-0.000 000 000 000 000 000 008 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 000 008 2(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
-0.000 000 000 000 000 000 008 2(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:

|-0.000 000 000 000 000 000 008 2| = 0.000 000 000 000 000 000 008 2


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: 0.000 000 000 000 000 000 008 2.

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) 0.000 000 000 000 000 000 008 2 × 2 = 0 + 0.000 000 000 000 000 000 016 4;
  • 2) 0.000 000 000 000 000 000 016 4 × 2 = 0 + 0.000 000 000 000 000 000 032 8;
  • 3) 0.000 000 000 000 000 000 032 8 × 2 = 0 + 0.000 000 000 000 000 000 065 6;
  • 4) 0.000 000 000 000 000 000 065 6 × 2 = 0 + 0.000 000 000 000 000 000 131 2;
  • 5) 0.000 000 000 000 000 000 131 2 × 2 = 0 + 0.000 000 000 000 000 000 262 4;
  • 6) 0.000 000 000 000 000 000 262 4 × 2 = 0 + 0.000 000 000 000 000 000 524 8;
  • 7) 0.000 000 000 000 000 000 524 8 × 2 = 0 + 0.000 000 000 000 000 001 049 6;
  • 8) 0.000 000 000 000 000 001 049 6 × 2 = 0 + 0.000 000 000 000 000 002 099 2;
  • 9) 0.000 000 000 000 000 002 099 2 × 2 = 0 + 0.000 000 000 000 000 004 198 4;
  • 10) 0.000 000 000 000 000 004 198 4 × 2 = 0 + 0.000 000 000 000 000 008 396 8;
  • 11) 0.000 000 000 000 000 008 396 8 × 2 = 0 + 0.000 000 000 000 000 016 793 6;
  • 12) 0.000 000 000 000 000 016 793 6 × 2 = 0 + 0.000 000 000 000 000 033 587 2;
  • 13) 0.000 000 000 000 000 033 587 2 × 2 = 0 + 0.000 000 000 000 000 067 174 4;
  • 14) 0.000 000 000 000 000 067 174 4 × 2 = 0 + 0.000 000 000 000 000 134 348 8;
  • 15) 0.000 000 000 000 000 134 348 8 × 2 = 0 + 0.000 000 000 000 000 268 697 6;
  • 16) 0.000 000 000 000 000 268 697 6 × 2 = 0 + 0.000 000 000 000 000 537 395 2;
  • 17) 0.000 000 000 000 000 537 395 2 × 2 = 0 + 0.000 000 000 000 001 074 790 4;
  • 18) 0.000 000 000 000 001 074 790 4 × 2 = 0 + 0.000 000 000 000 002 149 580 8;
  • 19) 0.000 000 000 000 002 149 580 8 × 2 = 0 + 0.000 000 000 000 004 299 161 6;
  • 20) 0.000 000 000 000 004 299 161 6 × 2 = 0 + 0.000 000 000 000 008 598 323 2;
  • 21) 0.000 000 000 000 008 598 323 2 × 2 = 0 + 0.000 000 000 000 017 196 646 4;
  • 22) 0.000 000 000 000 017 196 646 4 × 2 = 0 + 0.000 000 000 000 034 393 292 8;
  • 23) 0.000 000 000 000 034 393 292 8 × 2 = 0 + 0.000 000 000 000 068 786 585 6;
  • 24) 0.000 000 000 000 068 786 585 6 × 2 = 0 + 0.000 000 000 000 137 573 171 2;
  • 25) 0.000 000 000 000 137 573 171 2 × 2 = 0 + 0.000 000 000 000 275 146 342 4;
  • 26) 0.000 000 000 000 275 146 342 4 × 2 = 0 + 0.000 000 000 000 550 292 684 8;
  • 27) 0.000 000 000 000 550 292 684 8 × 2 = 0 + 0.000 000 000 001 100 585 369 6;
  • 28) 0.000 000 000 001 100 585 369 6 × 2 = 0 + 0.000 000 000 002 201 170 739 2;
  • 29) 0.000 000 000 002 201 170 739 2 × 2 = 0 + 0.000 000 000 004 402 341 478 4;
  • 30) 0.000 000 000 004 402 341 478 4 × 2 = 0 + 0.000 000 000 008 804 682 956 8;
  • 31) 0.000 000 000 008 804 682 956 8 × 2 = 0 + 0.000 000 000 017 609 365 913 6;
  • 32) 0.000 000 000 017 609 365 913 6 × 2 = 0 + 0.000 000 000 035 218 731 827 2;
  • 33) 0.000 000 000 035 218 731 827 2 × 2 = 0 + 0.000 000 000 070 437 463 654 4;
  • 34) 0.000 000 000 070 437 463 654 4 × 2 = 0 + 0.000 000 000 140 874 927 308 8;
  • 35) 0.000 000 000 140 874 927 308 8 × 2 = 0 + 0.000 000 000 281 749 854 617 6;
  • 36) 0.000 000 000 281 749 854 617 6 × 2 = 0 + 0.000 000 000 563 499 709 235 2;
  • 37) 0.000 000 000 563 499 709 235 2 × 2 = 0 + 0.000 000 001 126 999 418 470 4;
  • 38) 0.000 000 001 126 999 418 470 4 × 2 = 0 + 0.000 000 002 253 998 836 940 8;
  • 39) 0.000 000 002 253 998 836 940 8 × 2 = 0 + 0.000 000 004 507 997 673 881 6;
  • 40) 0.000 000 004 507 997 673 881 6 × 2 = 0 + 0.000 000 009 015 995 347 763 2;
  • 41) 0.000 000 009 015 995 347 763 2 × 2 = 0 + 0.000 000 018 031 990 695 526 4;
  • 42) 0.000 000 018 031 990 695 526 4 × 2 = 0 + 0.000 000 036 063 981 391 052 8;
  • 43) 0.000 000 036 063 981 391 052 8 × 2 = 0 + 0.000 000 072 127 962 782 105 6;
  • 44) 0.000 000 072 127 962 782 105 6 × 2 = 0 + 0.000 000 144 255 925 564 211 2;
  • 45) 0.000 000 144 255 925 564 211 2 × 2 = 0 + 0.000 000 288 511 851 128 422 4;
  • 46) 0.000 000 288 511 851 128 422 4 × 2 = 0 + 0.000 000 577 023 702 256 844 8;
  • 47) 0.000 000 577 023 702 256 844 8 × 2 = 0 + 0.000 001 154 047 404 513 689 6;
  • 48) 0.000 001 154 047 404 513 689 6 × 2 = 0 + 0.000 002 308 094 809 027 379 2;
  • 49) 0.000 002 308 094 809 027 379 2 × 2 = 0 + 0.000 004 616 189 618 054 758 4;
  • 50) 0.000 004 616 189 618 054 758 4 × 2 = 0 + 0.000 009 232 379 236 109 516 8;
  • 51) 0.000 009 232 379 236 109 516 8 × 2 = 0 + 0.000 018 464 758 472 219 033 6;
  • 52) 0.000 018 464 758 472 219 033 6 × 2 = 0 + 0.000 036 929 516 944 438 067 2;
  • 53) 0.000 036 929 516 944 438 067 2 × 2 = 0 + 0.000 073 859 033 888 876 134 4;
  • 54) 0.000 073 859 033 888 876 134 4 × 2 = 0 + 0.000 147 718 067 777 752 268 8;
  • 55) 0.000 147 718 067 777 752 268 8 × 2 = 0 + 0.000 295 436 135 555 504 537 6;
  • 56) 0.000 295 436 135 555 504 537 6 × 2 = 0 + 0.000 590 872 271 111 009 075 2;
  • 57) 0.000 590 872 271 111 009 075 2 × 2 = 0 + 0.001 181 744 542 222 018 150 4;
  • 58) 0.001 181 744 542 222 018 150 4 × 2 = 0 + 0.002 363 489 084 444 036 300 8;
  • 59) 0.002 363 489 084 444 036 300 8 × 2 = 0 + 0.004 726 978 168 888 072 601 6;
  • 60) 0.004 726 978 168 888 072 601 6 × 2 = 0 + 0.009 453 956 337 776 145 203 2;
  • 61) 0.009 453 956 337 776 145 203 2 × 2 = 0 + 0.018 907 912 675 552 290 406 4;
  • 62) 0.018 907 912 675 552 290 406 4 × 2 = 0 + 0.037 815 825 351 104 580 812 8;
  • 63) 0.037 815 825 351 104 580 812 8 × 2 = 0 + 0.075 631 650 702 209 161 625 6;
  • 64) 0.075 631 650 702 209 161 625 6 × 2 = 0 + 0.151 263 301 404 418 323 251 2;
  • 65) 0.151 263 301 404 418 323 251 2 × 2 = 0 + 0.302 526 602 808 836 646 502 4;
  • 66) 0.302 526 602 808 836 646 502 4 × 2 = 0 + 0.605 053 205 617 673 293 004 8;
  • 67) 0.605 053 205 617 673 293 004 8 × 2 = 1 + 0.210 106 411 235 346 586 009 6;
  • 68) 0.210 106 411 235 346 586 009 6 × 2 = 0 + 0.420 212 822 470 693 172 019 2;
  • 69) 0.420 212 822 470 693 172 019 2 × 2 = 0 + 0.840 425 644 941 386 344 038 4;
  • 70) 0.840 425 644 941 386 344 038 4 × 2 = 1 + 0.680 851 289 882 772 688 076 8;
  • 71) 0.680 851 289 882 772 688 076 8 × 2 = 1 + 0.361 702 579 765 545 376 153 6;
  • 72) 0.361 702 579 765 545 376 153 6 × 2 = 0 + 0.723 405 159 531 090 752 307 2;
  • 73) 0.723 405 159 531 090 752 307 2 × 2 = 1 + 0.446 810 319 062 181 504 614 4;
  • 74) 0.446 810 319 062 181 504 614 4 × 2 = 0 + 0.893 620 638 124 363 009 228 8;
  • 75) 0.893 620 638 124 363 009 228 8 × 2 = 1 + 0.787 241 276 248 726 018 457 6;
  • 76) 0.787 241 276 248 726 018 457 6 × 2 = 1 + 0.574 482 552 497 452 036 915 2;
  • 77) 0.574 482 552 497 452 036 915 2 × 2 = 1 + 0.148 965 104 994 904 073 830 4;
  • 78) 0.148 965 104 994 904 073 830 4 × 2 = 0 + 0.297 930 209 989 808 147 660 8;
