Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 100 0000 0001 - 0000 0100 0001 0011 0100 0000 0000 0010 0001 0000 0000 0000 0001 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 100 0000 0001 - 0000 0100 0001 0011 0100 0000 0000 0010 0001 0000 0000 0000 0001: 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 0001
The last 52 bits contain the mantissa:
0000 0100 0001 0011 0100 0000 0000 0010 0001 0000 0000 0000 0001
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
100 0000 0001(2) =
1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 1 × 20 =
1,024 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 1 =
1,024 + 1 =
1,025(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 = 1,025 - 1023 = 2
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).
0000 0100 0001 0011 0100 0000 0000 0010 0001 0000 0000 0000 0001(2) =
0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 1 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 1 × 2-12 + 0 × 2-13 + 0 × 2-14 + 1 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 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 + 1 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 2-35 + 1 × 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 + 1 × 2-52 =
0 + 0 + 0 + 0 + 0 + 0.015 625 + 0 + 0 + 0 + 0 + 0 + 0.000 244 140 625 + 0 + 0 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0 + 0 + 0 + 0 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.015 625 + 0.000 244 140 625 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.015 918 732 169 666 549 580 711 034 650 448 709 726 333 618 164 062 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 + 0.015 918 732 169 666 549 580 711 034 650 448 709 726 333 618 164 062 5) × 22 =
1.015 918 732 169 666 549 580 711 034 650 448 709 726 333 618 164 062 5 × 22 =
4.063 674 928 678 666 198 322 844 138 601 794 838 905 334 472 656 25
0 - 100 0000 0001 - 0000 0100 0001 0011 0100 0000 0000 0010 0001 0000 0000 0000 0001 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) = 4.063 674 928 678 666 198 322 844 138 601 794 838 905 334 472 656 25(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: