Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 100 0000 0110 - 0001 1000 0100 1110 0111 1001 0000 1001 1100 0110 0001 0101 0110 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 100 0000 0110 - 0001 1000 0100 1110 0111 1001 0000 1001 1100 0110 0001 0101 0110: 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 0000 0110
The last 52 bits contain the mantissa:
0001 1000 0100 1110 0111 1001 0000 1001 1100 0110 0001 0101 0110
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
100 0000 0110(2) =
1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
1,024 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 4 + 2 + 0 =
1,024 + 4 + 2 =
1,030(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 = 1,030 - 1023 = 7
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 1000 0100 1110 0111 1001 0000 1001 1100 0110 0001 0101 0110(2) =
0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 1 × 2-4 + 1 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 1 × 2-10 + 0 × 2-11 + 0 × 2-12 + 1 × 2-13 + 1 × 2-14 + 1 × 2-15 + 0 × 2-16 + 0 × 2-17 + 1 × 2-18 + 1 × 2-19 + 1 × 2-20 + 1 × 2-21 + 0 × 2-22 + 0 × 2-23 + 1 × 2-24 + 0 × 2-25 + 0 × 2-26 + 0 × 2-27 + 0 × 2-28 + 1 × 2-29 + 0 × 2-30 + 0 × 2-31 + 1 × 2-32 + 1 × 2-33 + 1 × 2-34 + 0 × 2-35 + 0 × 2-36 + 0 × 2-37 + 1 × 2-38 + 1 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 1 × 2-44 + 0 × 2-45 + 1 × 2-46 + 0 × 2-47 + 1 × 2-48 + 0 × 2-49 + 1 × 2-50 + 1 × 2-51 + 0 × 2-52 =
0 + 0 + 0 + 0.062 5 + 0.031 25 + 0 + 0 + 0 + 0 + 0.000 976 562 5 + 0 + 0 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0 + 0 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0 + 0 + 0.000 000 059 604 644 775 390 625 + 0 + 0 + 0 + 0 + 0.000 000 001 862 645 149 230 957 031 25 + 0 + 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 + 0 + 0 + 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 + 0 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0 + 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 =
0.062 5 + 0.031 25 + 0.000 976 562 5 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0.000 000 059 604 644 775 390 625 + 0.000 000 001 862 645 149 230 957 031 25 + 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 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 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 =
0.094 947 399 984 524 505 356 375 811 970 792 710 781 097 412 109 375(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 + 0.094 947 399 984 524 505 356 375 811 970 792 710 781 097 412 109 375) × 27 =
1.094 947 399 984 524 505 356 375 811 970 792 710 781 097 412 109 375 × 27 =
140.153 267 198 019 136 685 616 103 932 261 466 979 980 468 75
0 - 100 0000 0110 - 0001 1000 0100 1110 0111 1001 0000 1001 1100 0110 0001 0101 0110 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) = 140.153 267 198 019 136 685 616 103 932 261 466 979 980 468 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: