Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 111 1000 0000 - 1111 1111 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 111 1000 0000 - 1111 1111 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001: 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.
1
The next 11 bits contain the exponent:
111 1000 0000
The last 52 bits contain the mantissa:
1111 1111 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
111 1000 0000(2) =
1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
1,024 + 512 + 256 + 128 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =
1,024 + 512 + 256 + 128 =
1,920(10)
3. 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 = 1,920 - 1023 = 897
4. 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).
1111 1111 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001(2) =
1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 1 × 2-5 + 1 × 2-6 + 1 × 2-7 + 1 × 2-8 + 1 × 2-9 + 1 × 2-10 + 1 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 0 × 2-25 + 0 × 2-26 + 0 × 2-27 + 0 × 2-28 + 0 × 2-29 + 0 × 2-30 + 0 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 2-35 + 0 × 2-36 + 0 × 2-37 + 0 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 0 × 2-46 + 0 × 2-47 + 0 × 2-48 + 0 × 2-49 + 0 × 2-50 + 0 × 2-51 + 1 × 2-52 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.999 511 718 750 000 222 044 604 925 031 308 084 726 333 618 164 062 5(10)
5. Put all the numbers into expression to calculate the double precision floating point decimal value:
(-1)Sign × (1 + Mantissa) × 2(Adjusted exponent) =
(-1)1 × (1 + 0.999 511 718 750 000 222 044 604 925 031 308 084 726 333 618 164 062 5) × 2897 =
-1.999 511 718 750 000 222 044 604 925 031 308 084 726 333 618 164 062 5 × 2897 =
-2 112 662 211 914 599 046 927 119 082 991 741 076 925 513 726 966 930 854 580 377 047 898 772 270 178 289 847 609 324 869 800 166 834 712 712 892 644 523 563 568 117 018 717 517 872 157 116 268 006 325 705 023 248 625 472 121 963 809 513 883 874 798 680 420 785 392 021 691 189 463 627 283 934 796 583 674 081 166 098 369 490 978 698 918 266 484 807 971 110 912
1 - 111 1000 0000 - 1111 1111 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 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) = -2 112 662 211 914 599 046 927 119 082 991 741 076 925 513 726 966 930 854 580 377 047 898 772 270 178 289 847 609 324 869 800 166 834 712 712 892 644 523 563 568 117 018 717 517 872 157 116 268 006 325 705 023 248 625 472 121 963 809 513 883 874 798 680 420 785 392 021 691 189 463 627 283 934 796 583 674 081 166 098 369 490 978 698 918 266 484 807 971 110 912(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: