Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 101 0000 0001 - 1101 0110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 101 0000 0001 - 1101 0110 0000 0000 0000 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:
101 0000 0001
The last 52 bits contain the mantissa:
1101 0110 0000 0000 0000 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.
101 0000 0001(2) =
1 × 210 + 0 × 29 + 1 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 1 × 20 =
1,024 + 0 + 256 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 1 =
1,024 + 256 + 1 =
1,281(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 = 1,281 - 1023 = 258
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 0110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010(2) =
1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 1 × 2-6 + 1 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 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 + 1 × 2-51 + 0 × 2-52 =
0.5 + 0.25 + 0 + 0.062 5 + 0 + 0.015 625 + 0.007 812 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 + 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 + 0.015 625 + 0.007 812 5 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 =
0.835 937 500 000 000 444 089 209 850 062 616 169 452 667 236 328 125(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 + 0.835 937 500 000 000 444 089 209 850 062 616 169 452 667 236 328 125) × 2258 =
-1.835 937 500 000 000 444 089 209 850 062 616 169 452 667 236 328 125 × 2258 =
-850 348 155 336 541 015 829 919 086 308 307 092 668 599 269 557 111 015 006 749 836 755 594 964 369 408
1 - 101 0000 0001 - 1101 0110 0000 0000 0000 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) = -850 348 155 336 541 015 829 919 086 308 307 092 668 599 269 557 111 015 006 749 836 755 594 964 369 408(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: