Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 010 0001 1000 - 1111 0100 0011 1000 1101 1010 1010 0000 0110 0000 0000 0000 1000 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 010 0001 1000 - 1111 0100 0011 1000 1101 1010 1010 0000 0110 0000 0000 0000 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:
010 0001 1000
The last 52 bits contain the mantissa:
1111 0100 0011 1000 1101 1010 1010 0000 0110 0000 0000 0000 1000
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
010 0001 1000(2) =
0 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
0 + 512 + 0 + 0 + 0 + 0 + 16 + 8 + 0 + 0 + 0 =
512 + 16 + 8 =
536(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 = 536 - 1023 = -487
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).
1111 0100 0011 1000 1101 1010 1010 0000 0110 0000 0000 0000 1000(2) =
1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 0 × 2-5 + 1 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 1 × 2-11 + 1 × 2-12 + 1 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 1 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 1 × 2-21 + 0 × 2-22 + 1 × 2-23 + 0 × 2-24 + 1 × 2-25 + 0 × 2-26 + 1 × 2-27 + 0 × 2-28 + 0 × 2-29 + 0 × 2-30 + 0 × 2-31 + 0 × 2-32 + 0 × 2-33 + 1 × 2-34 + 1 × 2-35 + 0 × 2-36 + 0 × 2-37 + 0 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 0 × 2-46 + 0 × 2-47 + 0 × 2-48 + 1 × 2-49 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0 + 0.015 625 + 0 + 0 + 0 + 0 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0 + 0 + 0 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0 + 0.000 000 119 209 289 550 781 25 + 0 + 0.000 000 029 802 322 387 695 312 5 + 0 + 0.000 000 007 450 580 596 923 828 125 + 0 + 0 + 0 + 0 + 0 + 0 + 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 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0 + 0 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0.015 625 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0.000 000 119 209 289 550 781 25 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 007 450 580 596 923 828 125 + 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 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 =
0.953 992 523 340 277 287 502 431 136 090 308 427 810 668 945 312 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 + 0.953 992 523 340 277 287 502 431 136 090 308 427 810 668 945 312 5) × 2-487 =
1.953 992 523 340 277 287 502 431 136 090 308 427 810 668 945 312 5 × 2-487 =
0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 004 890 069 248 867 174 909 091 468 481 550 199 098 634 382 576 342 990 970 713 022 212 346 673 306 879 677 380 149 470 239 614 710 521 622 033 813 005 222 721 761 678 014 243 155 292 474 784 181 438 597 413 090 940 6
0 - 010 0001 1000 - 1111 0100 0011 1000 1101 1010 1010 0000 0110 0000 0000 0000 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) = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 004 890 069 248 867 174 909 091 468 481 550 199 098 634 382 576 342 990 970 713 022 212 346 673 306 879 677 380 149 470 239 614 710 521 622 033 813 005 222 721 761 678 014 243 155 292 474 784 181 438 597 413 090 940 6(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: