-0.016 738 891 601 562 496 530 552 76 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 552 76(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
-0.016 738 891 601 562 496 530 552 76(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:

|-0.016 738 891 601 562 496 530 552 76| = 0.016 738 891 601 562 496 530 552 76


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: 0.016 738 891 601 562 496 530 552 76.

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) 0.016 738 891 601 562 496 530 552 76 × 2 = 0 + 0.033 477 783 203 124 993 061 105 52;
  • 2) 0.033 477 783 203 124 993 061 105 52 × 2 = 0 + 0.066 955 566 406 249 986 122 211 04;
  • 3) 0.066 955 566 406 249 986 122 211 04 × 2 = 0 + 0.133 911 132 812 499 972 244 422 08;
  • 4) 0.133 911 132 812 499 972 244 422 08 × 2 = 0 + 0.267 822 265 624 999 944 488 844 16;
  • 5) 0.267 822 265 624 999 944 488 844 16 × 2 = 0 + 0.535 644 531 249 999 888 977 688 32;
  • 6) 0.535 644 531 249 999 888 977 688 32 × 2 = 1 + 0.071 289 062 499 999 777 955 376 64;
  • 7) 0.071 289 062 499 999 777 955 376 64 × 2 = 0 + 0.142 578 124 999 999 555 910 753 28;
  • 8) 0.142 578 124 999 999 555 910 753 28 × 2 = 0 + 0.285 156 249 999 999 111 821 506 56;
  • 9) 0.285 156 249 999 999 111 821 506 56 × 2 = 0 + 0.570 312 499 999 998 223 643 013 12;
  • 10) 0.570 312 499 999 998 223 643 013 12 × 2 = 1 + 0.140 624 999 999 996 447 286 026 24;
  • 11) 0.140 624 999 999 996 447 286 026 24 × 2 = 0 + 0.281 249 999 999 992 894 572 052 48;
  • 12) 0.281 249 999 999 992 894 572 052 48 × 2 = 0 + 0.562 499 999 999 985 789 144 104 96;
  • 13) 0.562 499 999 999 985 789 144 104 96 × 2 = 1 + 0.124 999 999 999 971 578 288 209 92;
  • 14) 0.124 999 999 999 971 578 288 209 92 × 2 = 0 + 0.249 999 999 999 943 156 576 419 84;
  • 15) 0.249 999 999 999 943 156 576 419 84 × 2 = 0 + 0.499 999 999 999 886 313 152 839 68;
  • 16) 0.499 999 999 999 886 313 152 839 68 × 2 = 0 + 0.999 999 999 999 772 626 305 679 36;
  • 17) 0.999 999 999 999 772 626 305 679 36 × 2 = 1 + 0.999 999 999 999 545 252 611 358 72;
  • 18) 0.999 999 999 999 545 252 611 358 72 × 2 = 1 + 0.999 999 999 999 090 505 222 717 44;
  • 19) 0.999 999 999 999 090 505 222 717 44 × 2 = 1 + 0.999 999 999 998 181 010 445 434 88;
  • 20) 0.999 999 999 998 181 010 445 434 88 × 2 = 1 + 0.999 999 999 996 362 020 890 869 76;
  • 21) 0.999 999 999 996 362 020 890 869 76 × 2 = 1 + 0.999 999 999 992 724 041 781 739 52;
  • 22) 0.999 999 999 992 724 041 781 739 52 × 2 = 1 + 0.999 999 999 985 448 083 563 479 04;
  • 23) 0.999 999 999 985 448 083 563 479 04 × 2 = 1 + 0.999 999 999 970 896 167 126 958 08;
  • 24) 0.999 999 999 970 896 167 126 958 08 × 2 = 1 + 0.999 999 999 941 792 334 253 916 16;
  • 25) 0.999 999 999 941 792 334 253 916 16 × 2 = 1 + 0.999 999 999 883 584 668 507 832 32;
  • 26) 0.999 999 999 883 584 668 507 832 32 × 2 = 1 + 0.999 999 999 767 169 337 015 664 64;
  • 27) 0.999 999 999 767 169 337 015 664 64 × 2 = 1 + 0.999 999 999 534 338 674 031 329 28;
  • 28) 0.999 999 999 534 338 674 031 329 28 × 2 = 1 + 0.999 999 999 068 677 348 062 658 56;
  • 29) 0.999 999 999 068 677 348 062 658 56 × 2 = 1 + 0.999 999 998 137 354 696 125 317 12;
  • 30) 0.999 999 998 137 354 696 125 317 12 × 2 = 1 + 0.999 999 996 274 709 392 250 634 24;
  • 31) 0.999 999 996 274 709 392 250 634 24 × 2 = 1 + 0.999 999 992 549 418 784 501 268 48;
  • 32) 0.999 999 992 549 418 784 501 268 48 × 2 = 1 + 0.999 999 985 098 837 569 002 536 96;
  • 33) 0.999 999 985 098 837 569 002 536 96 × 2 = 1 + 0.999 999 970 197 675 138 005 073 92;
  • 34) 0.999 999 970 197 675 138 005 073 92 × 2 = 1 + 0.999 999 940 395 350 276 010 147 84;
  • 35) 0.999 999 940 395 350 276 010 147 84 × 2 = 1 + 0.999 999 880 790 700 552 020 295 68;
  • 36) 0.999 999 880 790 700 552 020 295 68 × 2 = 1 + 0.999 999 761 581 401 104 040 591 36;
  • 37) 0.999 999 761 581 401 104 040 591 36 × 2 = 1 + 0.999 999 523 162 802 208 081 182 72;
  • 38) 0.999 999 523 162 802 208 081 182 72 × 2 = 1 + 0.999 999 046 325 604 416 162 365 44;
  • 39) 0.999 999 046 325 604 416 162 365 44 × 2 = 1 + 0.999 998 092 651 208 832 324 730 88;
  • 40) 0.999 998 092 651 208 832 324 730 88 × 2 = 1 + 0.999 996 185 302 417 664 649 461 76;
  • 41) 0.999 996 185 302 417 664 649 461 76 × 2 = 1 + 0.999 992 370 604 835 329 298 923 52;
  • 42) 0.999 992 370 604 835 329 298 923 52 × 2 = 1 + 0.999 984 741 209 670 658 597 847 04;
  • 43) 0.999 984 741 209 670 658 597 847 04 × 2 = 1 + 0.999 969 482 419 341 317 195 694 08;
  • 44) 0.999 969 482 419 341 317 195 694 08 × 2 = 1 + 0.999 938 964 838 682 634 391 388 16;
  • 45) 0.999 938 964 838 682 634 391 388 16 × 2 = 1 + 0.999 877 929 677 365 268 782 776 32;
  • 46) 0.999 877 929 677 365 268 782 776 32 × 2 = 1 + 0.999 755 859 354 730 537 565 552 64;
  • 47) 0.999 755 859 354 730 537 565 552 64 × 2 = 1 + 0.999 511 718 709 461 075 131 105 28;
  • 48) 0.999 511 718 709 461 075 131 105 28 × 2 = 1 + 0.999 023 437 418 922 150 262 210 56;
  • 49) 0.999 023 437 418 922 150 262 210 56 × 2 = 1 + 0.998 046 874 837 844 300 524 421 12;
  • 50) 0.998 046 874 837 844 300 524 421 12 × 2 = 1 + 0.996 093 749 675 688 601 048 842 24;
  • 51) 0.996 093 749 675 688 601 048 842 24 × 2 = 1 + 0.992 187 499 351 377 202 097 684 48;
  • 52) 0.992 187 499 351 377 202 097 684 48 × 2 = 1 + 0.984 374 998 702 754 404 195 368 96;
  • 53) 0.984 374 998 702 754 404 195 368 96 × 2 = 1 + 0.968 749 997 405 508 808 390 737 92;
  • 54) 0.968 749 997 405 508 808 390 737 92 × 2 = 1 + 0.937 499 994 811 017 616 781 475 84;
  • 55) 0.937 499 994 811 017 616 781 475 84 × 2 = 1 + 0.874 999 989 622 035 233 562 951 68;
  • 56) 0.874 999 989 622 035 233 562 951 68 × 2 = 1 + 0.749 999 979 244 070 467 125 903 36;
  • 57) 0.749 999 979 244 070 467 125 903 36 × 2 = 1 + 0.499 999 958 488 140 934 251 806 72;
  • 58) 0.499 999 958 488 140 934 251 806 72 × 2 = 0 + 0.999 999 916 976 281 868 503 613 44;

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:


0.016 738 891 601 562 496 530 552 76(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 10(2)

6. Positive number before normalization:

0.016 738 891 601 562 496 530 552 76(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 10(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:


0.016 738 891 601 562 496 530 552 76(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 10(2) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 10(2) × 20 =

1.0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110(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 1110


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 1110 =

0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110


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 1110


Decimal number -0.016 738 891 601 562 496 530 552 76 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 1110

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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