-0.016 738 891 601 562 531 086 244 689 504 381 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 531 086 244 689 504 381 5(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
-0.016 738 891 601 562 531 086 244 689 504 381 5(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:

|-0.016 738 891 601 562 531 086 244 689 504 381 5| = 0.016 738 891 601 562 531 086 244 689 504 381 5


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: 0.016 738 891 601 562 531 086 244 689 504 381 5.

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) 0.016 738 891 601 562 531 086 244 689 504 381 5 × 2 = 0 + 0.033 477 783 203 125 062 172 489 379 008 763;
  • 2) 0.033 477 783 203 125 062 172 489 379 008 763 × 2 = 0 + 0.066 955 566 406 250 124 344 978 758 017 526;
  • 3) 0.066 955 566 406 250 124 344 978 758 017 526 × 2 = 0 + 0.133 911 132 812 500 248 689 957 516 035 052;
  • 4) 0.133 911 132 812 500 248 689 957 516 035 052 × 2 = 0 + 0.267 822 265 625 000 497 379 915 032 070 104;
  • 5) 0.267 822 265 625 000 497 379 915 032 070 104 × 2 = 0 + 0.535 644 531 250 000 994 759 830 064 140 208;
  • 6) 0.535 644 531 250 000 994 759 830 064 140 208 × 2 = 1 + 0.071 289 062 500 001 989 519 660 128 280 416;
  • 7) 0.071 289 062 500 001 989 519 660 128 280 416 × 2 = 0 + 0.142 578 125 000 003 979 039 320 256 560 832;
  • 8) 0.142 578 125 000 003 979 039 320 256 560 832 × 2 = 0 + 0.285 156 250 000 007 958 078 640 513 121 664;
  • 9) 0.285 156 250 000 007 958 078 640 513 121 664 × 2 = 0 + 0.570 312 500 000 015 916 157 281 026 243 328;
  • 10) 0.570 312 500 000 015 916 157 281 026 243 328 × 2 = 1 + 0.140 625 000 000 031 832 314 562 052 486 656;
  • 11) 0.140 625 000 000 031 832 314 562 052 486 656 × 2 = 0 + 0.281 250 000 000 063 664 629 124 104 973 312;
  • 12) 0.281 250 000 000 063 664 629 124 104 973 312 × 2 = 0 + 0.562 500 000 000 127 329 258 248 209 946 624;
  • 13) 0.562 500 000 000 127 329 258 248 209 946 624 × 2 = 1 + 0.125 000 000 000 254 658 516 496 419 893 248;
  • 14) 0.125 000 000 000 254 658 516 496 419 893 248 × 2 = 0 + 0.250 000 000 000 509 317 032 992 839 786 496;
  • 15) 0.250 000 000 000 509 317 032 992 839 786 496 × 2 = 0 + 0.500 000 000 001 018 634 065 985 679 572 992;
  • 16) 0.500 000 000 001 018 634 065 985 679 572 992 × 2 = 1 + 0.000 000 000 002 037 268 131 971 359 145 984;
  • 17) 0.000 000 000 002 037 268 131 971 359 145 984 × 2 = 0 + 0.000 000 000 004 074 536 263 942 718 291 968;
  • 18) 0.000 000 000 004 074 536 263 942 718 291 968 × 2 = 0 + 0.000 000 000 008 149 072 527 885 436 583 936;
  • 19) 0.000 000 000 008 149 072 527 885 436 583 936 × 2 = 0 + 0.000 000 000 016 298 145 055 770 873 167 872;
  • 20) 0.000 000 000 016 298 145 055 770 873 167 872 × 2 = 0 + 0.000 000 000 032 596 290 111 541 746 335 744;
  • 21) 0.000 000 000 032 596 290 111 541 746 335 744 × 2 = 0 + 0.000 000 000 065 192 580 223 083 492 671 488;
  • 22) 0.000 000 000 065 192 580 223 083 492 671 488 × 2 = 0 + 0.000 000 000 130 385 160 446 166 985 342 976;
  • 23) 0.000 000 000 130 385 160 446 166 985 342 976 × 2 = 0 + 0.000 000 000 260 770 320 892 333 970 685 952;
  • 24) 0.000 000 000 260 770 320 892 333 970 685 952 × 2 = 0 + 0.000 000 000 521 540 641 784 667 941 371 904;
  • 25) 0.000 000 000 521 540 641 784 667 941 371 904 × 2 = 0 + 0.000 000 001 043 081 283 569 335 882 743 808;
  • 26) 0.000 000 001 043 081 283 569 335 882 743 808 × 2 = 0 + 0.000 000 002 086 162 567 138 671 765 487 616;
  • 27) 0.000 000 002 086 162 567 138 671 765 487 616 × 2 = 0 + 0.000 000 004 172 325 134 277 343 530 975 232;
  • 28) 0.000 000 004 172 325 134 277 343 530 975 232 × 2 = 0 + 0.000 000 008 344 650 268 554 687 061 950 464;
  • 29) 0.000 000 008 344 650 268 554 687 061 950 464 × 2 = 0 + 0.000 000 016 689 300 537 109 374 123 900 928;
  • 30) 0.000 000 016 689 300 537 109 374 123 900 928 × 2 = 0 + 0.000 000 033 378 601 074 218 748 247 801 856;
  • 31) 0.000 000 033 378 601 074 218 748 247 801 856 × 2 = 0 + 0.000 000 066 757 202 148 437 496 495 603 712;
  • 32) 0.000 000 066 757 202 148 437 496 495 603 712 × 2 = 0 + 0.000 000 133 514 404 296 874 992 991 207 424;
  • 33) 0.000 000 133 514 404 296 874 992 991 207 424 × 2 = 0 + 0.000 000 267 028 808 593 749 985 982 414 848;
