266.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 682 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 266.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 682(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
266.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 682(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 266.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 266 ÷ 2 = 133 + 0;
  • 133 ÷ 2 = 66 + 1;
  • 66 ÷ 2 = 33 + 0;
  • 33 ÷ 2 = 16 + 1;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

266(10) =

1 0000 1010(2)


3. Convert to binary (base 2) the fractional part: 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 682.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 682 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 335 364;
  • 2) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 335 364 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 670 728;
  • 3) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 670 728 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 341 456;
  • 4) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 341 456 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 682 912;
  • 5) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 682 912 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 365 824;
  • 6) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 365 824 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 731 648;
  • 7) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 731 648 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 463 296;
  • 8) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 463 296 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 926 592;
  • 9) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 926 592 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 853 184;
  • 10) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 853 184 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 706 368;
  • 11) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 706 368 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 335 412 736;
  • 12) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 335 412 736 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 670 825 472;
  • 13) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 670 825 472 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 341 650 944;
  • 14) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 341 650 944 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 683 301 888;
  • 15) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 683 301 888 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 366 603 776;
  • 16) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 366 603 776 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 733 207 552;
  • 17) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 733 207 552 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 466 415 104;
  • 18) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 466 415 104 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 932 830 208;
  • 19) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 932 830 208 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 865 660 416;
  • 20) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 865 660 416 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 731 320 832;
  • 21) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 731 320 832 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 335 462 641 664;
  • 22) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 335 462 641 664 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 670 925 283 328;
  • 23) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 670 925 283 328 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 341 850 566 656;
  • 24) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 341 850 566 656 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 683 701 133 312;
  • 25) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 683 701 133 312 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 367 402 266 624;
  • 26) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 367 402 266 624 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 734 804 533 248;
  • 27) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 734 804 533 248 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 469 609 066 496;
  • 28) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 469 609 066 496 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 939 218 132 992;
  • 29) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 939 218 132 992 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 878 436 265 984;
  • 30) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 878 436 265 984 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 756 872 531 968;
  • 31) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 756 872 531 968 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 335 513 745 063 936;
  • 32) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 335 513 745 063 936 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 671 027 490 127 872;
  • 33) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 671 027 490 127 872 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 342 054 980 255 744;
  • 34) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 342 054 980 255 744 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 684 109 960 511 488;
  • 35) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 684 109 960 511 488 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 368 219 921 022 976;
  • 36) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 368 219 921 022 976 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 736 439 842 045 952;
  • 37) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 736 439 842 045 952 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 472 879 684 091 904;
  • 38) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 472 879 684 091 904 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 945 759 368 183 808;
  • 39) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 945 759 368 183 808 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 891 518 736 367 616;
  • 40) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 891 518 736 367 616 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 667 783 037 472 735 232;
  • 41) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 667 783 037 472 735 232 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 335 566 074 945 470 464;
  • 42) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 335 566 074 945 470 464 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 671 132 149 890 940 928;
  • 43) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 671 132 149 890 940 928 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 342 264 299 781 881 856;
  • 44) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 342 264 299 781 881 856 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 684 528 599 563 763 712;
  • 45) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 684 528 599 563 763 712 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 369 057 199 127 527 424;
  • 46) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 369 057 199 127 527 424 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 738 114 398 255 054 848;
  • 47) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 738 114 398 255 054 848 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 476 228 796 510 109 696;
  • 48) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 476 228 796 510 109 696 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 952 457 593 020 219 392;
  • 49) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 952 457 593 020 219 392 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 904 915 186 040 438 784;
  • 50) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 904 915 186 040 438 784 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 667 809 830 372 080 877 568;
  • 51) 0.666 666 666 666 666 666 666 666 666 666 666 666 667 809 830 372 080 877 568 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 335 619 660 744 161 755 136;
  • 52) 0.333 333 333 333 333 333 333 333 333 333 333 333 335 619 660 744 161 755 136 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 671 239 321 488 323 510 272;
  • 53) 0.666 666 666 666 666 666 666 666 666 666 666 666 671 239 321 488 323 510 272 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 342 478 642 976 647 020 544;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 682(10) =

0.1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1(2)

5. Positive number before normalization:

266.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 682(10) =

1 0000 1010.1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 8 positions to the left, so that only one non zero digit remains to the left of it:


266.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 682(10) =

1 0000 1010.1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1(2) =

1 0000 1010.1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1(2) × 20 =

1.0000 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1(2) × 28


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): 8

Mantissa (not normalized):
1.0000 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

8 + 2(11-1) - 1 =

(8 + 1 023)(10) =

1 031(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 1 031 ÷ 2 = 515 + 1;
  • 515 ÷ 2 = 257 + 1;
  • 257 ÷ 2 = 128 + 1;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1031(10) =

100 0000 0111(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 0000 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1 0101 0101 =

0000 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
100 0000 0111

Mantissa (52 bits) =
0000 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010


Decimal number 266.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 667 682 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0111 - 0000 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100