  • 79) 0.297 930 209 989 808 147 660 8 × 2 = 0 + 0.595 860 419 979 616 295 321 6;
  • 80) 0.595 860 419 979 616 295 321 6 × 2 = 1 + 0.191 720 839 959 232 590 643 2;
  • 81) 0.191 720 839 959 232 590 643 2 × 2 = 0 + 0.383 441 679 918 465 181 286 4;
  • 82) 0.383 441 679 918 465 181 286 4 × 2 = 0 + 0.766 883 359 836 930 362 572 8;
  • 83) 0.766 883 359 836 930 362 572 8 × 2 = 1 + 0.533 766 719 673 860 725 145 6;
  • 84) 0.533 766 719 673 860 725 145 6 × 2 = 1 + 0.067 533 439 347 721 450 291 2;
  • 85) 0.067 533 439 347 721 450 291 2 × 2 = 0 + 0.135 066 878 695 442 900 582 4;
  • 86) 0.135 066 878 695 442 900 582 4 × 2 = 0 + 0.270 133 757 390 885 801 164 8;
  • 87) 0.270 133 757 390 885 801 164 8 × 2 = 0 + 0.540 267 514 781 771 602 329 6;
  • 88) 0.540 267 514 781 771 602 329 6 × 2 = 1 + 0.080 535 029 563 543 204 659 2;
  • 89) 0.080 535 029 563 543 204 659 2 × 2 = 0 + 0.161 070 059 127 086 409 318 4;
  • 90) 0.161 070 059 127 086 409 318 4 × 2 = 0 + 0.322 140 118 254 172 818 636 8;
  • 91) 0.322 140 118 254 172 818 636 8 × 2 = 0 + 0.644 280 236 508 345 637 273 6;
  • 92) 0.644 280 236 508 345 637 273 6 × 2 = 1 + 0.288 560 473 016 691 274 547 2;
  • 93) 0.288 560 473 016 691 274 547 2 × 2 = 0 + 0.577 120 946 033 382 549 094 4;
  • 94) 0.577 120 946 033 382 549 094 4 × 2 = 1 + 0.154 241 892 066 765 098 188 8;
  • 95) 0.154 241 892 066 765 098 188 8 × 2 = 0 + 0.308 483 784 133 530 196 377 6;
  • 96) 0.308 483 784 133 530 196 377 6 × 2 = 0 + 0.616 967 568 267 060 392 755 2;
  • 97) 0.616 967 568 267 060 392 755 2 × 2 = 1 + 0.233 935 136 534 120 785 510 4;
  • 98) 0.233 935 136 534 120 785 510 4 × 2 = 0 + 0.467 870 273 068 241 571 020 8;
  • 99) 0.467 870 273 068 241 571 020 8 × 2 = 0 + 0.935 740 546 136 483 142 041 6;
  • 100) 0.935 740 546 136 483 142 041 6 × 2 = 1 + 0.871 481 092 272 966 284 083 2;
  • 101) 0.871 481 092 272 966 284 083 2 × 2 = 1 + 0.742 962 184 545 932 568 166 4;
  • 102) 0.742 962 184 545 932 568 166 4 × 2 = 1 + 0.485 924 369 091 865 136 332 8;
  • 103) 0.485 924 369 091 865 136 332 8 × 2 = 0 + 0.971 848 738 183 730 272 665 6;
  • 104) 0.971 848 738 183 730 272 665 6 × 2 = 1 + 0.943 697 476 367 460 545 331 2;
  • 105) 0.943 697 476 367 460 545 331 2 × 2 = 1 + 0.887 394 952 734 921 090 662 4;
  • 106) 0.887 394 952 734 921 090 662 4 × 2 = 1 + 0.774 789 905 469 842 181 324 8;
  • 107) 0.774 789 905 469 842 181 324 8 × 2 = 1 + 0.549 579 810 939 684 362 649 6;
  • 108) 0.549 579 810 939 684 362 649 6 × 2 = 1 + 0.099 159 621 879 368 725 299 2;
  • 109) 0.099 159 621 879 368 725 299 2 × 2 = 0 + 0.198 319 243 758 737 450 598 4;
  • 110) 0.198 319 243 758 737 450 598 4 × 2 = 0 + 0.396 638 487 517 474 901 196 8;
  • 111) 0.396 638 487 517 474 901 196 8 × 2 = 0 + 0.793 276 975 034 949 802 393 6;
  • 112) 0.793 276 975 034 949 802 393 6 × 2 = 1 + 0.586 553 950 069 899 604 787 2;
  • 113) 0.586 553 950 069 899 604 787 2 × 2 = 1 + 0.173 107 900 139 799 209 574 4;
  • 114) 0.173 107 900 139 799 209 574 4 × 2 = 0 + 0.346 215 800 279 598 419 148 8;
  • 115) 0.346 215 800 279 598 419 148 8 × 2 = 0 + 0.692 431 600 559 196 838 297 6;
  • 116) 0.692 431 600 559 196 838 297 6 × 2 = 1 + 0.384 863 201 118 393 676 595 2;
  • 117) 0.384 863 201 118 393 676 595 2 × 2 = 0 + 0.769 726 402 236 787 353 190 4;
  • 118) 0.769 726 402 236 787 353 190 4 × 2 = 1 + 0.539 452 804 473 574 706 380 8;
  • 119) 0.539 452 804 473 574 706 380 8 × 2 = 1 + 0.078 905 608 947 149 412 761 6;

