are the significant bits?
Rules for determining significance (Binary)
=============================================
1. A 1 bit is always significant
2. The bit '0' is significant if it lies between other significant bits
3. The bit '0' is never significant if it precedes all 0s,
even if it follows an embedded radix/point.
4. The bit '0' is significant if it follows an embedded radix point
and other significant bits.

What then, are the "significant digits" in 00.010000?

  The answer is 10000

  The answer is 010000

  The answer is 1

  The answer is 00010000

 11. In the binary fixed-point notation, the radix position is fixed at a certain point within the bit pattern. In the following example where four bits are used for the integer part: a7,a6,a5,a4 contain __________________
Floatingpointrepresentation_quiz1_image2.png

  None of the above

  the integer part in two's complement form

  the exponent part in two's complement

  the fractional part in normal binary form

 12. In the binary fixed-point notation, where four bits are used for the exponent, the a3,a2,a1 and a0 contain ____________

  None of the above

  the exponent part in two's complement

  the fractional part in normal binary form

  the integer part in two's complement form

 13. In fixed point notation, if there are four bits for the integer part and four bits for the exponent what is the range of values available?

  Answer: -256.9375 to 255.93725

  Answer: -128.9375 to 127.93725

  Answer: -8.9375 to 7.93725

  Answer: -16.9375 to 17.93725

 14. The advantage of using floating-point numbers (over fixed) is that ________________

  the range of numbers is reduced, which saves space

  the precision is greatly increased even with just a single bit

  the range of numbers that can be represented with a set number of bits is far larger

  None of the above

 15. Floating point numbers consist of two parts:

  the whole number and the real number

  the mantissa and the exponent

  the sign bit and the fractional exponent

  the integer part and the non integer part

 16. In floating point binary negative numbers can be represented by:

  either using a sign bit or using the two's complement standard

  using a sign bit only

  using two's complement only

  using the negative bit notation which sets three bits to denote the sign

 17. In general, the mantissa is specified as a fixed-point binary number. The binary point is placed in between the _________________________________________________

  first and second bits (from the right)

  most significant bit and the last bit (on the right)

  least significant bit and the most significant bit

  most significant bit and the second most significant bit

 18. Based on the floating point 8 bit binary byte shown in the image below, note that the a7 decides whether the number is negative or positive and is called the ________
Floatingpointrepresentation_quiz1_image1.png

  exponent in excess-4 notation

  sign bit

  None of the above

  fractional part in normal binary

 19. The following excerpt shows the steps necessary to convert a decimal real number to floating point representation. Fill in the blanks for step 3 ………
Steps for converting a decimal real number to a 
floating point representation
=============================================
1. Convert the number from decimal to binary
2. Change binary format to the mantissa and exponent format
3. ____________________________________________________________?
4. Normalise the mantissa and adjust the exponent so that the number
represents the same value

  Perform binary subtraction if any of the numbers are to be negative

  Perform two's complement conversion if numbers are to be negative

  Perform the addition of a sign bit if any number is to be positive

  Perform binary addition if any of the numbers are to be negative

 20. One way of representing real numbers is to use the two's complement standard for both the mantissa and the exponent. In this way, a negative number could be represented by using a ___________________

  positive mantissa

  positive exponent

  negative mantissa

  negative exponent

 21. When representing a real number using the two's complement standard for both mantissa and exponent, a number smaller than 1 could be represented by using a _____________

  positive exponent

  negative mantissa

  positive mantissa

  negative exponent

 22. In this example, the mantissa is positive and the exponent is negative.
Floatingpointrepresentation_quiz1_image3.png

  TRUE

  FALSE

 23. Convert the floating point 10.001 in binary into its equivalent decimal notation

  2.25

  10.125

  2.001

  2.125

 24. Assuming a 16 bit register with 10 bits for the mantissa and 6 bits for the exponent, convert this into its Decimal (base 10) equivalent.
floatingpoint_answer1.png

  Answer: 6.5

  Answer: 4.5

  Answer: -6.5

  Answer: 3.25

 25. Assuming a 16 bit register with 10 bits for the mantissa and 6 bits for the exponent, convert this into its Decimal (base 10) equivalent..
floatingpoint_answer2.png

  Answer: 2.625

  Answer: -3.5

  Answer: -4.625

  Answer: - 2.625

 26. Assuming a 16 bit register with 10 bits for the mantissa and 6 bits for the exponent, convert this into its Decimal (base 10) equivalent...
floatingpoint_answer3.png

  Answer: 0.1625

  Answer: 0.5

  Answer: 0.125

  Answer: -0.125

 27. Assuming a 16 bit register with 10 bits for the mantissa and 6 bits for the exponent, convert this into its Decimal (base 10) equivalent....
floatingpoint_answer4.png

  Answer: 1.5

  Answer: -2.5

  Answer: -1.5

  Answer: -3.5

 28. Assuming a 16 bit register with 10 bits for the mantissa and 6 bits for the exponent, convert this into its Decimal (base 10) equivalent…..
floatingpoint_answer5.png

  Answer: -0.5

  Answer: – 0.125

  Answer: 0.125

  Answer: -2.25

 29. If the exponent is in a format called 'excess -127', this means that the number should be worked out as a normal binary number but then 127 should be subtracted from it. 00000000 would therefore be:

  Answer: +127

  Answer: -0.00000027

  Answer: -127

  None of the above

 30. The standard always includes ways to represent +0, -0 +infinity and -infinity. This means if a number is worked out that is larger than the storage available, it can be stored as infinity.

  FALSE

  TRUE