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  TRUE/FALSE

id sign biased expo                  mantissa 
1 0 00000001 00000000000000000000000 is first smallest no in IEEE 754 representationid
2 0 11111110 11111111111111111111111 is largest no represented through IEEE 754
3 0 00000000 00000000000000000000001 is smallest de-normalized no
4 0 00000000 11111111111111111111111 largest unnormalized no
5 0 11111111 11111111111111111111111111 largest no represented through IEEE 754 
6 0 00000000 10000000000000000000000000 smallest normalized no
7 0 00000000 10000000000000000000000001 2nd smallest normalized no
8 0 00000000 1000000000000000000000000 is first smallest no in IEEE 754 representationid
9 0 00000001 1000000000000000000000000 is 2nd smallest no in IEEE 754 representationid
10 0 00000001 000000000000000000000001 is 2nd smallest no in IEEE 754 representationid
11 0 11111111 11111111111111111111111111  largest no represented through IBM 370

For each id(row i.e exponent and mantissa) there is a corresponding red statement you hv to tell that it is true or false.

asked in CO & Architecture by Active (1.6k points)   | 605 views

2 Answers

+2 votes

All 0 exponent and non-zero mantissa means a number is denormalized in IEEE 754 - no implied 1 before ".". Also, bias for denormalized numbers is "-126" and that for normalized is "-127".

 

IEEE 754 Single precision
Exponent Sign Mantissa Value Remark
00000000 0 00...0 0 Special value 0
00000001 0 00...0 $1.0 \times 2^{-126}$ Smallest positive normalized number
00000000 0 00...001 $1.0 \times 2^{-149}$ Smallest positive denormalized number
11111111 0 100..0
...
111..1
QNAN When exponent is all 1's number is NAN. Mantissa here can be any non-zero value. If mantissa starts with 1, it is QNaN meaning the value is indeterminate.
11111111 0 000...1
0111...1
SNaN Same as above but mantissa bits starting with 0 and used to represent Signalling NaN which are used to represent invalid numbers. 
11111111 0 000...000 $+\infty$ Positive infinity
11111110 0 1111...1 $(2 -  2^{-23}) \times 2^{127}$ Largest normalized number
00000000 0 111...111 $\left(1 - 2^{-23}\right ) \times 2^{-126}$ Largest de-normalized number

 

Similarly, we can see for negative numbers. 

Ref: http://steve.hollasch.net/cgindex/coding/ieeefloat.html

answered by Veteran (285k points)  
well explained sir thank u

Sir i have doubt in smallest positive denormalized number 3rd row.. Is it correct?

Isnt it should be 2-23 * 2-126 ??

that is the only definition to represent denormalized no i.e biased exponent should be zero in any case and for mantissa there should atleast one 1. and in case of smallest no that 1 must be at the last position

2-23 * 2-126  = 1.0 * 2-149 

sir please correct the sign of power in the below parts.minus signs are missing.

(2−2-23 )×2127   Largest normalized number

(1−2-23)×2−126 Largest de-normalized number

Also please explain why there is a difference in the highlighted red part. For largest denormalized I understood its a geometric series of mantissa part but I didn't get why there is change in largest normalized part even if mantissa are same.

thanks. Corrected that. For normalized numbers in IEEE 754 representation there is an "implicit 1" before the decimal point. So, actual value will be (1.0 + mantissa) * 2^ {exp}
Ohh yess...got it ! Its like I counted all forgot myself :P thnks sir !
0 votes

1.T   2.T  3.T  4.T  5.F  6.T  7.T  8.F  9.F  10.T  11.T

answered by Active (1.6k points)  

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