0.000 000 000 000 000 000 008 534 88 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 534 88(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
0.000 000 000 000 000 000 008 534 88(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: 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;

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.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 000 008 534 88.

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) 0.000 000 000 000 000 000 008 534 88 × 2 = 0 + 0.000 000 000 000 000 000 017 069 76;
  • 2) 0.000 000 000 000 000 000 017 069 76 × 2 = 0 + 0.000 000 000 000 000 000 034 139 52;
  • 3) 0.000 000 000 000 000 000 034 139 52 × 2 = 0 + 0.000 000 000 000 000 000 068 279 04;
  • 4) 0.000 000 000 000 000 000 068 279 04 × 2 = 0 + 0.000 000 000 000 000 000 136 558 08;
  • 5) 0.000 000 000 000 000 000 136 558 08 × 2 = 0 + 0.000 000 000 000 000 000 273 116 16;
  • 6) 0.000 000 000 000 000 000 273 116 16 × 2 = 0 + 0.000 000 000 000 000 000 546 232 32;
  • 7) 0.000 000 000 000 000 000 546 232 32 × 2 = 0 + 0.000 000 000 000 000 001 092 464 64;
  • 8) 0.000 000 000 000 000 001 092 464 64 × 2 = 0 + 0.000 000 000 000 000 002 184 929 28;
  • 9) 0.000 000 000 000 000 002 184 929 28 × 2 = 0 + 0.000 000 000 000 000 004 369 858 56;
  • 10) 0.000 000 000 000 000 004 369 858 56 × 2 = 0 + 0.000 000 000 000 000 008 739 717 12;
  • 11) 0.000 000 000 000 000 008 739 717 12 × 2 = 0 + 0.000 000 000 000 000 017 479 434 24;
  • 12) 0.000 000 000 000 000 017 479 434 24 × 2 = 0 + 0.000 000 000 000 000 034 958 868 48;
  • 13) 0.000 000 000 000 000 034 958 868 48 × 2 = 0 + 0.000 000 000 000 000 069 917 736 96;
  • 14) 0.000 000 000 000 000 069 917 736 96 × 2 = 0 + 0.000 000 000 000 000 139 835 473 92;
  • 15) 0.000 000 000 000 000 139 835 473 92 × 2 = 0 + 0.000 000 000 000 000 279 670 947 84;
  • 16) 0.000 000 000 000 000 279 670 947 84 × 2 = 0 + 0.000 000 000 000 000 559 341 895 68;
  • 17) 0.000 000 000 000 000 559 341 895 68 × 2 = 0 + 0.000 000 000 000 001 118 683 791 36;
  • 18) 0.000 000 000 000 001 118 683 791 36 × 2 = 0 + 0.000 000 000 000 002 237 367 582 72;
  • 19) 0.000 000 000 000 002 237 367 582 72 × 2 = 0 + 0.000 000 000 000 004 474 735 165 44;
  • 20) 0.000 000 000 000 004 474 735 165 44 × 2 = 0 + 0.000 000 000 000 008 949 470 330 88;
  • 21) 0.000 000 000 000 008 949 470 330 88 × 2 = 0 + 0.000 000 000 000 017 898 940 661 76;
  • 22) 0.000 000 000 000 017 898 940 661 76 × 2 = 0 + 0.000 000 000 000 035 797 881 323 52;
  • 23) 0.000 000 000 000 035 797 881 323 52 × 2 = 0 + 0.000 000 000 000 071 595 762 647 04;
  • 24) 0.000 000 000 000 071 595 762 647 04 × 2 = 0 + 0.000 000 000 000 143 191 525 294 08;
  • 25) 0.000 000 000 000 143 191 525 294 08 × 2 = 0 + 0.000 000 000 000 286 383 050 588 16;
  • 26) 0.000 000 000 000 286 383 050 588 16 × 2 = 0 + 0.000 000 000 000 572 766 101 176 32;
  • 27) 0.000 000 000 000 572 766 101 176 32 × 2 = 0 + 0.000 000 000 001 145 532 202 352 64;
  • 28) 0.000 000 000 001 145 532 202 352 64 × 2 = 0 + 0.000 000 000 002 291 064 404 705 28;
  • 29) 0.000 000 000 002 291 064 404 705 28 × 2 = 0 + 0.000 000 000 004 582 128 809 410 56;
  • 30) 0.000 000 000 004 582 128 809 410 56 × 2 = 0 + 0.000 000 000 009 164 257 618 821 12;
  • 31) 0.000 000 000 009 164 257 618 821 12 × 2 = 0 + 0.000 000 000 018 328 515 237 642 24;
  • 32) 0.000 000 000 018 328 515 237 642 24 × 2 = 0 + 0.000 000 000 036 657 030 475 284 48;
  • 33) 0.000 000 000 036 657 030 475 284 48 × 2 = 0 + 0.000 000 000 073 314 060 950 568 96;
  • 34) 0.000 000 000 073 314 060 950 568 96 × 2 = 0 + 0.000 000 000 146 628 121 901 137 92;
  • 35) 0.000 000 000 146 628 121 901 137 92 × 2 = 0 + 0.000 000 000 293 256 243 802 275 84;
  • 36) 0.000 000 000 293 256 243 802 275 84 × 2 = 0 + 0.000 000 000 586 512 487 604 551 68;
