Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 110 0111 1111 - 0000 0000 0000 0000 0000 0100 0000 0000 0000 0000 0000 0000 0000 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 110 0111 1111 - 0000 0000 0000 0000 0000 0100 0000 0000 0000 0000 0000 0000 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:
110 0111 1111
The last 52 bits contain the mantissa:
0000 0000 0000 0000 0000 0100 0000 0000 0000 0000 0000 0000 0000
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
110 0111 1111(2) =
1 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20 =
1,024 + 512 + 0 + 0 + 64 + 32 + 16 + 8 + 4 + 2 + 1 =
1,024 + 512 + 64 + 32 + 16 + 8 + 4 + 2 + 1 =
1,663(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 = 1,663 - 1023 = 640
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 0100 0000 0000 0000 0000 0000 0000 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 + 1 × 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 + 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 + 0.000 000 238 418 579 101 562 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 =
0.000 000 238 418 579 101 562 5 =
0.000 000 238 418 579 101 562 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)1 × (1 + 0.000 000 238 418 579 101 562 5) × 2640 =
-1.000 000 238 418 579 101 562 5 × 2640 =
-4 562 441 705 392 804 507 380 189 721 976 938 344 348 977 278 255 265 035 540 571 230 360 870 201 635 167 981 531 305 941 122 176 038 968 219 359 458 941 195 143 522 784 151 169 921 510 997 087 630 419 130 702 222 891 455 326 707 620 968 186 769 225 809 920
1 - 110 0111 1111 - 0000 0000 0000 0000 0000 0100 0000 0000 0000 0000 0000 0000 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) = -4 562 441 705 392 804 507 380 189 721 976 938 344 348 977 278 255 265 035 540 571 230 360 870 201 635 167 981 531 305 941 122 176 038 968 219 359 458 941 195 143 522 784 151 169 921 510 997 087 630 419 130 702 222 891 455 326 707 620 968 186 769 225 809 920(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: