-0.016 738 891 601 562 496 530 553 072 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 553 072(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
-0.016 738 891 601 562 496 530 553 072(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:

|-0.016 738 891 601 562 496 530 553 072| = 0.016 738 891 601 562 496 530 553 072


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: 0.016 738 891 601 562 496 530 553 072.

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) 0.016 738 891 601 562 496 530 553 072 × 2 = 0 + 0.033 477 783 203 124 993 061 106 144;
  • 2) 0.033 477 783 203 124 993 061 106 144 × 2 = 0 + 0.066 955 566 406 249 986 122 212 288;
  • 3) 0.066 955 566 406 249 986 122 212 288 × 2 = 0 + 0.133 911 132 812 499 972 244 424 576;
  • 4) 0.133 911 132 812 499 972 244 424 576 × 2 = 0 + 0.267 822 265 624 999 944 488 849 152;
  • 5) 0.267 822 265 624 999 944 488 849 152 × 2 = 0 + 0.535 644 531 249 999 888 977 698 304;
  • 6) 0.535 644 531 249 999 888 977 698 304 × 2 = 1 + 0.071 289 062 499 999 777 955 396 608;
  • 7) 0.071 289 062 499 999 777 955 396 608 × 2 = 0 + 0.142 578 124 999 999 555 910 793 216;
  • 8) 0.142 578 124 999 999 555 910 793 216 × 2 = 0 + 0.285 156 249 999 999 111 821 586 432;
  • 9) 0.285 156 249 999 999 111 821 586 432 × 2 = 0 + 0.570 312 499 999 998 223 643 172 864;
  • 10) 0.570 312 499 999 998 223 643 172 864 × 2 = 1 + 0.140 624 999 999 996 447 286 345 728;
  • 11) 0.140 624 999 999 996 447 286 345 728 × 2 = 0 + 0.281 249 999 999 992 894 572 691 456;
  • 12) 0.281 249 999 999 992 894 572 691 456 × 2 = 0 + 0.562 499 999 999 985 789 145 382 912;
  • 13) 0.562 499 999 999 985 789 145 382 912 × 2 = 1 + 0.124 999 999 999 971 578 290 765 824;
  • 14) 0.124 999 999 999 971 578 290 765 824 × 2 = 0 + 0.249 999 999 999 943 156 581 531 648;
  • 15) 0.249 999 999 999 943 156 581 531 648 × 2 = 0 + 0.499 999 999 999 886 313 163 063 296;
  • 16) 0.499 999 999 999 886 313 163 063 296 × 2 = 0 + 0.999 999 999 999 772 626 326 126 592;
  • 17) 0.999 999 999 999 772 626 326 126 592 × 2 = 1 + 0.999 999 999 999 545 252 652 253 184;
  • 18) 0.999 999 999 999 545 252 652 253 184 × 2 = 1 + 0.999 999 999 999 090 505 304 506 368;
  • 19) 0.999 999 999 999 090 505 304 506 368 × 2 = 1 + 0.999 999 999 998 181 010 609 012 736;
  • 20) 0.999 999 999 998 181 010 609 012 736 × 2 = 1 + 0.999 999 999 996 362 021 218 025 472;
  • 21) 0.999 999 999 996 362 021 218 025 472 × 2 = 1 + 0.999 999 999 992 724 042 436 050 944;
  • 22) 0.999 999 999 992 724 042 436 050 944 × 2 = 1 + 0.999 999 999 985 448 084 872 101 888;
  • 23) 0.999 999 999 985 448 084 872 101 888 × 2 = 1 + 0.999 999 999 970 896 169 744 203 776;
  • 24) 0.999 999 999 970 896 169 744 203 776 × 2 = 1 + 0.999 999 999 941 792 339 488 407 552;
  • 25) 0.999 999 999 941 792 339 488 407 552 × 2 = 1 + 0.999 999 999 883 584 678 976 815 104;
  • 26) 0.999 999 999 883 584 678 976 815 104 × 2 = 1 + 0.999 999 999 767 169 357 953 630 208;
  • 27) 0.999 999 999 767 169 357 953 630 208 × 2 = 1 + 0.999 999 999 534 338 715 907 260 416;
  • 28) 0.999 999 999 534 338 715 907 260 416 × 2 = 1 + 0.999 999 999 068 677 431 814 520 832;
  • 29) 0.999 999 999 068 677 431 814 520 832 × 2 = 1 + 0.999 999 998 137 354 863 629 041 664;
  • 30) 0.999 999 998 137 354 863 629 041 664 × 2 = 1 + 0.999 999 996 274 709 727 258 083 328;
  • 31) 0.999 999 996 274 709 727 258 083 328 × 2 = 1 + 0.999 999 992 549 419 454 516 166 656;
  • 32) 0.999 999 992 549 419 454 516 166 656 × 2 = 1 + 0.999 999 985 098 838 909 032 333 312;
  • 33) 0.999 999 985 098 838 909 032 333 312 × 2 = 1 + 0.999 999 970 197 677 818 064 666 624;
  • 34) 0.999 999 970 197 677 818 064 666 624 × 2 = 1 + 0.999 999 940 395 355 636 129 333 248;
  • 35) 0.999 999 940 395 355 636 129 333 248 × 2 = 1 + 0.999 999 880 790 711 272 258 666 496;
  • 36) 0.999 999 880 790 711 272 258 666 496 × 2 = 1 + 0.999 999 761 581 422 544 517 332 992;
  • 37) 0.999 999 761 581 422 544 517 332 992 × 2 = 1 + 0.999 999 523 162 845 089 034 665 984;
  • 38) 0.999 999 523 162 845 089 034 665 984 × 2 = 1 + 0.999 999 046 325 690 178 069 331 968;
  • 39) 0.999 999 046 325 690 178 069 331 968 × 2 = 1 + 0.999 998 092 651 380 356 138 663 936;
  • 40) 0.999 998 092 651 380 356 138 663 936 × 2 = 1 + 0.999 996 185 302 760 712 277 327 872;
  • 41) 0.999 996 185 302 760 712 277 327 872 × 2 = 1 + 0.999 992 370 605 521 424 554 655 744;
  • 42) 0.999 992 370 605 521 424 554 655 744 × 2 = 1 + 0.999 984 741 211 042 849 109 311 488;
  • 43) 0.999 984 741 211 042 849 109 311 488 × 2 = 1 + 0.999 969 482 422 085 698 218 622 976;
  • 44) 0.999 969 482 422 085 698 218 622 976 × 2 = 1 + 0.999 938 964 844 171 396 437 245 952;
  • 45) 0.999 938 964 844 171 396 437 245 952 × 2 = 1 + 0.999 877 929 688 342 792 874 491 904;
  • 46) 0.999 877 929 688 342 792 874 491 904 × 2 = 1 + 0.999 755 859 376 685 585 748 983 808;
  • 47) 0.999 755 859 376 685 585 748 983 808 × 2 = 1 + 0.999 511 718 753 371 171 497 967 616;
  • 48) 0.999 511 718 753 371 171 497 967 616 × 2 = 1 + 0.999 023 437 506 742 342 995 935 232;
  • 49) 0.999 023 437 506 742 342 995 935 232 × 2 = 1 + 0.998 046 875 013 484 685 991 870 464;
  • 50) 0.998 046 875 013 484 685 991 870 464 × 2 = 1 + 0.996 093 750 026 969 371 983 740 928;
  • 51) 0.996 093 750 026 969 371 983 740 928 × 2 = 1 + 0.992 187 500 053 938 743 967 481 856;
  • 52) 0.992 187 500 053 938 743 967 481 856 × 2 = 1 + 0.984 375 000 107 877 487 934 963 712;
  • 53) 0.984 375 000 107 877 487 934 963 712 × 2 = 1 + 0.968 750 000 215 754 975 869 927 424;
  • 54) 0.968 750 000 215 754 975 869 927 424 × 2 = 1 + 0.937 500 000 431 509 951 739 854 848;
  • 55) 0.937 500 000 431 509 951 739 854 848 × 2 = 1 + 0.875 000 000 863 019 903 479 709 696;
  • 56) 0.875 000 000 863 019 903 479 709 696 × 2 = 1 + 0.750 000 001 726 039 806 959 419 392;
  • 57) 0.750 000 001 726 039 806 959 419 392 × 2 = 1 + 0.500 000 003 452 079 613 918 838 784;
  • 58) 0.500 000 003 452 079 613 918 838 784 × 2 = 1 + 0.000 000 006 904 159 227 837 677 568;

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:


0.016 738 891 601 562 496 530 553 072(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 530 553 072(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:


0.016 738 891 601 562 496 530 553 072(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 -0.016 738 891 601 562 496 530 553 072 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