-0.016 738 891 601 562 496 530 757 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 757(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
-0.016 738 891 601 562 496 530 757(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:

|-0.016 738 891 601 562 496 530 757| = 0.016 738 891 601 562 496 530 757


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: 0.016 738 891 601 562 496 530 757.

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) 0.016 738 891 601 562 496 530 757 × 2 = 0 + 0.033 477 783 203 124 993 061 514;
  • 2) 0.033 477 783 203 124 993 061 514 × 2 = 0 + 0.066 955 566 406 249 986 123 028;
  • 3) 0.066 955 566 406 249 986 123 028 × 2 = 0 + 0.133 911 132 812 499 972 246 056;
  • 4) 0.133 911 132 812 499 972 246 056 × 2 = 0 + 0.267 822 265 624 999 944 492 112;
  • 5) 0.267 822 265 624 999 944 492 112 × 2 = 0 + 0.535 644 531 249 999 888 984 224;
  • 6) 0.535 644 531 249 999 888 984 224 × 2 = 1 + 0.071 289 062 499 999 777 968 448;
  • 7) 0.071 289 062 499 999 777 968 448 × 2 = 0 + 0.142 578 124 999 999 555 936 896;
  • 8) 0.142 578 124 999 999 555 936 896 × 2 = 0 + 0.285 156 249 999 999 111 873 792;
  • 9) 0.285 156 249 999 999 111 873 792 × 2 = 0 + 0.570 312 499 999 998 223 747 584;
  • 10) 0.570 312 499 999 998 223 747 584 × 2 = 1 + 0.140 624 999 999 996 447 495 168;
  • 11) 0.140 624 999 999 996 447 495 168 × 2 = 0 + 0.281 249 999 999 992 894 990 336;
  • 12) 0.281 249 999 999 992 894 990 336 × 2 = 0 + 0.562 499 999 999 985 789 980 672;
  • 13) 0.562 499 999 999 985 789 980 672 × 2 = 1 + 0.124 999 999 999 971 579 961 344;
  • 14) 0.124 999 999 999 971 579 961 344 × 2 = 0 + 0.249 999 999 999 943 159 922 688;
  • 15) 0.249 999 999 999 943 159 922 688 × 2 = 0 + 0.499 999 999 999 886 319 845 376;
  • 16) 0.499 999 999 999 886 319 845 376 × 2 = 0 + 0.999 999 999 999 772 639 690 752;
  • 17) 0.999 999 999 999 772 639 690 752 × 2 = 1 + 0.999 999 999 999 545 279 381 504;
  • 18) 0.999 999 999 999 545 279 381 504 × 2 = 1 + 0.999 999 999 999 090 558 763 008;
  • 19) 0.999 999 999 999 090 558 763 008 × 2 = 1 + 0.999 999 999 998 181 117 526 016;
  • 20) 0.999 999 999 998 181 117 526 016 × 2 = 1 + 0.999 999 999 996 362 235 052 032;
  • 21) 0.999 999 999 996 362 235 052 032 × 2 = 1 + 0.999 999 999 992 724 470 104 064;
  • 22) 0.999 999 999 992 724 470 104 064 × 2 = 1 + 0.999 999 999 985 448 940 208 128;
  • 23) 0.999 999 999 985 448 940 208 128 × 2 = 1 + 0.999 999 999 970 897 880 416 256;
  • 24) 0.999 999 999 970 897 880 416 256 × 2 = 1 + 0.999 999 999 941 795 760 832 512;
  • 25) 0.999 999 999 941 795 760 832 512 × 2 = 1 + 0.999 999 999 883 591 521 665 024;
  • 26) 0.999 999 999 883 591 521 665 024 × 2 = 1 + 0.999 999 999 767 183 043 330 048;
  • 27) 0.999 999 999 767 183 043 330 048 × 2 = 1 + 0.999 999 999 534 366 086 660 096;
  • 28) 0.999 999 999 534 366 086 660 096 × 2 = 1 + 0.999 999 999 068 732 173 320 192;
  • 29) 0.999 999 999 068 732 173 320 192 × 2 = 1 + 0.999 999 998 137 464 346 640 384;
  • 30) 0.999 999 998 137 464 346 640 384 × 2 = 1 + 0.999 999 996 274 928 693 280 768;
  • 31) 0.999 999 996 274 928 693 280 768 × 2 = 1 + 0.999 999 992 549 857 386 561 536;
  • 32) 0.999 999 992 549 857 386 561 536 × 2 = 1 + 0.999 999 985 099 714 773 123 072;
  • 33) 0.999 999 985 099 714 773 123 072 × 2 = 1 + 0.999 999 970 199 429 546 246 144;
  • 34) 0.999 999 970 199 429 546 246 144 × 2 = 1 + 0.999 999 940 398 859 092 492 288;
  • 35) 0.999 999 940 398 859 092 492 288 × 2 = 1 + 0.999 999 880 797 718 184 984 576;
  • 36) 0.999 999 880 797 718 184 984 576 × 2 = 1 + 0.999 999 761 595 436 369 969 152;
