# 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:

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

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?```

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 __________________ 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?

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 ________ 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. 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. 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.. 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... 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.... 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….. 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: