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+3 votes

**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.

+3 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".

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

0

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

**Isnt it should be 2 ^{-23 }* 2^{-126 }??**

0

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

+1

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

**(2−2 ^{-}^{23 }**)

**(1−2 ^{-}^{23})**×2

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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