-0.016 738 891 601 562 496 530 552 29 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 552 29(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 29(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 29| = 0.016 738 891 601 562 496 530 552 29


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 29.

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 29 × 2 = 0 + 0.033 477 783 203 124 993 061 104 58;
  • 2) 0.033 477 783 203 124 993 061 104 58 × 2 = 0 + 0.066 955 566 406 249 986 122 209 16;
  • 3) 0.066 955 566 406 249 986 122 209 16 × 2 = 0 + 0.133 911 132 812 499 972 244 418 32;
  • 4) 0.133 911 132 812 499 972 244 418 32 × 2 = 0 + 0.267 822 265 624 999 944 488 836 64;
  • 5) 0.267 822 265 624 999 944 488 836 64 × 2 = 0 + 0.535 644 531 249 999 888 977 673 28;
  • 6) 0.535 644 531 249 999 888 977 673 28 × 2 = 1 + 0.071 289 062 499 999 777 955 346 56;
  • 7) 0.071 289 062 499 999 777 955 346 56 × 2 = 0 + 0.142 578 124 999 999 555 910 693 12;
  • 8) 0.142 578 124 999 999 555 910 693 12 × 2 = 0 + 0.285 156 249 999 999 111 821 386 24;
  • 9) 0.285 156 249 999 999 111 821 386 24 × 2 = 0 + 0.570 312 499 999 998 223 642 772 48;
  • 10) 0.570 312 499 999 998 223 642 772 48 × 2 = 1 + 0.140 624 999 999 996 447 285 544 96;
  • 11) 0.140 624 999 999 996 447 285 544 96 × 2 = 0 + 0.281 249 999 999 992 894 571 089 92;
  • 12) 0.281 249 999 999 992 894 571 089 92 × 2 = 0 + 0.562 499 999 999 985 789 142 179 84;
  • 13) 0.562 499 999 999 985 789 142 179 84 × 2 = 1 + 0.124 999 999 999 971 578 284 359 68;
  • 14) 0.124 999 999 999 971 578 284 359 68 × 2 = 0 + 0.249 999 999 999 943 156 568 719 36;
  • 15) 0.249 999 999 999 943 156 568 719 36 × 2 = 0 + 0.499 999 999 999 886 313 137 438 72;
  • 16) 0.499 999 999 999 886 313 137 438 72 × 2 = 0 + 0.999 999 999 999 772 626 274 877 44;
  • 17) 0.999 999 999 999 772 626 274 877 44 × 2 = 1 + 0.999 999 999 999 545 252 549 754 88;
  • 18) 0.999 999 999 999 545 252 549 754 88 × 2 = 1 + 0.999 999 999 999 090 505 099 509 76;
  • 19) 0.999 999 999 999 090 505 099 509 76 × 2 = 1 + 0.999 999 999 998 181 010 199 019 52;
  • 20) 0.999 999 999 998 181 010 199 019 52 × 2 = 1 + 0.999 999 999 996 362 020 398 039 04;
  • 21) 0.999 999 999 996 362 020 398 039 04 × 2 = 1 + 0.999 999 999 992 724 040 796 078 08;
  • 22) 0.999 999 999 992 724 040 796 078 08 × 2 = 1 + 0.999 999 999 985 448 081 592 156 16;
  • 23) 0.999 999 999 985 448 081 592 156 16 × 2 = 1 + 0.999 999 999 970 896 163 184 312 32;
  • 24) 0.999 999 999 970 896 163 184 312 32 × 2 = 1 + 0.999 999 999 941 792 326 368 624 64;
  • 25) 0.999 999 999 941 792 326 368 624 64 × 2 = 1 + 0.999 999 999 883 584 652 737 249 28;
  • 26) 0.999 999 999 883 584 652 737 249 28 × 2 = 1 + 0.999 999 999 767 169 305 474 498 56;
  • 27) 0.999 999 999 767 169 305 474 498 56 × 2 = 1 + 0.999 999 999 534 338 610 948 997 12;
  • 28) 0.999 999 999 534 338 610 948 997 12 × 2 = 1 + 0.999 999 999 068 677 221 897 994 24;
  • 29) 0.999 999 999 068 677 221 897 994 24 × 2 = 1 + 0.999 999 998 137 354 443 795 988 48;
  • 30) 0.999 999 998 137 354 443 795 988 48 × 2 = 1 + 0.999 999 996 274 708 887 591 976 96;
  • 31) 0.999 999 996 274 708 887 591 976 96 × 2 = 1 + 0.999 999 992 549 417 775 183 953 92;
  • 32) 0.999 999 992 549 417 775 183 953 92 × 2 = 1 + 0.999 999 985 098 835 550 367 907 84;
  • 33) 0.999 999 985 098 835 550 367 907 84 × 2 = 1 + 0.999 999 970 197 671 100 735 815 68;
  • 34) 0.999 999 970 197 671 100 735 815 68 × 2 = 1 + 0.999 999 940 395 342 201 471 631 36;
  • 35) 0.999 999 940 395 342 201 471 631 36 × 2 = 1 + 0.999 999 880 790 684 402 943 262 72;
  • 36) 0.999 999 880 790 684 402 943 262 72 × 2 = 1 + 0.999 999 761 581 368 805 886 525 44;
  • 37) 0.999 999 761 581 368 805 886 525 44 × 2 = 1 + 0.999 999 523 162 737 611 773 050 88;
  • 38) 0.999 999 523 162 737 611 773 050 88 × 2 = 1 + 0.999 999 046 325 475 223 546 101 76;
  • 39) 0.999 999 046 325 475 223 546 101 76 × 2 = 1 + 0.999 998 092 650 950 447 092 203 52;
  • 40) 0.999 998 092 650 950 447 092 203 52 × 2 = 1 + 0.999 996 185 301 900 894 184 407 04;
  • 41) 0.999 996 185 301 900 894 184 407 04 × 2 = 1 + 0.999 992 370 603 801 788 368 814 08;
  • 42) 0.999 992 370 603 801 788 368 814 08 × 2 = 1 + 0.999 984 741 207 603 576 737 628 16;
  • 43) 0.999 984 741 207 603 576 737 628 16 × 2 = 1 + 0.999 969 482 415 207 153 475 256 32;
  • 44) 0.999 969 482 415 207 153 475 256 32 × 2 = 1 + 0.999 938 964 830 414 306 950 512 64;
  • 45) 0.999 938 964 830 414 306 950 512 64 × 2 = 1 + 0.999 877 929 660 828 613 901 025 28;
  • 46) 0.999 877 929 660 828 613 901 025 28 × 2 = 1 + 0.999 755 859 321 657 227 802 050 56;
  • 47) 0.999 755 859 321 657 227 802 050 56 × 2 = 1 + 0.999 511 718 643 314 455 604 101 12;
  • 48) 0.999 511 718 643 314 455 604 101 12 × 2 = 1 + 0.999 023 437 286 628 911 208 202 24;
  • 49) 0.999 023 437 286 628 911 208 202 24 × 2 = 1 + 0.998 046 874 573 257 822 416 404 48;
  • 50) 0.998 046 874 573 257 822 416 404 48 × 2 = 1 + 0.996 093 749 146 515 644 832 808 96;
  • 51) 0.996 093 749 146 515 644 832 808 96 × 2 = 1 + 0.992 187 498 293 031 289 665 617 92;
  • 52) 0.992 187 498 293 031 289 665 617 92 × 2 = 1 + 0.984 374 996 586 062 579 331 235 84;
  • 53) 0.984 374 996 586 062 579 331 235 84 × 2 = 1 + 0.968 749 993 172 125 158 662 471 68;
  • 54) 0.968 749 993 172 125 158 662 471 68 × 2 = 1 + 0.937 499 986 344 250 317 324 943 36;
  • 55) 0.937 499 986 344 250 317 324 943 36 × 2 = 1 + 0.874 999 972 688 500 634 649 886 72;
  • 56) 0.874 999 972 688 500 634 649 886 72 × 2 = 1 + 0.749 999 945 377 001 269 299 773 44;
  • 57) 0.749 999 945 377 001 269 299 773 44 × 2 = 1 + 0.499 999 890 754 002 538 599 546 88;
  • 58) 0.499 999 890 754 002 538 599 546 88 × 2 = 0 + 0.999 999 781 508 005 077 199 093 76;

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 29(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 29(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 29(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 29 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