Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 000 0000 0000 - 0000 1000 0001 0000 0001 1000 0010 0000 0010 1000 0010 1110 0010 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 000 0000 0000 - 0000 1000 0001 0000 0001 1000 0010 0000 0010 1000 0010 1110 0010: 64 bit double precision IEEE 754 binary floating point standard representation number converted to decimal system (base ten)
1. Identify the elements that make up the binary representation of the number:
The first bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.
0
The next 11 bits contain the exponent:
000 0000 0000
The last 52 bits contain the mantissa:
0000 1000 0001 0000 0001 1000 0010 0000 0010 1000 0010 1110 0010
2. Reserved bitpattern.
We notice that all the bits that make up the exponent are on 0 (clear) and at least one bit of the mantissa is set on 1 (set).
This is one of the reserved bitpatterns of the special values of: Denormalized.
Denormalized numbers are too small to be correctly represented so they approximate to zero.
Depending on the sign bit, -0 and +0 are two distinct values though they both compare as equal (0).
3. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
000 0000 0000(2) =
0 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =
0(10)
4. Adjust the exponent.
Subtract the excess bits: 2(11 - 1) - 1 = 1023,
that is due to the 11 bit excess/bias notation.
The exponent, adjusted = 0 - 1023 = -1023
5. Convert the mantissa from binary (from base 2) to decimal (in base 10).
The mantissa represents the fractional part of the number (what comes after the whole part of the number, separated from it by a comma).
0000 1000 0001 0000 0001 1000 0010 0000 0010 1000 0010 1110 0010(2) =
0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 1 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 1 × 2-20 + 1 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 0 × 2-25 + 0 × 2-26 + 1 × 2-27 + 0 × 2-28 + 0 × 2-29 + 0 × 2-30 + 0 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 1 × 2-35 + 0 × 2-36 + 1 × 2-37 + 0 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 1 × 2-43 + 0 × 2-44 + 1 × 2-45 + 1 × 2-46 + 1 × 2-47 + 0 × 2-48 + 0 × 2-49 + 0 × 2-50 + 1 × 2-51 + 0 × 2-52 =
0 + 0 + 0 + 0 + 0.031 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 244 140 625 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0 + 0 + 0 + 0 + 0 + 0.000 000 007 450 580 596 923 828 125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0 + 0 + 0 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0 =
0.031 25 + 0.000 244 140 625 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 =
0.031 495 578 623 598 863 288 179 927 621 968 090 534 210 205 078 125(10)
6. Put all the numbers into expression to calculate the double precision floating point decimal value:
(-1)Sign × (1 + Mantissa) × 2(Adjusted exponent) =
(-1)0 × (1 + 0.031 495 578 623 598 863 288 179 927 621 968 090 534 210 205 078 125) × 2-1023 =
1.031 495 578 623 598 863 288 179 927 621 968 090 534 210 205 078 125 × 2-1023 =
0
0 - 000 0000 0000 - 0000 1000 0001 0000 0001 1000 0010 0000 0010 1000 0010 1110 0010 converted from a 64 bit double precision IEEE 754 binary floating point standard representation number to a decimal system number, written in base ten (double) = 0(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.
More operations with 64 bit double precision IEEE 754 binary floating point standard representation numbers: