Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 110 0001 0000 - 1101 1000 0011 0111 0000 0000 0000 0000 0000 0000 0000 0000 0000 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 110 0001 0000 - 1101 1000 0011 0111 0000 0000 0000 0000 0000 0000 0000 0000 0000: 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 0001 0000
The last 52 bits contain the mantissa:
1101 1000 0011 0111 0000 0000 0000 0000 0000 0000 0000 0000 0000
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
110 0001 0000(2) =
1 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
1,024 + 512 + 0 + 0 + 0 + 0 + 16 + 0 + 0 + 0 + 0 =
1,024 + 512 + 16 =
1,552(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 = 1,552 - 1023 = 529
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).
1101 1000 0011 0111 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) =
1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 1 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 1 × 2-11 + 1 × 2-12 + 0 × 2-13 + 1 × 2-14 + 1 × 2-15 + 1 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 0 × 2-25 + 0 × 2-26 + 0 × 2-27 + 0 × 2-28 + 0 × 2-29 + 0 × 2-30 + 0 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 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 + 0 × 2-49 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0.5 + 0.25 + 0 + 0.062 5 + 0.031 25 + 0 + 0 + 0 + 0 + 0 + 0.000 488 281 25 + 0.000 244 140 625 + 0 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =
0.5 + 0.25 + 0.062 5 + 0.031 25 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 =
0.844 589 233 398 437 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 + 0.844 589 233 398 437 5) × 2529 =
1.844 589 233 398 437 5 × 2529 =
3 241 659 354 453 941 471 152 410 319 916 220 241 625 196 191 907 905 316 505 736 344 494 385 776 607 001 746 769 296 356 565 004 909 326 329 762 481 179 478 459 115 491 262 975 570 002 229 049 054 796 976 226 304
0 - 110 0001 0000 - 1101 1000 0011 0111 0000 0000 0000 0000 0000 0000 0000 0000 0000 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) = 3 241 659 354 453 941 471 152 410 319 916 220 241 625 196 191 907 905 316 505 736 344 494 385 776 607 001 746 769 296 356 565 004 909 326 329 762 481 179 478 459 115 491 262 975 570 002 229 049 054 796 976 226 304(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: