Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 001 1011 1000 - 1111 0000 0110 0100 0010 0000 0000 0000 0000 0000 0001 0101 0000 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 001 1011 1000 - 1111 0000 0110 0100 0010 0000 0000 0000 0000 0000 0001 0101 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.
1
The next 11 bits contain the exponent:
001 1011 1000
The last 52 bits contain the mantissa:
1111 0000 0110 0100 0010 0000 0000 0000 0000 0000 0001 0101 0000
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
001 1011 1000(2) =
0 × 210 + 0 × 29 + 1 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
0 + 0 + 256 + 128 + 0 + 32 + 16 + 8 + 0 + 0 + 0 =
256 + 128 + 32 + 16 + 8 =
440(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 = 440 - 1023 = -583
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 0000 0110 0100 0010 0000 0000 0000 0000 0000 0001 0101 0000(2) =
1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 1 × 2-10 + 1 × 2-11 + 0 × 2-12 + 0 × 2-13 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 1 × 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 + 1 × 2-44 + 0 × 2-45 + 1 × 2-46 + 0 × 2-47 + 1 × 2-48 + 0 × 2-49 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0 + 0 + 0 + 0 + 0 + 0.000 976 562 5 + 0.000 488 281 25 + 0 + 0 + 0.000 061 035 156 25 + 0 + 0 + 0 + 0 + 0.000 001 907 348 632 812 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 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0 + 0 + 0 + 0 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 061 035 156 25 + 0.000 001 907 348 632 812 5 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 =
0.939 027 786 254 957 419 487 254 810 519 516 468 048 095 703 125(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 + 0.939 027 786 254 957 419 487 254 810 519 516 468 048 095 703 125) × 2-583 =
-1.939 027 786 254 957 419 487 254 810 519 516 468 048 095 703 125 × 2-583 =
-0
1 - 001 1011 1000 - 1111 0000 0110 0100 0010 0000 0000 0000 0000 0000 0001 0101 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) = -0(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: