-284.011 100 000 001 110 001 110 000 000 010 100 010 704 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -284.011 100 000 001 110 001 110 000 000 010 100 010 704(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
-284.011 100 000 001 110 001 110 000 000 010 100 010 704(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:

|-284.011 100 000 001 110 001 110 000 000 010 100 010 704| = 284.011 100 000 001 110 001 110 000 000 010 100 010 704


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: 0.011 100 000 001 110 001 110 000 000 010 100 010 704.

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) 0.011 100 000 001 110 001 110 000 000 010 100 010 704 × 2 = 0 + 0.022 200 000 002 220 002 220 000 000 020 200 021 408;
  • 2) 0.022 200 000 002 220 002 220 000 000 020 200 021 408 × 2 = 0 + 0.044 400 000 004 440 004 440 000 000 040 400 042 816;
  • 3) 0.044 400 000 004 440 004 440 000 000 040 400 042 816 × 2 = 0 + 0.088 800 000 008 880 008 880 000 000 080 800 085 632;
  • 4) 0.088 800 000 008 880 008 880 000 000 080 800 085 632 × 2 = 0 + 0.177 600 000 017 760 017 760 000 000 161 600 171 264;
  • 5) 0.177 600 000 017 760 017 760 000 000 161 600 171 264 × 2 = 0 + 0.355 200 000 035 520 035 520 000 000 323 200 342 528;
  • 6) 0.355 200 000 035 520 035 520 000 000 323 200 342 528 × 2 = 0 + 0.710 400 000 071 040 071 040 000 000 646 400 685 056;
  • 7) 0.710 400 000 071 040 071 040 000 000 646 400 685 056 × 2 = 1 + 0.420 800 000 142 080 142 080 000 001 292 801 370 112;
  • 8) 0.420 800 000 142 080 142 080 000 001 292 801 370 112 × 2 = 0 + 0.841 600 000 284 160 284 160 000 002 585 602 740 224;
  • 9) 0.841 600 000 284 160 284 160 000 002 585 602 740 224 × 2 = 1 + 0.683 200 000 568 320 568 320 000 005 171 205 480 448;
  • 10) 0.683 200 000 568 320 568 320 000 005 171 205 480 448 × 2 = 1 + 0.366 400 001 136 641 136 640 000 010 342 410 960 896;
  • 11) 0.366 400 001 136 641 136 640 000 010 342 410 960 896 × 2 = 0 + 0.732 800 002 273 282 273 280 000 020 684 821 921 792;
  • 12) 0.732 800 002 273 282 273 280 000 020 684 821 921 792 × 2 = 1 + 0.465 600 004 546 564 546 560 000 041 369 643 843 584;
  • 13) 0.465 600 004 546 564 546 560 000 041 369 643 843 584 × 2 = 0 + 0.931 200 009 093 129 093 120 000 082 739 287 687 168;
  • 14) 0.931 200 009 093 129 093 120 000 082 739 287 687 168 × 2 = 1 + 0.862 400 018 186 258 186 240 000 165 478 575 374 336;
  • 15) 0.862 400 018 186 258 186 240 000 165 478 575 374 336 × 2 = 1 + 0.724 800 036 372 516 372 480 000 330 957 150 748 672;
  • 16) 0.724 800 036 372 516 372 480 000 330 957 150 748 672 × 2 = 1 + 0.449 600 072 745 032 744 960 000 661 914 301 497 344;
  • 17) 0.449 600 072 745 032 744 960 000 661 914 301 497 344 × 2 = 0 + 0.899 200 145 490 065 489 920 001 323 828 602 994 688;
  • 18) 0.899 200 145 490 065 489 920 001 323 828 602 994 688 × 2 = 1 + 0.798 400 290 980 130 979 840 002 647 657 205 989 376;
  • 19) 0.798 400 290 980 130 979 840 002 647 657 205 989 376 × 2 = 1 + 0.596 800 581 960 261 959 680 005 295 314 411 978 752;
  • 20) 0.596 800 581 960 261 959 680 005 295 314 411 978 752 × 2 = 1 + 0.193 601 163 920 523 919 360 010 590 628 823 957 504;
  • 21) 0.193 601 163 920 523 919 360 010 590 628 823 957 504 × 2 = 0 + 0.387 202 327 841 047 838 720 021 181 257 647 915 008;
  • 22) 0.387 202 327 841 047 838 720 021 181 257 647 915 008 × 2 = 0 + 0.774 404 655 682 095 677 440 042 362 515 295 830 016;
  • 23) 0.774 404 655 682 095 677 440 042 362 515 295 830 016 × 2 = 1 + 0.548 809 311 364 191 354 880 084 725 030 591 660 032;
  • 24) 0.548 809 311 364 191 354 880 084 725 030 591 660 032 × 2 = 1 + 0.097 618 622 728 382 709 760 169 450 061 183 320 064;
  • 25) 0.097 618 622 728 382 709 760 169 450 061 183 320 064 × 2 = 0 + 0.195 237 245 456 765 419 520 338 900 122 366 640 128;
  • 26) 0.195 237 245 456 765 419 520 338 900 122 366 640 128 × 2 = 0 + 0.390 474 490 913 530 839 040 677 800 244 733 280 256;
  • 27) 0.390 474 490 913 530 839 040 677 800 244 733 280 256 × 2 = 0 + 0.780 948 981 827 061 678 081 355 600 489 466 560 512;
  • 28) 0.780 948 981 827 061 678 081 355 600 489 466 560 512 × 2 = 1 + 0.561 897 963 654 123 356 162 711 200 978 933 121 024;
  • 29) 0.561 897 963 654 123 356 162 711 200 978 933 121 024 × 2 = 1 + 0.123 795 927 308 246 712 325 422 401 957 866 242 048;
  • 30) 0.123 795 927 308 246 712 325 422 401 957 866 242 048 × 2 = 0 + 0.247 591 854 616 493 424 650 844 803 915 732 484 096;
  • 31) 0.247 591 854 616 493 424 650 844 803 915 732 484 096 × 2 = 0 + 0.495 183 709 232 986 849 301 689 607 831 464 968 192;
  • 32) 0.495 183 709 232 986 849 301 689 607 831 464 968 192 × 2 = 0 + 0.990 367 418 465 973 698 603 379 215 662 929 936 384;
  • 33) 0.990 367 418 465 973 698 603 379 215 662 929 936 384 × 2 = 1 + 0.980 734 836 931 947 397 206 758 431 325 859 872 768;
  • 34) 0.980 734 836 931 947 397 206 758 431 325 859 872 768 × 2 = 1 + 0.961 469 673 863 894 794 413 516 862 651 719 745 536;
  • 35) 0.961 469 673 863 894 794 413 516 862 651 719 745 536 × 2 = 1 + 0.922 939 347 727 789 588 827 033 725 303 439 491 072;
  • 36) 0.922 939 347 727 789 588 827 033 725 303 439 491 072 × 2 = 1 + 0.845 878 695 455 579 177 654 067 450 606 878 982 144;
  • 37) 0.845 878 695 455 579 177 654 067 450 606 878 982 144 × 2 = 1 + 0.691 757 390 911 158 355 308 134 901 213 757 964 288;
  • 38) 0.691 757 390 911 158 355 308 134 901 213 757 964 288 × 2 = 1 + 0.383 514 781 822 316 710 616 269 802 427 515 928 576;
  • 39) 0.383 514 781 822 316 710 616 269 802 427 515 928 576 × 2 = 0 + 0.767 029 563 644 633 421 232 539 604 855 031 857 152;
  • 40) 0.767 029 563 644 633 421 232 539 604 855 031 857 152 × 2 = 1 + 0.534 059 127 289 266 842 465 079 209 710 063 714 304;
  • 41) 0.534 059 127 289 266 842 465 079 209 710 063 714 304 × 2 = 1 + 0.068 118 254 578 533 684 930 158 419 420 127 428 608;
  • 42) 0.068 118 254 578 533 684 930 158 419 420 127 428 608 × 2 = 0 + 0.136 236 509 157 067 369 860 316 838 840 254 857 216;
  • 43) 0.136 236 509 157 067 369 860 316 838 840 254 857 216 × 2 = 0 + 0.272 473 018 314 134 739 720 633 677 680 509 714 432;
  • 44) 0.272 473 018 314 134 739 720 633 677 680 509 714 432 × 2 = 0 + 0.544 946 036 628 269 479 441 267 355 361 019 428 864;
  • 45) 0.544 946 036 628 269 479 441 267 355 361 019 428 864 × 2 = 1 + 0.089 892 073 256 538 958 882 534 710 722 038 857 728;
  • 46) 0.089 892 073 256 538 958 882 534 710 722 038 857 728 × 2 = 0 + 0.179 784 146 513 077 917 765 069 421 444 077 715 456;
  • 47) 0.179 784 146 513 077 917 765 069 421 444 077 715 456 × 2 = 0 + 0.359 568 293 026 155 835 530 138 842 888 155 430 912;
  • 48) 0.359 568 293 026 155 835 530 138 842 888 155 430 912 × 2 = 0 + 0.719 136 586 052 311 671 060 277 685 776 310 861 824;
  • 49) 0.719 136 586 052 311 671 060 277 685 776 310 861 824 × 2 = 1 + 0.438 273 172 104 623 342 120 555 371 552 621 723 648;
  • 50) 0.438 273 172 104 623 342 120 555 371 552 621 723 648 × 2 = 0 + 0.876 546 344 209 246 684 241 110 743 105 243 447 296;
  • 51) 0.876 546 344 209 246 684 241 110 743 105 243 447 296 × 2 = 1 + 0.753 092 688 418 493 368 482 221 486 210 486 894 592;
  • 52) 0.753 092 688 418 493 368 482 221 486 210 486 894 592 × 2 = 1 + 0.506 185 376 836 986 736 964 442 972 420 973 789 184;
  • 53) 0.506 185 376 836 986 736 964 442 972 420 973 789 184 × 2 = 1 + 0.012 370 753 673 973 473 928 885 944 841 947 578 368;

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:


0.011 100 000 001 110 001 110 000 000 010 100 010 704(10) =

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

6. Positive number before normalization:

284.011 100 000 001 110 001 110 000 000 010 100 010 704(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:


284.011 100 000 001 110 001 110 000 000 010 100 010 704(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 -284.011 100 000 001 110 001 110 000 000 010 100 010 704 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