  • 37) 0.000 000 000 586 512 487 604 551 68 × 2 = 0 + 0.000 000 001 173 024 975 209 103 36;
  • 38) 0.000 000 001 173 024 975 209 103 36 × 2 = 0 + 0.000 000 002 346 049 950 418 206 72;
  • 39) 0.000 000 002 346 049 950 418 206 72 × 2 = 0 + 0.000 000 004 692 099 900 836 413 44;
  • 40) 0.000 000 004 692 099 900 836 413 44 × 2 = 0 + 0.000 000 009 384 199 801 672 826 88;
  • 41) 0.000 000 009 384 199 801 672 826 88 × 2 = 0 + 0.000 000 018 768 399 603 345 653 76;
  • 42) 0.000 000 018 768 399 603 345 653 76 × 2 = 0 + 0.000 000 037 536 799 206 691 307 52;
  • 43) 0.000 000 037 536 799 206 691 307 52 × 2 = 0 + 0.000 000 075 073 598 413 382 615 04;
  • 44) 0.000 000 075 073 598 413 382 615 04 × 2 = 0 + 0.000 000 150 147 196 826 765 230 08;
  • 45) 0.000 000 150 147 196 826 765 230 08 × 2 = 0 + 0.000 000 300 294 393 653 530 460 16;
  • 46) 0.000 000 300 294 393 653 530 460 16 × 2 = 0 + 0.000 000 600 588 787 307 060 920 32;
  • 47) 0.000 000 600 588 787 307 060 920 32 × 2 = 0 + 0.000 001 201 177 574 614 121 840 64;
  • 48) 0.000 001 201 177 574 614 121 840 64 × 2 = 0 + 0.000 002 402 355 149 228 243 681 28;
  • 49) 0.000 002 402 355 149 228 243 681 28 × 2 = 0 + 0.000 004 804 710 298 456 487 362 56;
  • 50) 0.000 004 804 710 298 456 487 362 56 × 2 = 0 + 0.000 009 609 420 596 912 974 725 12;
  • 51) 0.000 009 609 420 596 912 974 725 12 × 2 = 0 + 0.000 019 218 841 193 825 949 450 24;
  • 52) 0.000 019 218 841 193 825 949 450 24 × 2 = 0 + 0.000 038 437 682 387 651 898 900 48;
  • 53) 0.000 038 437 682 387 651 898 900 48 × 2 = 0 + 0.000 076 875 364 775 303 797 800 96;
  • 54) 0.000 076 875 364 775 303 797 800 96 × 2 = 0 + 0.000 153 750 729 550 607 595 601 92;
  • 55) 0.000 153 750 729 550 607 595 601 92 × 2 = 0 + 0.000 307 501 459 101 215 191 203 84;
  • 56) 0.000 307 501 459 101 215 191 203 84 × 2 = 0 + 0.000 615 002 918 202 430 382 407 68;
  • 57) 0.000 615 002 918 202 430 382 407 68 × 2 = 0 + 0.001 230 005 836 404 860 764 815 36;
  • 58) 0.001 230 005 836 404 860 764 815 36 × 2 = 0 + 0.002 460 011 672 809 721 529 630 72;
  • 59) 0.002 460 011 672 809 721 529 630 72 × 2 = 0 + 0.004 920 023 345 619 443 059 261 44;
  • 60) 0.004 920 023 345 619 443 059 261 44 × 2 = 0 + 0.009 840 046 691 238 886 118 522 88;
  • 61) 0.009 840 046 691 238 886 118 522 88 × 2 = 0 + 0.019 680 093 382 477 772 237 045 76;
  • 62) 0.019 680 093 382 477 772 237 045 76 × 2 = 0 + 0.039 360 186 764 955 544 474 091 52;
  • 63) 0.039 360 186 764 955 544 474 091 52 × 2 = 0 + 0.078 720 373 529 911 088 948 183 04;
  • 64) 0.078 720 373 529 911 088 948 183 04 × 2 = 0 + 0.157 440 747 059 822 177 896 366 08;
  • 65) 0.157 440 747 059 822 177 896 366 08 × 2 = 0 + 0.314 881 494 119 644 355 792 732 16;
  • 66) 0.314 881 494 119 644 355 792 732 16 × 2 = 0 + 0.629 762 988 239 288 711 585 464 32;
  • 67) 0.629 762 988 239 288 711 585 464 32 × 2 = 1 + 0.259 525 976 478 577 423 170 928 64;
  • 68) 0.259 525 976 478 577 423 170 928 64 × 2 = 0 + 0.519 051 952 957 154 846 341 857 28;
  • 69) 0.519 051 952 957 154 846 341 857 28 × 2 = 1 + 0.038 103 905 914 309 692 683 714 56;
  • 70) 0.038 103 905 914 309 692 683 714 56 × 2 = 0 + 0.076 207 811 828 619 385 367 429 12;
  • 71) 0.076 207 811 828 619 385 367 429 12 × 2 = 0 + 0.152 415 623 657 238 770 734 858 24;
  • 72) 0.152 415 623 657 238 770 734 858 24 × 2 = 0 + 0.304 831 247 314 477 541 469 716 48;
  • 73) 0.304 831 247 314 477 541 469 716 48 × 2 = 0 + 0.609 662 494 628 955 082 939 432 96;
  • 74) 0.609 662 494 628 955 082 939 432 96 × 2 = 1 + 0.219 324 989 257 910 165 878 865 92;
  • 75) 0.219 324 989 257 910 165 878 865 92 × 2 = 0 + 0.438 649 978 515 820 331 757 731 84;
  • 76) 0.438 649 978 515 820 331 757 731 84 × 2 = 0 + 0.877 299 957 031 640 663 515 463 68;
  • 77) 0.877 299 957 031 640 663 515 463 68 × 2 = 1 + 0.754 599 914 063 281 327 030 927 36;
  • 78) 0.754 599 914 063 281 327 030 927 36 × 2 = 1 + 0.509 199 828 126 562 654 061 854 72;
  • 79) 0.509 199 828 126 562 654 061 854 72 × 2 = 1 + 0.018 399 656 253 125 308 123 709 44;
  • 80) 0.018 399 656 253 125 308 123 709 44 × 2 = 0 + 0.036 799 312 506 250 616 247 418 88;
  • 81) 0.036 799 312 506 250 616 247 418 88 × 2 = 0 + 0.073 598 625 012 501 232 494 837 76;
  • 82) 0.073 598 625 012 501 232 494 837 76 × 2 = 0 + 0.147 197 250 025 002 464 989 675 52;
  • 83) 0.147 197 250 025 002 464 989 675 52 × 2 = 0 + 0.294 394 500 050 004 929 979 351 04;
  • 84) 0.294 394 500 050 004 929 979 351 04 × 2 = 0 + 0.588 789 000 100 009 859 958 702 08;
  • 85) 0.588 789 000 100 009 859 958 702 08 × 2 = 1 + 0.177 578 000 200 019 719 917 404 16;
  • 86) 0.177 578 000 200 019 719 917 404 16 × 2 = 0 + 0.355 156 000 400 039 439 834 808 32;
  • 87) 0.355 156 000 400 039 439 834 808 32 × 2 = 0 + 0.710 312 000 800 078 879 669 616 64;
  • 88) 0.710 312 000 800 078 879 669 616 64 × 2 = 1 + 0.420 624 001 600 157 759 339 233 28;
  • 89) 0.420 624 001 600 157 759 339 233 28 × 2 = 0 + 0.841 248 003 200 315 518 678 466 56;
  • 90) 0.841 248 003 200 315 518 678 466 56 × 2 = 1 + 0.682 496 006 400 631 037 356 933 12;
  • 91) 0.682 496 006 400 631 037 356 933 12 × 2 = 1 + 0.364 992 012 801 262 074 713 866 24;
  • 92) 0.364 992 012 801 262 074 713 866 24 × 2 = 0 + 0.729 984 025 602 524 149 427 732 48;
  • 93) 0.729 984 025 602 524 149 427 732 48 × 2 = 1 + 0.459 968 051 205 048 298 855 464 96;
  • 94) 0.459 968 051 205 048 298 855 464 96 × 2 = 0 + 0.919 936 102 410 096 597 710 929 92;
  • 95) 0.919 936 102 410 096 597 710 929 92 × 2 = 1 + 0.839 872 204 820 193 195 421 859 84;
  • 96) 0.839 872 204 820 193 195 421 859 84 × 2 = 1 + 0.679 744 409 640 386 390 843 719 68;
  • 97) 0.679 744 409 640 386 390 843 719 68 × 2 = 1 + 0.359 488 819 280 772 781 687 439 36;
  • 98) 0.359 488 819 280 772 781 687 439 36 × 2 = 0 + 0.718 977 638 561 545 563 374 878 72;
  • 99) 0.718 977 638 561 545 563 374 878 72 × 2 = 1 + 0.437 955 277 123 091 126 749 757 44;
  • 100) 0.437 955 277 123 091 126 749 757 44 × 2 = 0 + 0.875 910 554 246 182 253 499 514 88;
  • 101) 0.875 910 554 246 182 253 499 514 88 × 2 = 1 + 0.751 821 108 492 364 506 999 029 76;
  • 102) 0.751 821 108 492 364 506 999 029 76 × 2 = 1 + 0.503 642 216 984 729 013 998 059 52;
  • 103) 0.503 642 216 984 729 013 998 059 52 × 2 = 1 + 0.007 284 433 969 458 027 996 119 04;
  • 104) 0.007 284 433 969 458 027 996 119 04 × 2 = 0 + 0.014 568 867 938 916 055 992 238 08;
  • 105) 0.014 568 867 938 916 055 992 238 08 × 2 = 0 + 0.029 137 735 877 832 111 984 476 16;
  • 106) 0.029 137 735 877 832 111 984 476 16 × 2 = 0 + 0.058 275 471 755 664 223 968 952 32;
  • 107) 0.058 275 471 755 664 223 968 952 32 × 2 = 0 + 0.116 550 943 511 328 447 937 904 64;
  • 108) 0.116 550 943 511 328 447 937 904 64 × 2 = 0 + 0.233 101 887 022 656 895 875 809 28;
  • 109) 0.233 101 887 022 656 895 875 809 28 × 2 = 0 + 0.466 203 774 045 313 791 751 618 56;
  • 110) 0.466 203 774 045 313 791 751 618 56 × 2 = 0 + 0.932 407 548 090 627 583 503 237 12;
  • 111) 0.932 407 548 090 627 583 503 237 12 × 2 = 1 + 0.864 815 096 181 255 167 006 474 24;
  • 112) 0.864 815 096 181 255 167 006 474 24 × 2 = 1 + 0.729 630 192 362 510 334 012 948 48;
  • 113) 0.729 630 192 362 510 334 012 948 48 × 2 = 1 + 0.459 260 384 725 020 668 025 896 96;
  • 114) 0.459 260 384 725 020 668 025 896 96 × 2 = 0 + 0.918 520 769 450 041 336 051 793 92;
  • 115) 0.918 520 769 450 041 336 051 793 92 × 2 = 1 + 0.837 041 538 900 082 672 103 587 84;
  • 116) 0.837 041 538 900 082 672 103 587 84 × 2 = 1 + 0.674 083 077 800 165 344 207 175 68;
  • 117) 0.674 083 077 800 165 344 207 175 68 × 2 = 1 + 0.348 166 155 600 330 688 414 351 36;
  • 118) 0.348 166 155 600 330 688 414 351 36 × 2 = 0 + 0.696 332 311 200 661 376 828 702 72;
  • 119) 0.696 332 311 200 661 376 828 702 72 × 2 = 1 + 0.392 664 622 401 322 753 657 405 44;

