-0.016 738 891 601 562 496 530 563 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 563 1(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
-0.016 738 891 601 562 496 530 563 1(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:

|-0.016 738 891 601 562 496 530 563 1| = 0.016 738 891 601 562 496 530 563 1


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: 0.016 738 891 601 562 496 530 563 1.

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) 0.016 738 891 601 562 496 530 563 1 × 2 = 0 + 0.033 477 783 203 124 993 061 126 2;
  • 2) 0.033 477 783 203 124 993 061 126 2 × 2 = 0 + 0.066 955 566 406 249 986 122 252 4;
  • 3) 0.066 955 566 406 249 986 122 252 4 × 2 = 0 + 0.133 911 132 812 499 972 244 504 8;
  • 4) 0.133 911 132 812 499 972 244 504 8 × 2 = 0 + 0.267 822 265 624 999 944 489 009 6;
  • 5) 0.267 822 265 624 999 944 489 009 6 × 2 = 0 + 0.535 644 531 249 999 888 978 019 2;
  • 6) 0.535 644 531 249 999 888 978 019 2 × 2 = 1 + 0.071 289 062 499 999 777 956 038 4;
  • 7) 0.071 289 062 499 999 777 956 038 4 × 2 = 0 + 0.142 578 124 999 999 555 912 076 8;
  • 8) 0.142 578 124 999 999 555 912 076 8 × 2 = 0 + 0.285 156 249 999 999 111 824 153 6;
  • 9) 0.285 156 249 999 999 111 824 153 6 × 2 = 0 + 0.570 312 499 999 998 223 648 307 2;
  • 10) 0.570 312 499 999 998 223 648 307 2 × 2 = 1 + 0.140 624 999 999 996 447 296 614 4;
  • 11) 0.140 624 999 999 996 447 296 614 4 × 2 = 0 + 0.281 249 999 999 992 894 593 228 8;
  • 12) 0.281 249 999 999 992 894 593 228 8 × 2 = 0 + 0.562 499 999 999 985 789 186 457 6;
  • 13) 0.562 499 999 999 985 789 186 457 6 × 2 = 1 + 0.124 999 999 999 971 578 372 915 2;
  • 14) 0.124 999 999 999 971 578 372 915 2 × 2 = 0 + 0.249 999 999 999 943 156 745 830 4;
  • 15) 0.249 999 999 999 943 156 745 830 4 × 2 = 0 + 0.499 999 999 999 886 313 491 660 8;
  • 16) 0.499 999 999 999 886 313 491 660 8 × 2 = 0 + 0.999 999 999 999 772 626 983 321 6;
  • 17) 0.999 999 999 999 772 626 983 321 6 × 2 = 1 + 0.999 999 999 999 545 253 966 643 2;
  • 18) 0.999 999 999 999 545 253 966 643 2 × 2 = 1 + 0.999 999 999 999 090 507 933 286 4;
  • 19) 0.999 999 999 999 090 507 933 286 4 × 2 = 1 + 0.999 999 999 998 181 015 866 572 8;
  • 20) 0.999 999 999 998 181 015 866 572 8 × 2 = 1 + 0.999 999 999 996 362 031 733 145 6;
  • 21) 0.999 999 999 996 362 031 733 145 6 × 2 = 1 + 0.999 999 999 992 724 063 466 291 2;
  • 22) 0.999 999 999 992 724 063 466 291 2 × 2 = 1 + 0.999 999 999 985 448 126 932 582 4;
  • 23) 0.999 999 999 985 448 126 932 582 4 × 2 = 1 + 0.999 999 999 970 896 253 865 164 8;
  • 24) 0.999 999 999 970 896 253 865 164 8 × 2 = 1 + 0.999 999 999 941 792 507 730 329 6;
  • 25) 0.999 999 999 941 792 507 730 329 6 × 2 = 1 + 0.999 999 999 883 585 015 460 659 2;
  • 26) 0.999 999 999 883 585 015 460 659 2 × 2 = 1 + 0.999 999 999 767 170 030 921 318 4;
  • 27) 0.999 999 999 767 170 030 921 318 4 × 2 = 1 + 0.999 999 999 534 340 061 842 636 8;
  • 28) 0.999 999 999 534 340 061 842 636 8 × 2 = 1 + 0.999 999 999 068 680 123 685 273 6;
  • 29) 0.999 999 999 068 680 123 685 273 6 × 2 = 1 + 0.999 999 998 137 360 247 370 547 2;
  • 30) 0.999 999 998 137 360 247 370 547 2 × 2 = 1 + 0.999 999 996 274 720 494 741 094 4;
  • 31) 0.999 999 996 274 720 494 741 094 4 × 2 = 1 + 0.999 999 992 549 440 989 482 188 8;
  • 32) 0.999 999 992 549 440 989 482 188 8 × 2 = 1 + 0.999 999 985 098 881 978 964 377 6;
  • 33) 0.999 999 985 098 881 978 964 377 6 × 2 = 1 + 0.999 999 970 197 763 957 928 755 2;
  • 34) 0.999 999 970 197 763 957 928 755 2 × 2 = 1 + 0.999 999 940 395 527 915 857 510 4;
  • 35) 0.999 999 940 395 527 915 857 510 4 × 2 = 1 + 0.999 999 880 791 055 831 715 020 8;
  • 36) 0.999 999 880 791 055 831 715 020 8 × 2 = 1 + 0.999 999 761 582 111 663 430 041 6;
  • 37) 0.999 999 761 582 111 663 430 041 6 × 2 = 1 + 0.999 999 523 164 223 326 860 083 2;
  • 38) 0.999 999 523 164 223 326 860 083 2 × 2 = 1 + 0.999 999 046 328 446 653 720 166 4;
  • 39) 0.999 999 046 328 446 653 720 166 4 × 2 = 1 + 0.999 998 092 656 893 307 440 332 8;
  • 40) 0.999 998 092 656 893 307 440 332 8 × 2 = 1 + 0.999 996 185 313 786 614 880 665 6;
  • 41) 0.999 996 185 313 786 614 880 665 6 × 2 = 1 + 0.999 992 370 627 573 229 761 331 2;
  • 42) 0.999 992 370 627 573 229 761 331 2 × 2 = 1 + 0.999 984 741 255 146 459 522 662 4;
  • 43) 0.999 984 741 255 146 459 522 662 4 × 2 = 1 + 0.999 969 482 510 292 919 045 324 8;
  • 44) 0.999 969 482 510 292 919 045 324 8 × 2 = 1 + 0.999 938 965 020 585 838 090 649 6;
  • 45) 0.999 938 965 020 585 838 090 649 6 × 2 = 1 + 0.999 877 930 041 171 676 181 299 2;
  • 46) 0.999 877 930 041 171 676 181 299 2 × 2 = 1 + 0.999 755 860 082 343 352 362 598 4;
  • 47) 0.999 755 860 082 343 352 362 598 4 × 2 = 1 + 0.999 511 720 164 686 704 725 196 8;
  • 48) 0.999 511 720 164 686 704 725 196 8 × 2 = 1 + 0.999 023 440 329 373 409 450 393 6;
  • 49) 0.999 023 440 329 373 409 450 393 6 × 2 = 1 + 0.998 046 880 658 746 818 900 787 2;
  • 50) 0.998 046 880 658 746 818 900 787 2 × 2 = 1 + 0.996 093 761 317 493 637 801 574 4;
  • 51) 0.996 093 761 317 493 637 801 574 4 × 2 = 1 + 0.992 187 522 634 987 275 603 148 8;
  • 52) 0.992 187 522 634 987 275 603 148 8 × 2 = 1 + 0.984 375 045 269 974 551 206 297 6;
  • 53) 0.984 375 045 269 974 551 206 297 6 × 2 = 1 + 0.968 750 090 539 949 102 412 595 2;
  • 54) 0.968 750 090 539 949 102 412 595 2 × 2 = 1 + 0.937 500 181 079 898 204 825 190 4;
  • 55) 0.937 500 181 079 898 204 825 190 4 × 2 = 1 + 0.875 000 362 159 796 409 650 380 8;
  • 56) 0.875 000 362 159 796 409 650 380 8 × 2 = 1 + 0.750 000 724 319 592 819 300 761 6;
  • 57) 0.750 000 724 319 592 819 300 761 6 × 2 = 1 + 0.500 001 448 639 185 638 601 523 2;
  • 58) 0.500 001 448 639 185 638 601 523 2 × 2 = 1 + 0.000 002 897 278 371 277 203 046 4;

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:


0.016 738 891 601 562 496 530 563 1(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 530 563 1(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:


0.016 738 891 601 562 496 530 563 1(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 -0.016 738 891 601 562 496 530 563 1 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