Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 100 1101 0110 - 1000 1100 0000 0100 0011 0000 0100 0000 1110 0010 1100 0100 0100 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 100 1101 0110 - 1000 1100 0000 0100 0011 0000 0100 0000 1110 0010 1100 0100 0100: 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:
100 1101 0110
The last 52 bits contain the mantissa:
1000 1100 0000 0100 0011 0000 0100 0000 1110 0010 1100 0100 0100
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
100 1101 0110(2) =
1 × 210 + 0 × 29 + 0 × 28 + 1 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
1,024 + 0 + 0 + 128 + 64 + 0 + 16 + 0 + 4 + 2 + 0 =
1,024 + 128 + 64 + 16 + 4 + 2 =
1,238(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 = 1,238 - 1023 = 215
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).
1000 1100 0000 0100 0011 0000 0100 0000 1110 0010 1100 0100 0100(2) =
1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 1 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 1 × 2-19 + 1 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 0 × 2-25 + 1 × 2-26 + 0 × 2-27 + 0 × 2-28 + 0 × 2-29 + 0 × 2-30 + 0 × 2-31 + 0 × 2-32 + 1 × 2-33 + 1 × 2-34 + 1 × 2-35 + 0 × 2-36 + 0 × 2-37 + 0 × 2-38 + 1 × 2-39 + 0 × 2-40 + 1 × 2-41 + 1 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 1 × 2-46 + 0 × 2-47 + 0 × 2-48 + 0 × 2-49 + 1 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0.5 + 0 + 0 + 0 + 0.031 25 + 0.015 625 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 061 035 156 25 + 0 + 0 + 0 + 0 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0 + 0 + 0 + 0 + 0 + 0.000 000 014 901 161 193 847 656 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 116 415 321 826 934 814 453 125 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0 + 0 + 0 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0 + 0 + 0 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0 + 0 + 0 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0 =
0.5 + 0.031 25 + 0.015 625 + 0.000 061 035 156 25 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 000 116 415 321 826 934 814 453 125 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =
0.546 938 911 286 603 435 257 802 630 076 184 868 812 561 035 156 25(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 + 0.546 938 911 286 603 435 257 802 630 076 184 868 812 561 035 156 25) × 2215 =
-1.546 938 911 286 603 435 257 802 630 076 184 868 812 561 035 156 25 × 2215 =
-81 455 840 909 427 545 949 390 578 342 654 469 649 049 649 862 935 968 592 160 620 544
1 - 100 1101 0110 - 1000 1100 0000 0100 0011 0000 0100 0000 1110 0010 1100 0100 0100 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) = -81 455 840 909 427 545 949 390 578 342 654 469 649 049 649 862 935 968 592 160 620 544(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: