Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 100 0001 1000 - 0010 0110 1111 1100 0110 1110 0000 0101 1111 1000 0001 0001 1110 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 100 0001 1000 - 0010 0110 1111 1100 0110 1110 0000 0101 1111 1000 0001 0001 1110: 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:
100 0001 1000
The last 52 bits contain the mantissa:
0010 0110 1111 1100 0110 1110 0000 0101 1111 1000 0001 0001 1110
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
100 0001 1000(2) =
1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
1,024 + 0 + 0 + 0 + 0 + 0 + 16 + 8 + 0 + 0 + 0 =
1,024 + 16 + 8 =
1,048(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 = 1,048 - 1023 = 25
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).
0010 0110 1111 1100 0110 1110 0000 0101 1111 1000 0001 0001 1110(2) =
0 × 2-1 + 0 × 2-2 + 1 × 2-3 + 0 × 2-4 + 0 × 2-5 + 1 × 2-6 + 1 × 2-7 + 0 × 2-8 + 1 × 2-9 + 1 × 2-10 + 1 × 2-11 + 1 × 2-12 + 1 × 2-13 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 1 × 2-18 + 1 × 2-19 + 0 × 2-20 + 1 × 2-21 + 1 × 2-22 + 1 × 2-23 + 0 × 2-24 + 0 × 2-25 + 0 × 2-26 + 0 × 2-27 + 0 × 2-28 + 0 × 2-29 + 1 × 2-30 + 0 × 2-31 + 1 × 2-32 + 1 × 2-33 + 1 × 2-34 + 1 × 2-35 + 1 × 2-36 + 1 × 2-37 + 0 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 1 × 2-44 + 0 × 2-45 + 0 × 2-46 + 0 × 2-47 + 1 × 2-48 + 1 × 2-49 + 1 × 2-50 + 1 × 2-51 + 0 × 2-52 =
0 + 0 + 0.125 + 0 + 0 + 0.015 625 + 0.007 812 5 + 0 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0 + 0 + 0 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 931 322 574 615 478 515 625 + 0 + 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 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0 + 0 + 0 + 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 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0 =
0.125 + 0.015 625 + 0.007 812 5 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 + 0.000 000 000 931 322 574 615 478 515 625 + 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 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 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 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 =
0.152 289 272 744 446 702 034 792 906 488 291 919 231 414 794 921 875(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 + 0.152 289 272 744 446 702 034 792 906 488 291 919 231 414 794 921 875) × 225 =
1.152 289 272 744 446 702 034 792 906 488 291 919 231 414 794 921 875 × 225 =
38 664 412.046 632 990 241 050 720 214 843 75
0 - 100 0001 1000 - 0010 0110 1111 1100 0110 1110 0000 0101 1111 1000 0001 0001 1110 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) = 38 664 412.046 632 990 241 050 720 214 843 75(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: