Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 100 0001 1110 - 1101 1100 1101 0110 0101 0000 0000 0000 0000 0000 0000 0000 0010 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 100 0001 1110 - 1101 1100 1101 0110 0101 0000 0000 0000 0000 0000 0000 0000 0010: 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.
1
The next 11 bits contain the exponent:
100 0001 1110
The last 52 bits contain the mantissa:
1101 1100 1101 0110 0101 0000 0000 0000 0000 0000 0000 0000 0010
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
100 0001 1110(2) =
1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
1,024 + 0 + 0 + 0 + 0 + 0 + 16 + 8 + 4 + 2 + 0 =
1,024 + 16 + 8 + 4 + 2 =
1,054(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 = 1,054 - 1023 = 31
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 1100 1101 0110 0101 0000 0000 0000 0000 0000 0000 0000 0010(2) =
1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 1 × 2-5 + 1 × 2-6 + 0 × 2-7 + 0 × 2-8 + 1 × 2-9 + 1 × 2-10 + 0 × 2-11 + 1 × 2-12 + 0 × 2-13 + 1 × 2-14 + 1 × 2-15 + 0 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 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 + 1 × 2-51 + 0 × 2-52 =
0.5 + 0.25 + 0 + 0.062 5 + 0.031 25 + 0.015 625 + 0 + 0 + 0.001 953 125 + 0.000 976 562 5 + 0 + 0.000 244 140 625 + 0 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0 + 0 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 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 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0 =
0.5 + 0.25 + 0.062 5 + 0.031 25 + 0.015 625 + 0.001 953 125 + 0.000 976 562 5 + 0.000 244 140 625 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 =
0.862 645 149 230 957 475 339 209 850 062 616 169 452 667 236 328 125(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)1 × (1 + 0.862 645 149 230 957 475 339 209 850 062 616 169 452 667 236 328 125) × 231 =
-1.862 645 149 230 957 475 339 209 850 062 616 169 452 667 236 328 125 × 231 =
-4 000 000 000.000 000 953 674 316 406 25
1 - 100 0001 1110 - 1101 1100 1101 0110 0101 0000 0000 0000 0000 0000 0000 0000 0010 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) = -4 000 000 000.000 000 953 674 316 406 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: