Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 110 0011 1000 - 1000 0110 0000 0010 0000 0001 1110 1111 1100 0001 0010 0011 1100 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 110 0011 1000 - 1000 0110 0000 0010 0000 0001 1110 1111 1100 0001 0010 0011 1100: 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:
110 0011 1000
The last 52 bits contain the mantissa:
1000 0110 0000 0010 0000 0001 1110 1111 1100 0001 0010 0011 1100
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
110 0011 1000(2) =
1 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
1,024 + 512 + 0 + 0 + 0 + 32 + 16 + 8 + 0 + 0 + 0 =
1,024 + 512 + 32 + 16 + 8 =
1,592(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 = 1,592 - 1023 = 569
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 0110 0000 0010 0000 0001 1110 1111 1100 0001 0010 0011 1100(2) =
1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 1 × 2-6 + 1 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 1 × 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 + 1 × 2-24 + 1 × 2-25 + 1 × 2-26 + 1 × 2-27 + 0 × 2-28 + 1 × 2-29 + 1 × 2-30 + 1 × 2-31 + 1 × 2-32 + 1 × 2-33 + 1 × 2-34 + 0 × 2-35 + 0 × 2-36 + 0 × 2-37 + 0 × 2-38 + 0 × 2-39 + 1 × 2-40 + 0 × 2-41 + 0 × 2-42 + 1 × 2-43 + 0 × 2-44 + 0 × 2-45 + 0 × 2-46 + 1 × 2-47 + 1 × 2-48 + 1 × 2-49 + 1 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0.5 + 0 + 0 + 0 + 0 + 0.015 625 + 0.007 812 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 030 517 578 125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 059 604 644 775 390 625 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 007 450 580 596 923 828 125 + 0 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 232 830 643 653 869 628 906 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 + 0 + 0 + 0 + 0 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0 + 0 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0 + 0 + 0 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0 =
0.5 + 0.015 625 + 0.007 812 5 + 0.000 030 517 578 125 + 0.000 000 059 604 644 775 390 625 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 232 830 643 653 869 628 906 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 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =
0.523 468 133 004 953 095 621 658 576 419 577 002 525 329 589 843 75(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)0 × (1 + 0.523 468 133 004 953 095 621 658 576 419 577 002 525 329 589 843 75) × 2569 =
1.523 468 133 004 953 095 621 658 576 419 577 002 525 329 589 843 75 × 2569 =
2 943 749 882 559 344 260 528 345 130 677 970 164 707 301 778 755 265 538 581 468 187 738 176 531 489 478 925 965 760 128 588 303 509 976 140 390 541 099 425 562 544 455 200 108 300 869 411 626 901 957 140 114 066 641 313 071 104
0 - 110 0011 1000 - 1000 0110 0000 0010 0000 0001 1110 1111 1100 0001 0010 0011 1100 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) = 2 943 749 882 559 344 260 528 345 130 677 970 164 707 301 778 755 265 538 581 468 187 738 176 531 489 478 925 965 760 128 588 303 509 976 140 390 541 099 425 562 544 455 200 108 300 869 411 626 901 957 140 114 066 641 313 071 104(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: