-0.000 000 000 000 000 004 573 63 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 004 573 63(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
-0.000 000 000 000 000 004 573 63(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. Start with the positive version of the number:

|-0.000 000 000 000 000 004 573 63| = 0.000 000 000 000 000 004 573 63


2. First, convert to binary (in base 2) the integer part: 0.
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;
  • 0 ÷ 2 = 0 + 0;

3. 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.

0(10) =

0(2)


4. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 004 573 63.

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) 0.000 000 000 000 000 004 573 63 × 2 = 0 + 0.000 000 000 000 000 009 147 26;
  • 2) 0.000 000 000 000 000 009 147 26 × 2 = 0 + 0.000 000 000 000 000 018 294 52;
  • 3) 0.000 000 000 000 000 018 294 52 × 2 = 0 + 0.000 000 000 000 000 036 589 04;
  • 4) 0.000 000 000 000 000 036 589 04 × 2 = 0 + 0.000 000 000 000 000 073 178 08;
  • 5) 0.000 000 000 000 000 073 178 08 × 2 = 0 + 0.000 000 000 000 000 146 356 16;
  • 6) 0.000 000 000 000 000 146 356 16 × 2 = 0 + 0.000 000 000 000 000 292 712 32;
  • 7) 0.000 000 000 000 000 292 712 32 × 2 = 0 + 0.000 000 000 000 000 585 424 64;
  • 8) 0.000 000 000 000 000 585 424 64 × 2 = 0 + 0.000 000 000 000 001 170 849 28;
  • 9) 0.000 000 000 000 001 170 849 28 × 2 = 0 + 0.000 000 000 000 002 341 698 56;
  • 10) 0.000 000 000 000 002 341 698 56 × 2 = 0 + 0.000 000 000 000 004 683 397 12;
  • 11) 0.000 000 000 000 004 683 397 12 × 2 = 0 + 0.000 000 000 000 009 366 794 24;
  • 12) 0.000 000 000 000 009 366 794 24 × 2 = 0 + 0.000 000 000 000 018 733 588 48;
  • 13) 0.000 000 000 000 018 733 588 48 × 2 = 0 + 0.000 000 000 000 037 467 176 96;
  • 14) 0.000 000 000 000 037 467 176 96 × 2 = 0 + 0.000 000 000 000 074 934 353 92;
  • 15) 0.000 000 000 000 074 934 353 92 × 2 = 0 + 0.000 000 000 000 149 868 707 84;
  • 16) 0.000 000 000 000 149 868 707 84 × 2 = 0 + 0.000 000 000 000 299 737 415 68;
  • 17) 0.000 000 000 000 299 737 415 68 × 2 = 0 + 0.000 000 000 000 599 474 831 36;
  • 18) 0.000 000 000 000 599 474 831 36 × 2 = 0 + 0.000 000 000 001 198 949 662 72;
  • 19) 0.000 000 000 001 198 949 662 72 × 2 = 0 + 0.000 000 000 002 397 899 325 44;
  • 20) 0.000 000 000 002 397 899 325 44 × 2 = 0 + 0.000 000 000 004 795 798 650 88;
  • 21) 0.000 000 000 004 795 798 650 88 × 2 = 0 + 0.000 000 000 009 591 597 301 76;
  • 22) 0.000 000 000 009 591 597 301 76 × 2 = 0 + 0.000 000 000 019 183 194 603 52;
  • 23) 0.000 000 000 019 183 194 603 52 × 2 = 0 + 0.000 000 000 038 366 389 207 04;
  • 24) 0.000 000 000 038 366 389 207 04 × 2 = 0 + 0.000 000 000 076 732 778 414 08;
  • 25) 0.000 000 000 076 732 778 414 08 × 2 = 0 + 0.000 000 000 153 465 556 828 16;
  • 26) 0.000 000 000 153 465 556 828 16 × 2 = 0 + 0.000 000 000 306 931 113 656 32;
  • 27) 0.000 000 000 306 931 113 656 32 × 2 = 0 + 0.000 000 000 613 862 227 312 64;
  • 28) 0.000 000 000 613 862 227 312 64 × 2 = 0 + 0.000 000 001 227 724 454 625 28;
  • 29) 0.000 000 001 227 724 454 625 28 × 2 = 0 + 0.000 000 002 455 448 909 250 56;
  • 30) 0.000 000 002 455 448 909 250 56 × 2 = 0 + 0.000 000 004 910 897 818 501 12;
  • 31) 0.000 000 004 910 897 818 501 12 × 2 = 0 + 0.000 000 009 821 795 637 002 24;
  • 32) 0.000 000 009 821 795 637 002 24 × 2 = 0 + 0.000 000 019 643 591 274 004 48;
  • 33) 0.000 000 019 643 591 274 004 48 × 2 = 0 + 0.000 000 039 287 182 548 008 96;
  • 34) 0.000 000 039 287 182 548 008 96 × 2 = 0 + 0.000 000 078 574 365 096 017 92;
  • 35) 0.000 000 078 574 365 096 017 92 × 2 = 0 + 0.000 000 157 148 730 192 035 84;
  • 36) 0.000 000 157 148 730 192 035 84 × 2 = 0 + 0.000 000 314 297 460 384 071 68;
  • 37) 0.000 000 314 297 460 384 071 68 × 2 = 0 + 0.000 000 628 594 920 768 143 36;
  • 38) 0.000 000 628 594 920 768 143 36 × 2 = 0 + 0.000 001 257 189 841 536 286 72;
  • 39) 0.000 001 257 189 841 536 286 72 × 2 = 0 + 0.000 002 514 379 683 072 573 44;
  • 40) 0.000 002 514 379 683 072 573 44 × 2 = 0 + 0.000 005 028 759 366 145 146 88;
  • 41) 0.000 005 028 759 366 145 146 88 × 2 = 0 + 0.000 010 057 518 732 290 293 76;
  • 42) 0.000 010 057 518 732 290 293 76 × 2 = 0 + 0.000 020 115 037 464 580 587 52;
  • 43) 0.000 020 115 037 464 580 587 52 × 2 = 0 + 0.000 040 230 074 929 161 175 04;
  • 44) 0.000 040 230 074 929 161 175 04 × 2 = 0 + 0.000 080 460 149 858 322 350 08;
  • 45) 0.000 080 460 149 858 322 350 08 × 2 = 0 + 0.000 160 920 299 716 644 700 16;
  • 46) 0.000 160 920 299 716 644 700 16 × 2 = 0 + 0.000 321 840 599 433 289 400 32;
  • 47) 0.000 321 840 599 433 289 400 32 × 2 = 0 + 0.000 643 681 198 866 578 800 64;
  • 48) 0.000 643 681 198 866 578 800 64 × 2 = 0 + 0.001 287 362 397 733 157 601 28;
  • 49) 0.001 287 362 397 733 157 601 28 × 2 = 0 + 0.002 574 724 795 466 315 202 56;
  • 50) 0.002 574 724 795 466 315 202 56 × 2 = 0 + 0.005 149 449 590 932 630 405 12;
  • 51) 0.005 149 449 590 932 630 405 12 × 2 = 0 + 0.010 298 899 181 865 260 810 24;
  • 52) 0.010 298 899 181 865 260 810 24 × 2 = 0 + 0.020 597 798 363 730 521 620 48;
  • 53) 0.020 597 798 363 730 521 620 48 × 2 = 0 + 0.041 195 596 727 461 043 240 96;
  • 54) 0.041 195 596 727 461 043 240 96 × 2 = 0 + 0.082 391 193 454 922 086 481 92;
  • 55) 0.082 391 193 454 922 086 481 92 × 2 = 0 + 0.164 782 386 909 844 172 963 84;
  • 56) 0.164 782 386 909 844 172 963 84 × 2 = 0 + 0.329 564 773 819 688 345 927 68;
  • 57) 0.329 564 773 819 688 345 927 68 × 2 = 0 + 0.659 129 547 639 376 691 855 36;
  • 58) 0.659 129 547 639 376 691 855 36 × 2 = 1 + 0.318 259 095 278 753 383 710 72;
  • 59) 0.318 259 095 278 753 383 710 72 × 2 = 0 + 0.636 518 190 557 506 767 421 44;
  • 60) 0.636 518 190 557 506 767 421 44 × 2 = 1 + 0.273 036 381 115 013 534 842 88;
  • 61) 0.273 036 381 115 013 534 842 88 × 2 = 0 + 0.546 072 762 230 027 069 685 76;
  • 62) 0.546 072 762 230 027 069 685 76 × 2 = 1 + 0.092 145 524 460 054 139 371 52;
  • 63) 0.092 145 524 460 054 139 371 52 × 2 = 0 + 0.184 291 048 920 108 278 743 04;
  • 64) 0.184 291 048 920 108 278 743 04 × 2 = 0 + 0.368 582 097 840 216 557 486 08;
  • 65) 0.368 582 097 840 216 557 486 08 × 2 = 0 + 0.737 164 195 680 433 114 972 16;
  • 66) 0.737 164 195 680 433 114 972 16 × 2 = 1 + 0.474 328 391 360 866 229 944 32;
  • 67) 0.474 328 391 360 866 229 944 32 × 2 = 0 + 0.948 656 782 721 732 459 888 64;
  • 68) 0.948 656 782 721 732 459 888 64 × 2 = 1 + 0.897 313 565 443 464 919 777 28;
  • 69) 0.897 313 565 443 464 919 777 28 × 2 = 1 + 0.794 627 130 886 929 839 554 56;
  • 70) 0.794 627 130 886 929 839 554 56 × 2 = 1 + 0.589 254 261 773 859 679 109 12;
  • 71) 0.589 254 261 773 859 679 109 12 × 2 = 1 + 0.178 508 523 547 719 358 218 24;