  • 37) 0.999 999 761 595 436 369 969 152 × 2 = 1 + 0.999 999 523 190 872 739 938 304;
  • 38) 0.999 999 523 190 872 739 938 304 × 2 = 1 + 0.999 999 046 381 745 479 876 608;
  • 39) 0.999 999 046 381 745 479 876 608 × 2 = 1 + 0.999 998 092 763 490 959 753 216;
  • 40) 0.999 998 092 763 490 959 753 216 × 2 = 1 + 0.999 996 185 526 981 919 506 432;
  • 41) 0.999 996 185 526 981 919 506 432 × 2 = 1 + 0.999 992 371 053 963 839 012 864;
  • 42) 0.999 992 371 053 963 839 012 864 × 2 = 1 + 0.999 984 742 107 927 678 025 728;
  • 43) 0.999 984 742 107 927 678 025 728 × 2 = 1 + 0.999 969 484 215 855 356 051 456;
  • 44) 0.999 969 484 215 855 356 051 456 × 2 = 1 + 0.999 938 968 431 710 712 102 912;
  • 45) 0.999 938 968 431 710 712 102 912 × 2 = 1 + 0.999 877 936 863 421 424 205 824;
  • 46) 0.999 877 936 863 421 424 205 824 × 2 = 1 + 0.999 755 873 726 842 848 411 648;
  • 47) 0.999 755 873 726 842 848 411 648 × 2 = 1 + 0.999 511 747 453 685 696 823 296;
  • 48) 0.999 511 747 453 685 696 823 296 × 2 = 1 + 0.999 023 494 907 371 393 646 592;
  • 49) 0.999 023 494 907 371 393 646 592 × 2 = 1 + 0.998 046 989 814 742 787 293 184;
  • 50) 0.998 046 989 814 742 787 293 184 × 2 = 1 + 0.996 093 979 629 485 574 586 368;
  • 51) 0.996 093 979 629 485 574 586 368 × 2 = 1 + 0.992 187 959 258 971 149 172 736;
  • 52) 0.992 187 959 258 971 149 172 736 × 2 = 1 + 0.984 375 918 517 942 298 345 472;
  • 53) 0.984 375 918 517 942 298 345 472 × 2 = 1 + 0.968 751 837 035 884 596 690 944;
  • 54) 0.968 751 837 035 884 596 690 944 × 2 = 1 + 0.937 503 674 071 769 193 381 888;
  • 55) 0.937 503 674 071 769 193 381 888 × 2 = 1 + 0.875 007 348 143 538 386 763 776;
  • 56) 0.875 007 348 143 538 386 763 776 × 2 = 1 + 0.750 014 696 287 076 773 527 552;
  • 57) 0.750 014 696 287 076 773 527 552 × 2 = 1 + 0.500 029 392 574 153 547 055 104;
  • 58) 0.500 029 392 574 153 547 055 104 × 2 = 1 + 0.000 058 785 148 307 094 110 208;

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:


0.016 738 891 601 562 496 530 757(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 11(2)

6. Positive number before normalization:

0.016 738 891 601 562 496 530 757(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 11(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 6 positions to the right, so that only one non zero digit remains to the left of it:


0.016 738 891 601 562 496 530 757(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 11(2) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 11(2) × 20 =

1.0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111(2) × 2-6


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): -6

Mantissa (not normalized):
1.0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-6 + 2(11-1) - 1 =

(-6 + 1 023)(10) =

1 017(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;
  • 1 017 ÷ 2 = 508 + 1;
  • 508 ÷ 2 = 254 + 0;
  • 254 ÷ 2 = 127 + 0;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 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) =

1017(10) =

011 1111 1001(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. 0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 =

0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


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 1111 1001

Mantissa (52 bits) =
0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


Decimal number -0.016 738 891 601 562 496 530 757 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1001 - 0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111

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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