-0.016 738 891 601 562 496 553 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 553 4(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
-0.016 738 891 601 562 496 553 4(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:

|-0.016 738 891 601 562 496 553 4| = 0.016 738 891 601 562 496 553 4


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: 0.016 738 891 601 562 496 553 4.

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) 0.016 738 891 601 562 496 553 4 × 2 = 0 + 0.033 477 783 203 124 993 106 8;
  • 2) 0.033 477 783 203 124 993 106 8 × 2 = 0 + 0.066 955 566 406 249 986 213 6;
  • 3) 0.066 955 566 406 249 986 213 6 × 2 = 0 + 0.133 911 132 812 499 972 427 2;
  • 4) 0.133 911 132 812 499 972 427 2 × 2 = 0 + 0.267 822 265 624 999 944 854 4;
  • 5) 0.267 822 265 624 999 944 854 4 × 2 = 0 + 0.535 644 531 249 999 889 708 8;
  • 6) 0.535 644 531 249 999 889 708 8 × 2 = 1 + 0.071 289 062 499 999 779 417 6;
  • 7) 0.071 289 062 499 999 779 417 6 × 2 = 0 + 0.142 578 124 999 999 558 835 2;
  • 8) 0.142 578 124 999 999 558 835 2 × 2 = 0 + 0.285 156 249 999 999 117 670 4;
  • 9) 0.285 156 249 999 999 117 670 4 × 2 = 0 + 0.570 312 499 999 998 235 340 8;
  • 10) 0.570 312 499 999 998 235 340 8 × 2 = 1 + 0.140 624 999 999 996 470 681 6;
  • 11) 0.140 624 999 999 996 470 681 6 × 2 = 0 + 0.281 249 999 999 992 941 363 2;
  • 12) 0.281 249 999 999 992 941 363 2 × 2 = 0 + 0.562 499 999 999 985 882 726 4;
  • 13) 0.562 499 999 999 985 882 726 4 × 2 = 1 + 0.124 999 999 999 971 765 452 8;
  • 14) 0.124 999 999 999 971 765 452 8 × 2 = 0 + 0.249 999 999 999 943 530 905 6;
  • 15) 0.249 999 999 999 943 530 905 6 × 2 = 0 + 0.499 999 999 999 887 061 811 2;
  • 16) 0.499 999 999 999 887 061 811 2 × 2 = 0 + 0.999 999 999 999 774 123 622 4;
  • 17) 0.999 999 999 999 774 123 622 4 × 2 = 1 + 0.999 999 999 999 548 247 244 8;
  • 18) 0.999 999 999 999 548 247 244 8 × 2 = 1 + 0.999 999 999 999 096 494 489 6;
  • 19) 0.999 999 999 999 096 494 489 6 × 2 = 1 + 0.999 999 999 998 192 988 979 2;
  • 20) 0.999 999 999 998 192 988 979 2 × 2 = 1 + 0.999 999 999 996 385 977 958 4;
  • 21) 0.999 999 999 996 385 977 958 4 × 2 = 1 + 0.999 999 999 992 771 955 916 8;
  • 22) 0.999 999 999 992 771 955 916 8 × 2 = 1 + 0.999 999 999 985 543 911 833 6;
  • 23) 0.999 999 999 985 543 911 833 6 × 2 = 1 + 0.999 999 999 971 087 823 667 2;
  • 24) 0.999 999 999 971 087 823 667 2 × 2 = 1 + 0.999 999 999 942 175 647 334 4;
  • 25) 0.999 999 999 942 175 647 334 4 × 2 = 1 + 0.999 999 999 884 351 294 668 8;
  • 26) 0.999 999 999 884 351 294 668 8 × 2 = 1 + 0.999 999 999 768 702 589 337 6;
  • 27) 0.999 999 999 768 702 589 337 6 × 2 = 1 + 0.999 999 999 537 405 178 675 2;
  • 28) 0.999 999 999 537 405 178 675 2 × 2 = 1 + 0.999 999 999 074 810 357 350 4;
  • 29) 0.999 999 999 074 810 357 350 4 × 2 = 1 + 0.999 999 998 149 620 714 700 8;
  • 30) 0.999 999 998 149 620 714 700 8 × 2 = 1 + 0.999 999 996 299 241 429 401 6;
  • 31) 0.999 999 996 299 241 429 401 6 × 2 = 1 + 0.999 999 992 598 482 858 803 2;
  • 32) 0.999 999 992 598 482 858 803 2 × 2 = 1 + 0.999 999 985 196 965 717 606 4;
  • 33) 0.999 999 985 196 965 717 606 4 × 2 = 1 + 0.999 999 970 393 931 435 212 8;
  • 34) 0.999 999 970 393 931 435 212 8 × 2 = 1 + 0.999 999 940 787 862 870 425 6;
  • 35) 0.999 999 940 787 862 870 425 6 × 2 = 1 + 0.999 999 881 575 725 740 851 2;
  • 36) 0.999 999 881 575 725 740 851 2 × 2 = 1 + 0.999 999 763 151 451 481 702 4;
  • 37) 0.999 999 763 151 451 481 702 4 × 2 = 1 + 0.999 999 526 302 902 963 404 8;
  • 38) 0.999 999 526 302 902 963 404 8 × 2 = 1 + 0.999 999 052 605 805 926 809 6;
  • 39) 0.999 999 052 605 805 926 809 6 × 2 = 1 + 0.999 998 105 211 611 853 619 2;
  • 40) 0.999 998 105 211 611 853 619 2 × 2 = 1 + 0.999 996 210 423 223 707 238 4;
  • 41) 0.999 996 210 423 223 707 238 4 × 2 = 1 + 0.999 992 420 846 447 414 476 8;
  • 42) 0.999 992 420 846 447 414 476 8 × 2 = 1 + 0.999 984 841 692 894 828 953 6;
  • 43) 0.999 984 841 692 894 828 953 6 × 2 = 1 + 0.999 969 683 385 789 657 907 2;
  • 44) 0.999 969 683 385 789 657 907 2 × 2 = 1 + 0.999 939 366 771 579 315 814 4;
  • 45) 0.999 939 366 771 579 315 814 4 × 2 = 1 + 0.999 878 733 543 158 631 628 8;
  • 46) 0.999 878 733 543 158 631 628 8 × 2 = 1 + 0.999 757 467 086 317 263 257 6;
  • 47) 0.999 757 467 086 317 263 257 6 × 2 = 1 + 0.999 514 934 172 634 526 515 2;
  • 48) 0.999 514 934 172 634 526 515 2 × 2 = 1 + 0.999 029 868 345 269 053 030 4;
  • 49) 0.999 029 868 345 269 053 030 4 × 2 = 1 + 0.998 059 736 690 538 106 060 8;
  • 50) 0.998 059 736 690 538 106 060 8 × 2 = 1 + 0.996 119 473 381 076 212 121 6;
  • 51) 0.996 119 473 381 076 212 121 6 × 2 = 1 + 0.992 238 946 762 152 424 243 2;
  • 52) 0.992 238 946 762 152 424 243 2 × 2 = 1 + 0.984 477 893 524 304 848 486 4;
  • 53) 0.984 477 893 524 304 848 486 4 × 2 = 1 + 0.968 955 787 048 609 696 972 8;
  • 54) 0.968 955 787 048 609 696 972 8 × 2 = 1 + 0.937 911 574 097 219 393 945 6;
  • 55) 0.937 911 574 097 219 393 945 6 × 2 = 1 + 0.875 823 148 194 438 787 891 2;
  • 56) 0.875 823 148 194 438 787 891 2 × 2 = 1 + 0.751 646 296 388 877 575 782 4;
  • 57) 0.751 646 296 388 877 575 782 4 × 2 = 1 + 0.503 292 592 777 755 151 564 8;
  • 58) 0.503 292 592 777 755 151 564 8 × 2 = 1 + 0.006 585 185 555 510 303 129 6;

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:


0.016 738 891 601 562 496 553 4(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 553 4(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:


0.016 738 891 601 562 496 553 4(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 -0.016 738 891 601 562 496 553 4 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