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:


0.000 000 000 000 000 000 008 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0110 1011 1001 0011 0001 0001 0100 1001 1101 1111 0001 1001 011(2)

6. Positive number before normalization:

0.000 000 000 000 000 000 008 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0110 1011 1001 0011 0001 0001 0100 1001 1101 1111 0001 1001 011(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 67 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 008 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0110 1011 1001 0011 0001 0001 0100 1001 1101 1111 0001 1001 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0110 1011 1001 0011 0001 0001 0100 1001 1101 1111 0001 1001 011(2) × 20 =

1.0011 0101 1100 1001 1000 1000 1010 0100 1110 1111 1000 1100 1011(2) × 2-67


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): -67

Mantissa (not normalized):
1.0011 0101 1100 1001 1000 1000 1010 0100 1110 1111 1000 1100 1011


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-67 + 2(11-1) - 1 =

(-67 + 1 023)(10) =

956(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;
  • 956 ÷ 2 = 478 + 0;
  • 478 ÷ 2 = 239 + 0;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 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) =

956(10) =

011 1011 1100(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. 0011 0101 1100 1001 1000 1000 1010 0100 1110 1111 1000 1100 1011 =

0011 0101 1100 1001 1000 1000 1010 0100 1110 1111 1000 1100 1011


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 1011 1100

Mantissa (52 bits) =
0011 0101 1100 1001 1000 1000 1010 0100 1110 1111 1000 1100 1011


Decimal number -0.000 000 000 000 000 000 008 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1011 1100 - 0011 0101 1100 1001 1000 1000 1010 0100 1110 1111 1000 1100 1011

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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