Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 101 0011 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 0000 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 101 0011 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 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:
101 0011 0000
The last 52 bits contain the mantissa:
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 0000
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
101 0011 0000(2) =
1 × 210 + 0 × 29 + 1 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
1,024 + 0 + 256 + 0 + 0 + 32 + 16 + 0 + 0 + 0 + 0 =
1,024 + 256 + 32 + 16 =
1,328(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 = 1,328 - 1023 = 305
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 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 0000(2) =
0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 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 + 1 × 2-46 + 1 × 2-47 + 0 × 2-48 + 0 × 2-49 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =
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 + 0 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0 + 0 + 0 + 0 + 0 =
0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 =
0.000 000 000 000 021 316 282 072 803 005 576 133 728 027 343 75(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 + 0.000 000 000 000 021 316 282 072 803 005 576 133 728 027 343 75) × 2305 =
-1.000 000 000 000 021 316 282 072 803 005 576 133 728 027 343 75 × 2305 =
-65 185 151 242 704 944 265 661 109 823 445 184 005 467 092 852 204 199 260 172 482 066 108 675 083 434 869 555 424 395 264
1 - 101 0011 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 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) = -65 185 151 242 704 944 265 661 109 823 445 184 005 467 092 852 204 199 260 172 482 066 108 675 083 434 869 555 424 395 264(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: