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

Question

 

  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.

Answer 1

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

Comments

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 !