consider the floating point representation, for each part provide ans as true or false...
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3 votes
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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.

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1 Answer

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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

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5 Comments

2-23 * 2-126  = 1.0 * 2-149 

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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.

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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}
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Ohh yess...got it ! Its like I counted all forgot myself :P thnks sir !
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