  • 34) 0.000 000 267 028 808 593 749 985 982 414 848 × 2 = 0 + 0.000 000 534 057 617 187 499 971 964 829 696;
  • 35) 0.000 000 534 057 617 187 499 971 964 829 696 × 2 = 0 + 0.000 001 068 115 234 374 999 943 929 659 392;
  • 36) 0.000 001 068 115 234 374 999 943 929 659 392 × 2 = 0 + 0.000 002 136 230 468 749 999 887 859 318 784;
  • 37) 0.000 002 136 230 468 749 999 887 859 318 784 × 2 = 0 + 0.000 004 272 460 937 499 999 775 718 637 568;
  • 38) 0.000 004 272 460 937 499 999 775 718 637 568 × 2 = 0 + 0.000 008 544 921 874 999 999 551 437 275 136;
  • 39) 0.000 008 544 921 874 999 999 551 437 275 136 × 2 = 0 + 0.000 017 089 843 749 999 999 102 874 550 272;
  • 40) 0.000 017 089 843 749 999 999 102 874 550 272 × 2 = 0 + 0.000 034 179 687 499 999 998 205 749 100 544;
  • 41) 0.000 034 179 687 499 999 998 205 749 100 544 × 2 = 0 + 0.000 068 359 374 999 999 996 411 498 201 088;
  • 42) 0.000 068 359 374 999 999 996 411 498 201 088 × 2 = 0 + 0.000 136 718 749 999 999 992 822 996 402 176;
  • 43) 0.000 136 718 749 999 999 992 822 996 402 176 × 2 = 0 + 0.000 273 437 499 999 999 985 645 992 804 352;
  • 44) 0.000 273 437 499 999 999 985 645 992 804 352 × 2 = 0 + 0.000 546 874 999 999 999 971 291 985 608 704;
  • 45) 0.000 546 874 999 999 999 971 291 985 608 704 × 2 = 0 + 0.001 093 749 999 999 999 942 583 971 217 408;
  • 46) 0.001 093 749 999 999 999 942 583 971 217 408 × 2 = 0 + 0.002 187 499 999 999 999 885 167 942 434 816;
  • 47) 0.002 187 499 999 999 999 885 167 942 434 816 × 2 = 0 + 0.004 374 999 999 999 999 770 335 884 869 632;
  • 48) 0.004 374 999 999 999 999 770 335 884 869 632 × 2 = 0 + 0.008 749 999 999 999 999 540 671 769 739 264;
  • 49) 0.008 749 999 999 999 999 540 671 769 739 264 × 2 = 0 + 0.017 499 999 999 999 999 081 343 539 478 528;
  • 50) 0.017 499 999 999 999 999 081 343 539 478 528 × 2 = 0 + 0.034 999 999 999 999 998 162 687 078 957 056;
  • 51) 0.034 999 999 999 999 998 162 687 078 957 056 × 2 = 0 + 0.069 999 999 999 999 996 325 374 157 914 112;
  • 52) 0.069 999 999 999 999 996 325 374 157 914 112 × 2 = 0 + 0.139 999 999 999 999 992 650 748 315 828 224;
  • 53) 0.139 999 999 999 999 992 650 748 315 828 224 × 2 = 0 + 0.279 999 999 999 999 985 301 496 631 656 448;
  • 54) 0.279 999 999 999 999 985 301 496 631 656 448 × 2 = 0 + 0.559 999 999 999 999 970 602 993 263 312 896;
  • 55) 0.559 999 999 999 999 970 602 993 263 312 896 × 2 = 1 + 0.119 999 999 999 999 941 205 986 526 625 792;
  • 56) 0.119 999 999 999 999 941 205 986 526 625 792 × 2 = 0 + 0.239 999 999 999 999 882 411 973 053 251 584;
  • 57) 0.239 999 999 999 999 882 411 973 053 251 584 × 2 = 0 + 0.479 999 999 999 999 764 823 946 106 503 168;
  • 58) 0.479 999 999 999 999 764 823 946 106 503 168 × 2 = 0 + 0.959 999 999 999 999 529 647 892 213 006 336;

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:


0.016 738 891 601 562 531 086 244 689 504 381 5(10) =

0.0000 0100 0100 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 00(2)

6. Positive number before normalization:

0.016 738 891 601 562 531 086 244 689 504 381 5(10) =

0.0000 0100 0100 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 00(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:


0.016 738 891 601 562 531 086 244 689 504 381 5(10) =

0.0000 0100 0100 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 00(2) =

0.0000 0100 0100 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 00(2) × 20 =

1.0001 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000(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 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000


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 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 =

0001 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000


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 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000


Decimal number -0.016 738 891 601 562 531 086 244 689 504 381 5 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1001 - 0001 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000

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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