Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 100 0001 1000 - 0001 1011 1101 1100 0000 1001 1100 1010 0100 1110 0100 0111 1000 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 100 0001 1000 - 0001 1011 1101 1100 0000 1001 1100 1010 0100 1110 0100 0111 1000: 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:
0001 1011 1101 1100 0000 1001 1100 1010 0100 1110 0100 0111 1000
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).
0001 1011 1101 1100 0000 1001 1100 1010 0100 1110 0100 0111 1000(2) =
0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 1 × 2-4 + 1 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 2-8 + 1 × 2-9 + 1 × 2-10 + 0 × 2-11 + 1 × 2-12 + 1 × 2-13 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 0 × 2-20 + 1 × 2-21 + 0 × 2-22 + 0 × 2-23 + 1 × 2-24 + 1 × 2-25 + 1 × 2-26 + 0 × 2-27 + 0 × 2-28 + 1 × 2-29 + 0 × 2-30 + 1 × 2-31 + 0 × 2-32 + 0 × 2-33 + 1 × 2-34 + 0 × 2-35 + 0 × 2-36 + 1 × 2-37 + 1 × 2-38 + 1 × 2-39 + 0 × 2-40 + 0 × 2-41 + 1 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 1 × 2-46 + 1 × 2-47 + 1 × 2-48 + 1 × 2-49 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0 + 0 + 0 + 0.062 5 + 0.031 25 + 0 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 476 837 158 203 125 + 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 + 0 + 0.000 000 001 862 645 149 230 957 031 25 + 0 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0 + 0 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0 + 0 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0 + 0 + 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 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 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 + 0 + 0 =
0.062 5 + 0.031 25 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 000 476 837 158 203 125 + 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 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 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 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 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 =
0.108 826 267 138 537 602 363 612 677 436 321 973 800 659 179 687 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)0 × (1 + 0.108 826 267 138 537 602 363 612 677 436 321 973 800 659 179 687 5) × 225 =
1.108 826 267 138 537 602 363 612 677 436 321 973 800 659 179 687 5 × 225 =
37 206 035.580 513 894 557 952 880 859 375
0 - 100 0001 1000 - 0001 1011 1101 1100 0000 1001 1100 1010 0100 1110 0100 0111 1000 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) = 37 206 035.580 513 894 557 952 880 859 375(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: