-284.011 100 000 001 110 001 110 000 000 010 100 010 621 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -284.011 100 000 001 110 001 110 000 000 010 100 010 621(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
-284.011 100 000 001 110 001 110 000 000 010 100 010 621(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:

|-284.011 100 000 001 110 001 110 000 000 010 100 010 621| = 284.011 100 000 001 110 001 110 000 000 010 100 010 621


2. First, convert to binary (in base 2) the integer part: 284.
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;
  • 284 ÷ 2 = 142 + 0;
  • 142 ÷ 2 = 71 + 0;
  • 71 ÷ 2 = 35 + 1;
  • 35 ÷ 2 = 17 + 1;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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.

284(10) =

1 0001 1100(2)


4. Convert to binary (base 2) the fractional part: 0.011 100 000 001 110 001 110 000 000 010 100 010 621.

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) 0.011 100 000 001 110 001 110 000 000 010 100 010 621 × 2 = 0 + 0.022 200 000 002 220 002 220 000 000 020 200 021 242;
  • 2) 0.022 200 000 002 220 002 220 000 000 020 200 021 242 × 2 = 0 + 0.044 400 000 004 440 004 440 000 000 040 400 042 484;
  • 3) 0.044 400 000 004 440 004 440 000 000 040 400 042 484 × 2 = 0 + 0.088 800 000 008 880 008 880 000 000 080 800 084 968;
  • 4) 0.088 800 000 008 880 008 880 000 000 080 800 084 968 × 2 = 0 + 0.177 600 000 017 760 017 760 000 000 161 600 169 936;
  • 5) 0.177 600 000 017 760 017 760 000 000 161 600 169 936 × 2 = 0 + 0.355 200 000 035 520 035 520 000 000 323 200 339 872;
  • 6) 0.355 200 000 035 520 035 520 000 000 323 200 339 872 × 2 = 0 + 0.710 400 000 071 040 071 040 000 000 646 400 679 744;
  • 7) 0.710 400 000 071 040 071 040 000 000 646 400 679 744 × 2 = 1 + 0.420 800 000 142 080 142 080 000 001 292 801 359 488;
  • 8) 0.420 800 000 142 080 142 080 000 001 292 801 359 488 × 2 = 0 + 0.841 600 000 284 160 284 160 000 002 585 602 718 976;
  • 9) 0.841 600 000 284 160 284 160 000 002 585 602 718 976 × 2 = 1 + 0.683 200 000 568 320 568 320 000 005 171 205 437 952;
  • 10) 0.683 200 000 568 320 568 320 000 005 171 205 437 952 × 2 = 1 + 0.366 400 001 136 641 136 640 000 010 342 410 875 904;
  • 11) 0.366 400 001 136 641 136 640 000 010 342 410 875 904 × 2 = 0 + 0.732 800 002 273 282 273 280 000 020 684 821 751 808;
  • 12) 0.732 800 002 273 282 273 280 000 020 684 821 751 808 × 2 = 1 + 0.465 600 004 546 564 546 560 000 041 369 643 503 616;
  • 13) 0.465 600 004 546 564 546 560 000 041 369 643 503 616 × 2 = 0 + 0.931 200 009 093 129 093 120 000 082 739 287 007 232;
  • 14) 0.931 200 009 093 129 093 120 000 082 739 287 007 232 × 2 = 1 + 0.862 400 018 186 258 186 240 000 165 478 574 014 464;
  • 15) 0.862 400 018 186 258 186 240 000 165 478 574 014 464 × 2 = 1 + 0.724 800 036 372 516 372 480 000 330 957 148 028 928;
  • 16) 0.724 800 036 372 516 372 480 000 330 957 148 028 928 × 2 = 1 + 0.449 600 072 745 032 744 960 000 661 914 296 057 856;
  • 17) 0.449 600 072 745 032 744 960 000 661 914 296 057 856 × 2 = 0 + 0.899 200 145 490 065 489 920 001 323 828 592 115 712;
  • 18) 0.899 200 145 490 065 489 920 001 323 828 592 115 712 × 2 = 1 + 0.798 400 290 980 130 979 840 002 647 657 184 231 424;
  • 19) 0.798 400 290 980 130 979 840 002 647 657 184 231 424 × 2 = 1 + 0.596 800 581 960 261 959 680 005 295 314 368 462 848;
  • 20) 0.596 800 581 960 261 959 680 005 295 314 368 462 848 × 2 = 1 + 0.193 601 163 920 523 919 360 010 590 628 736 925 696;
  • 21) 0.193 601 163 920 523 919 360 010 590 628 736 925 696 × 2 = 0 + 0.387 202 327 841 047 838 720 021 181 257 473 851 392;
  • 22) 0.387 202 327 841 047 838 720 021 181 257 473 851 392 × 2 = 0 + 0.774 404 655 682 095 677 440 042 362 514 947 702 784;
  • 23) 0.774 404 655 682 095 677 440 042 362 514 947 702 784 × 2 = 1 + 0.548 809 311 364 191 354 880 084 725 029 895 405 568;
  • 24) 0.548 809 311 364 191 354 880 084 725 029 895 405 568 × 2 = 1 + 0.097 618 622 728 382 709 760 169 450 059 790 811 136;
  • 25) 0.097 618 622 728 382 709 760 169 450 059 790 811 136 × 2 = 0 + 0.195 237 245 456 765 419 520 338 900 119 581 622 272;
  • 26) 0.195 237 245 456 765 419 520 338 900 119 581 622 272 × 2 = 0 + 0.390 474 490 913 530 839 040 677 800 239 163 244 544;
  • 27) 0.390 474 490 913 530 839 040 677 800 239 163 244 544 × 2 = 0 + 0.780 948 981 827 061 678 081 355 600 478 326 489 088;
  • 28) 0.780 948 981 827 061 678 081 355 600 478 326 489 088 × 2 = 1 + 0.561 897 963 654 123 356 162 711 200 956 652 978 176;
  • 29) 0.561 897 963 654 123 356 162 711 200 956 652 978 176 × 2 = 1 + 0.123 795 927 308 246 712 325 422 401 913 305 956 352;
  • 30) 0.123 795 927 308 246 712 325 422 401 913 305 956 352 × 2 = 0 + 0.247 591 854 616 493 424 650 844 803 826 611 912 704;
  • 31) 0.247 591 854 616 493 424 650 844 803 826 611 912 704 × 2 = 0 + 0.495 183 709 232 986 849 301 689 607 653 223 825 408;
  • 32) 0.495 183 709 232 986 849 301 689 607 653 223 825 408 × 2 = 0 + 0.990 367 418 465 973 698 603 379 215 306 447 650 816;
  • 33) 0.990 367 418 465 973 698 603 379 215 306 447 650 816 × 2 = 1 + 0.980 734 836 931 947 397 206 758 430 612 895 301 632;
  • 34) 0.980 734 836 931 947 397 206 758 430 612 895 301 632 × 2 = 1 + 0.961 469 673 863 894 794 413 516 861 225 790 603 264;
  • 35) 0.961 469 673 863 894 794 413 516 861 225 790 603 264 × 2 = 1 + 0.922 939 347 727 789 588 827 033 722 451 581 206 528;
  • 36) 0.922 939 347 727 789 588 827 033 722 451 581 206 528 × 2 = 1 + 0.845 878 695 455 579 177 654 067 444 903 162 413 056;
  • 37) 0.845 878 695 455 579 177 654 067 444 903 162 413 056 × 2 = 1 + 0.691 757 390 911 158 355 308 134 889 806 324 826 112;
  • 38) 0.691 757 390 911 158 355 308 134 889 806 324 826 112 × 2 = 1 + 0.383 514 781 822 316 710 616 269 779 612 649 652 224;
  • 39) 0.383 514 781 822 316 710 616 269 779 612 649 652 224 × 2 = 0 + 0.767 029 563 644 633 421 232 539 559 225 299 304 448;
  • 40) 0.767 029 563 644 633 421 232 539 559 225 299 304 448 × 2 = 1 + 0.534 059 127 289 266 842 465 079 118 450 598 608 896;
  • 41) 0.534 059 127 289 266 842 465 079 118 450 598 608 896 × 2 = 1 + 0.068 118 254 578 533 684 930 158 236 901 197 217 792;
  • 42) 0.068 118 254 578 533 684 930 158 236 901 197 217 792 × 2 = 0 + 0.136 236 509 157 067 369 860 316 473 802 394 435 584;
  • 43) 0.136 236 509 157 067 369 860 316 473 802 394 435 584 × 2 = 0 + 0.272 473 018 314 134 739 720 632 947 604 788 871 168;
  • 44) 0.272 473 018 314 134 739 720 632 947 604 788 871 168 × 2 = 0 + 0.544 946 036 628 269 479 441 265 895 209 577 742 336;
  • 45) 0.544 946 036 628 269 479 441 265 895 209 577 742 336 × 2 = 1 + 0.089 892 073 256 538 958 882 531 790 419 155 484 672;
  • 46) 0.089 892 073 256 538 958 882 531 790 419 155 484 672 × 2 = 0 + 0.179 784 146 513 077 917 765 063 580 838 310 969 344;
  • 47) 0.179 784 146 513 077 917 765 063 580 838 310 969 344 × 2 = 0 + 0.359 568 293 026 155 835 530 127 161 676 621 938 688;
  • 48) 0.359 568 293 026 155 835 530 127 161 676 621 938 688 × 2 = 0 + 0.719 136 586 052 311 671 060 254 323 353 243 877 376;
  • 49) 0.719 136 586 052 311 671 060 254 323 353 243 877 376 × 2 = 1 + 0.438 273 172 104 623 342 120 508 646 706 487 754 752;
  • 50) 0.438 273 172 104 623 342 120 508 646 706 487 754 752 × 2 = 0 + 0.876 546 344 209 246 684 241 017 293 412 975 509 504;
  • 51) 0.876 546 344 209 246 684 241 017 293 412 975 509 504 × 2 = 1 + 0.753 092 688 418 493 368 482 034 586 825 951 019 008;
  • 52) 0.753 092 688 418 493 368 482 034 586 825 951 019 008 × 2 = 1 + 0.506 185 376 836 986 736 964 069 173 651 902 038 016;
  • 53) 0.506 185 376 836 986 736 964 069 173 651 902 038 016 × 2 = 1 + 0.012 370 753 673 973 473 928 138 347 303 804 076 032;

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:


0.011 100 000 001 110 001 110 000 000 010 100 010 621(10) =

0.0000 0010 1101 0111 0111 0011 0001 1000 1111 1101 1000 1000 1011 1(2)

6. Positive number before normalization:

284.011 100 000 001 110 001 110 000 000 010 100 010 621(10) =

1 0001 1100.0000 0010 1101 0111 0111 0011 0001 1000 1111 1101 1000 1000 1011 1(2)

7. Normalize the binary representation of the number.

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


284.011 100 000 001 110 001 110 000 000 010 100 010 621(10) =

1 0001 1100.0000 0010 1101 0111 0111 0011 0001 1000 1111 1101 1000 1000 1011 1(2) =

1 0001 1100.0000 0010 1101 0111 0111 0011 0001 1000 1111 1101 1000 1000 1011 1(2) × 20 =

1.0001 1100 0000 0010 1101 0111 0111 0011 0001 1000 1111 1101 1000 1000 1011 1(2) × 28


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): 8

Mantissa (not normalized):
1.0001 1100 0000 0010 1101 0111 0111 0011 0001 1000 1111 1101 1000 1000 1011 1


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

8 + 2(11-1) - 1 =

(8 + 1 023)(10) =

1 031(10)


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

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) =

1031(10) =

100 0000 0111(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, 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. 0001 1100 0000 0010 1101 0111 0111 0011 0001 1000 1111 1101 1000 1 0001 0111 =

0001 1100 0000 0010 1101 0111 0111 0011 0001 1000 1111 1101 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) =
100 0000 0111

Mantissa (52 bits) =
0001 1100 0000 0010 1101 0111 0111 0011 0001 1000 1111 1101 1000


Decimal number -284.011 100 000 001 110 001 110 000 000 010 100 010 621 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 100 0000 0111 - 0001 1100 0000 0010 1101 0111 0111 0011 0001 1000 1111 1101 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