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:


0.000 000 000 000 000 000 008 534 88(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0000 1001 0110 1011 1010 1110 0000 0011 1011 101(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 534 88(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0000 1001 0110 1011 1010 1110 0000 0011 1011 101(2)

6. 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:


0.000 000 000 000 000 000 008 534 88(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0000 1001 0110 1011 1010 1110 0000 0011 1011 101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0000 1001 0110 1011 1010 1110 0000 0011 1011 101(2) × 20 =

1.0100 0010 0111 0000 0100 1011 0101 1101 0111 0000 0001 1101 1101(2) × 2-67


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): -67

Mantissa (not normalized):
1.0100 0010 0111 0000 0100 1011 0101 1101 0111 0000 0001 1101 1101


8. 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)


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;
  • 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;

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) =

956(10) =

011 1011 1100(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, only if necessary (not the case here).


Mantissa (normalized) =

1. 0100 0010 0111 0000 0100 1011 0101 1101 0111 0000 0001 1101 1101 =

0100 0010 0111 0000 0100 1011 0101 1101 0111 0000 0001 1101 1101


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) =
011 1011 1100

Mantissa (52 bits) =
0100 0010 0111 0000 0100 1011 0101 1101 0111 0000 0001 1101 1101


Decimal number 0.000 000 000 000 000 000 008 534 88 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 0000 0100 1011 0101 1101 0111 0000 0001 1101 1101

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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