  • 72) 0.178 508 523 547 719 358 218 24 × 2 = 0 + 0.357 017 047 095 438 716 436 48;
  • 73) 0.357 017 047 095 438 716 436 48 × 2 = 0 + 0.714 034 094 190 877 432 872 96;
  • 74) 0.714 034 094 190 877 432 872 96 × 2 = 1 + 0.428 068 188 381 754 865 745 92;
  • 75) 0.428 068 188 381 754 865 745 92 × 2 = 0 + 0.856 136 376 763 509 731 491 84;
  • 76) 0.856 136 376 763 509 731 491 84 × 2 = 1 + 0.712 272 753 527 019 462 983 68;
  • 77) 0.712 272 753 527 019 462 983 68 × 2 = 1 + 0.424 545 507 054 038 925 967 36;
  • 78) 0.424 545 507 054 038 925 967 36 × 2 = 0 + 0.849 091 014 108 077 851 934 72;
  • 79) 0.849 091 014 108 077 851 934 72 × 2 = 1 + 0.698 182 028 216 155 703 869 44;
  • 80) 0.698 182 028 216 155 703 869 44 × 2 = 1 + 0.396 364 056 432 311 407 738 88;
  • 81) 0.396 364 056 432 311 407 738 88 × 2 = 0 + 0.792 728 112 864 622 815 477 76;
  • 82) 0.792 728 112 864 622 815 477 76 × 2 = 1 + 0.585 456 225 729 245 630 955 52;
  • 83) 0.585 456 225 729 245 630 955 52 × 2 = 1 + 0.170 912 451 458 491 261 911 04;
  • 84) 0.170 912 451 458 491 261 911 04 × 2 = 0 + 0.341 824 902 916 982 523 822 08;
  • 85) 0.341 824 902 916 982 523 822 08 × 2 = 0 + 0.683 649 805 833 965 047 644 16;
  • 86) 0.683 649 805 833 965 047 644 16 × 2 = 1 + 0.367 299 611 667 930 095 288 32;
  • 87) 0.367 299 611 667 930 095 288 32 × 2 = 0 + 0.734 599 223 335 860 190 576 64;
  • 88) 0.734 599 223 335 860 190 576 64 × 2 = 1 + 0.469 198 446 671 720 381 153 28;
  • 89) 0.469 198 446 671 720 381 153 28 × 2 = 0 + 0.938 396 893 343 440 762 306 56;
  • 90) 0.938 396 893 343 440 762 306 56 × 2 = 1 + 0.876 793 786 686 881 524 613 12;
  • 91) 0.876 793 786 686 881 524 613 12 × 2 = 1 + 0.753 587 573 373 763 049 226 24;
  • 92) 0.753 587 573 373 763 049 226 24 × 2 = 1 + 0.507 175 146 747 526 098 452 48;
  • 93) 0.507 175 146 747 526 098 452 48 × 2 = 1 + 0.014 350 293 495 052 196 904 96;
  • 94) 0.014 350 293 495 052 196 904 96 × 2 = 0 + 0.028 700 586 990 104 393 809 92;
  • 95) 0.028 700 586 990 104 393 809 92 × 2 = 0 + 0.057 401 173 980 208 787 619 84;
  • 96) 0.057 401 173 980 208 787 619 84 × 2 = 0 + 0.114 802 347 960 417 575 239 68;
  • 97) 0.114 802 347 960 417 575 239 68 × 2 = 0 + 0.229 604 695 920 835 150 479 36;
  • 98) 0.229 604 695 920 835 150 479 36 × 2 = 0 + 0.459 209 391 841 670 300 958 72;
  • 99) 0.459 209 391 841 670 300 958 72 × 2 = 0 + 0.918 418 783 683 340 601 917 44;
  • 100) 0.918 418 783 683 340 601 917 44 × 2 = 1 + 0.836 837 567 366 681 203 834 88;
  • 101) 0.836 837 567 366 681 203 834 88 × 2 = 1 + 0.673 675 134 733 362 407 669 76;
  • 102) 0.673 675 134 733 362 407 669 76 × 2 = 1 + 0.347 350 269 466 724 815 339 52;
  • 103) 0.347 350 269 466 724 815 339 52 × 2 = 0 + 0.694 700 538 933 449 630 679 04;
  • 104) 0.694 700 538 933 449 630 679 04 × 2 = 1 + 0.389 401 077 866 899 261 358 08;
  • 105) 0.389 401 077 866 899 261 358 08 × 2 = 0 + 0.778 802 155 733 798 522 716 16;
  • 106) 0.778 802 155 733 798 522 716 16 × 2 = 1 + 0.557 604 311 467 597 045 432 32;
  • 107) 0.557 604 311 467 597 045 432 32 × 2 = 1 + 0.115 208 622 935 194 090 864 64;
  • 108) 0.115 208 622 935 194 090 864 64 × 2 = 0 + 0.230 417 245 870 388 181 729 28;
  • 109) 0.230 417 245 870 388 181 729 28 × 2 = 0 + 0.460 834 491 740 776 363 458 56;
  • 110) 0.460 834 491 740 776 363 458 56 × 2 = 0 + 0.921 668 983 481 552 726 917 12;

