24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 9(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
24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 9(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: 24.
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;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 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.

24(10) =

1 1000(2)


3. Convert to binary (base 2) the fractional part: 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 9.

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) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 9 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 559 8;
  • 2) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 559 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 119 6;
  • 3) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 119 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 239 2;
  • 4) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 239 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 478 4;
  • 5) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 478 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 956 8;
  • 6) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 956 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 913 6;
  • 7) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 913 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 827 2;
  • 8) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 827 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 654 4;
  • 9) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 654 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 223 308 8;
  • 10) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 223 308 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 446 617 6;
  • 11) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 446 617 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 893 235 2;
  • 12) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 893 235 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 786 470 4;
  • 13) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 786 470 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 572 940 8;
  • 14) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 572 940 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 145 881 6;
  • 15) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 145 881 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 291 763 2;
  • 16) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 291 763 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 583 526 4;
  • 17) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 583 526 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 889 167 052 8;
  • 18) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 889 167 052 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 778 334 105 6;
  • 19) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 778 334 105 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 556 668 211 2;
  • 20) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 556 668 211 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 113 336 422 4;
  • 21) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 113 336 422 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 226 672 844 8;
  • 22) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 226 672 844 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 453 345 689 6;
  • 23) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 453 345 689 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 906 691 379 2;
  • 24) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 906 691 379 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 813 382 758 4;
  • 25) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 813 382 758 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 626 765 516 8;
  • 26) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 626 765 516 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 253 531 033 6;
  • 27) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 253 531 033 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 507 062 067 2;
  • 28) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 507 062 067 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 445 014 124 134 4;
  • 29) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 445 014 124 134 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 890 028 248 268 8;
  • 30) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 890 028 248 268 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 780 056 496 537 6;
  • 31) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 780 056 496 537 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 560 112 993 075 2;
  • 32) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 560 112 993 075 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 120 225 986 150 4;
  • 33) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 120 225 986 150 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 240 451 972 300 8;
  • 34) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 240 451 972 300 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 480 903 944 601 6;
  • 35) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 480 903 944 601 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 961 807 889 203 2;
  • 36) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 961 807 889 203 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 923 615 778 406 4;
  • 37) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 923 615 778 406 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 847 231 556 812 8;
  • 38) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 847 231 556 812 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 694 463 113 625 6;
  • 39) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 694 463 113 625 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 223 388 926 227 251 2;
  • 40) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 223 388 926 227 251 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 446 777 852 454 502 4;
  • 41) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 446 777 852 454 502 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 893 555 704 909 004 8;
  • 42) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 893 555 704 909 004 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 787 111 409 818 009 6;
  • 43) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 787 111 409 818 009 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 574 222 819 636 019 2;
  • 44) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 574 222 819 636 019 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 148 445 639 272 038 4;
  • 45) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 148 445 639 272 038 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 296 891 278 544 076 8;
  • 46) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 296 891 278 544 076 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 593 782 557 088 153 6;
  • 47) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 593 782 557 088 153 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 889 187 565 114 176 307 2;
  • 48) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 889 187 565 114 176 307 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 778 375 130 228 352 614 4;
  • 49) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 778 375 130 228 352 614 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 556 750 260 456 705 228 8;
  • 50) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 556 750 260 456 705 228 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 113 500 520 913 410 457 6;
  • 51) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 113 500 520 913 410 457 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 227 001 041 826 820 915 2;
  • 52) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 227 001 041 826 820 915 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 454 002 083 653 641 830 4;
  • 53) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 454 002 083 653 641 830 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 908 004 167 307 283 660 8;

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:


0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 9(10) =

0.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2)

5. Positive number before normalization:

24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 9(10) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 4 positions to the left, so that only one non zero digit remains to the left of it:


24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 9(10) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2) × 20 =

1.1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2) × 24


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): 4

Mantissa (not normalized):
1.1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

4 + 2(11-1) - 1 =

(4 + 1 023)(10) =

1 027(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;
  • 1 027 ÷ 2 = 513 + 1;
  • 513 ÷ 2 = 256 + 1;
  • 256 ÷ 2 = 128 + 0;
  • 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) =

1027(10) =

100 0000 0011(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. 1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1 1000 =

1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001


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 0011

Mantissa (52 bits) =
1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001


Decimal number 24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 779 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0011 - 1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001

» Convert decimal 24.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 782 9 to 64 bit double precision IEEE 754 binary floating point representation standard

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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