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).


5. 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:


0.000 000 000 000 000 004 573 63(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1110 0101 1011 0110 0101 0111 1000 0001 1101 0110 00(2)

6. Positive number before normalization:

0.000 000 000 000 000 004 573 63(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1110 0101 1011 0110 0101 0111 1000 0001 1101 0110 00(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 58 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 004 573 63(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1110 0101 1011 0110 0101 0111 1000 0001 1101 0110 00(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1110 0101 1011 0110 0101 0111 1000 0001 1101 0110 00(2) × 20 =

1.0101 0001 0111 1001 0110 1101 1001 0101 1110 0000 0111 0101 1000(2) × 2-58


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): -58

Mantissa (not normalized):
1.0101 0001 0111 1001 0110 1101 1001 0101 1110 0000 0111 0101 1000


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-58 + 2(11-1) - 1 =

(-58 + 1 023)(10) =

965(10)


10. 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;
  • 965 ÷ 2 = 482 + 1;
  • 482 ÷ 2 = 241 + 0;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =

965(10) =

011 1100 0101(2)


12. 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, only if necessary (not the case here).


Mantissa (normalized) =

1. 0101 0001 0111 1001 0110 1101 1001 0101 1110 0000 0111 0101 1000 =

0101 0001 0111 1001 0110 1101 1001 0101 1110 0000 0111 0101 1000


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

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

Exponent (11 bits) =
011 1100 0101

Mantissa (52 bits) =
0101 0001 0111 1001 0110 1101 1001 0101 1110 0000 0111 0101 1000


Decimal number -0.000 000 000 000 000 004 573 63 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1100 0101 - 0101 0001 0111 1001 0110 1101 1001 0101 1110 0000 0111 0101 1000

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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