-0.016 738 891 601 562 531 086 244 689 504 383 032 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 531 086 244 689 504 383 032(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
-0.016 738 891 601 562 531 086 244 689 504 383 032(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:

|-0.016 738 891 601 562 531 086 244 689 504 383 032| = 0.016 738 891 601 562 531 086 244 689 504 383 032


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: 0.016 738 891 601 562 531 086 244 689 504 383 032.

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) 0.016 738 891 601 562 531 086 244 689 504 383 032 × 2 = 0 + 0.033 477 783 203 125 062 172 489 379 008 766 064;
  • 2) 0.033 477 783 203 125 062 172 489 379 008 766 064 × 2 = 0 + 0.066 955 566 406 250 124 344 978 758 017 532 128;
  • 3) 0.066 955 566 406 250 124 344 978 758 017 532 128 × 2 = 0 + 0.133 911 132 812 500 248 689 957 516 035 064 256;
  • 4) 0.133 911 132 812 500 248 689 957 516 035 064 256 × 2 = 0 + 0.267 822 265 625 000 497 379 915 032 070 128 512;
  • 5) 0.267 822 265 625 000 497 379 915 032 070 128 512 × 2 = 0 + 0.535 644 531 250 000 994 759 830 064 140 257 024;
  • 6) 0.535 644 531 250 000 994 759 830 064 140 257 024 × 2 = 1 + 0.071 289 062 500 001 989 519 660 128 280 514 048;
  • 7) 0.071 289 062 500 001 989 519 660 128 280 514 048 × 2 = 0 + 0.142 578 125 000 003 979 039 320 256 561 028 096;
  • 8) 0.142 578 125 000 003 979 039 320 256 561 028 096 × 2 = 0 + 0.285 156 250 000 007 958 078 640 513 122 056 192;
  • 9) 0.285 156 250 000 007 958 078 640 513 122 056 192 × 2 = 0 + 0.570 312 500 000 015 916 157 281 026 244 112 384;
  • 10) 0.570 312 500 000 015 916 157 281 026 244 112 384 × 2 = 1 + 0.140 625 000 000 031 832 314 562 052 488 224 768;
  • 11) 0.140 625 000 000 031 832 314 562 052 488 224 768 × 2 = 0 + 0.281 250 000 000 063 664 629 124 104 976 449 536;
  • 12) 0.281 250 000 000 063 664 629 124 104 976 449 536 × 2 = 0 + 0.562 500 000 000 127 329 258 248 209 952 899 072;
  • 13) 0.562 500 000 000 127 329 258 248 209 952 899 072 × 2 = 1 + 0.125 000 000 000 254 658 516 496 419 905 798 144;
  • 14) 0.125 000 000 000 254 658 516 496 419 905 798 144 × 2 = 0 + 0.250 000 000 000 509 317 032 992 839 811 596 288;
  • 15) 0.250 000 000 000 509 317 032 992 839 811 596 288 × 2 = 0 + 0.500 000 000 001 018 634 065 985 679 623 192 576;
  • 16) 0.500 000 000 001 018 634 065 985 679 623 192 576 × 2 = 1 + 0.000 000 000 002 037 268 131 971 359 246 385 152;
  • 17) 0.000 000 000 002 037 268 131 971 359 246 385 152 × 2 = 0 + 0.000 000 000 004 074 536 263 942 718 492 770 304;
  • 18) 0.000 000 000 004 074 536 263 942 718 492 770 304 × 2 = 0 + 0.000 000 000 008 149 072 527 885 436 985 540 608;
  • 19) 0.000 000 000 008 149 072 527 885 436 985 540 608 × 2 = 0 + 0.000 000 000 016 298 145 055 770 873 971 081 216;
  • 20) 0.000 000 000 016 298 145 055 770 873 971 081 216 × 2 = 0 + 0.000 000 000 032 596 290 111 541 747 942 162 432;
  • 21) 0.000 000 000 032 596 290 111 541 747 942 162 432 × 2 = 0 + 0.000 000 000 065 192 580 223 083 495 884 324 864;
  • 22) 0.000 000 000 065 192 580 223 083 495 884 324 864 × 2 = 0 + 0.000 000 000 130 385 160 446 166 991 768 649 728;
  • 23) 0.000 000 000 130 385 160 446 166 991 768 649 728 × 2 = 0 + 0.000 000 000 260 770 320 892 333 983 537 299 456;
  • 24) 0.000 000 000 260 770 320 892 333 983 537 299 456 × 2 = 0 + 0.000 000 000 521 540 641 784 667 967 074 598 912;
  • 25) 0.000 000 000 521 540 641 784 667 967 074 598 912 × 2 = 0 + 0.000 000 001 043 081 283 569 335 934 149 197 824;
  • 26) 0.000 000 001 043 081 283 569 335 934 149 197 824 × 2 = 0 + 0.000 000 002 086 162 567 138 671 868 298 395 648;
  • 27) 0.000 000 002 086 162 567 138 671 868 298 395 648 × 2 = 0 + 0.000 000 004 172 325 134 277 343 736 596 791 296;
  • 28) 0.000 000 004 172 325 134 277 343 736 596 791 296 × 2 = 0 + 0.000 000 008 344 650 268 554 687 473 193 582 592;
  • 29) 0.000 000 008 344 650 268 554 687 473 193 582 592 × 2 = 0 + 0.000 000 016 689 300 537 109 374 946 387 165 184;
  • 30) 0.000 000 016 689 300 537 109 374 946 387 165 184 × 2 = 0 + 0.000 000 033 378 601 074 218 749 892 774 330 368;
  • 31) 0.000 000 033 378 601 074 218 749 892 774 330 368 × 2 = 0 + 0.000 000 066 757 202 148 437 499 785 548 660 736;
  • 32) 0.000 000 066 757 202 148 437 499 785 548 660 736 × 2 = 0 + 0.000 000 133 514 404 296 874 999 571 097 321 472;
  • 33) 0.000 000 133 514 404 296 874 999 571 097 321 472 × 2 = 0 + 0.000 000 267 028 808 593 749 999 142 194 642 944;
  • 34) 0.000 000 267 028 808 593 749 999 142 194 642 944 × 2 = 0 + 0.000 000 534 057 617 187 499 998 284 389 285 888;
  • 35) 0.000 000 534 057 617 187 499 998 284 389 285 888 × 2 = 0 + 0.000 001 068 115 234 374 999 996 568 778 571 776;
  • 36) 0.000 001 068 115 234 374 999 996 568 778 571 776 × 2 = 0 + 0.000 002 136 230 468 749 999 993 137 557 143 552;
  • 37) 0.000 002 136 230 468 749 999 993 137 557 143 552 × 2 = 0 + 0.000 004 272 460 937 499 999 986 275 114 287 104;
  • 38) 0.000 004 272 460 937 499 999 986 275 114 287 104 × 2 = 0 + 0.000 008 544 921 874 999 999 972 550 228 574 208;
  • 39) 0.000 008 544 921 874 999 999 972 550 228 574 208 × 2 = 0 + 0.000 017 089 843 749 999 999 945 100 457 148 416;
  • 40) 0.000 017 089 843 749 999 999 945 100 457 148 416 × 2 = 0 + 0.000 034 179 687 499 999 999 890 200 914 296 832;
  • 41) 0.000 034 179 687 499 999 999 890 200 914 296 832 × 2 = 0 + 0.000 068 359 374 999 999 999 780 401 828 593 664;
  • 42) 0.000 068 359 374 999 999 999 780 401 828 593 664 × 2 = 0 + 0.000 136 718 749 999 999 999 560 803 657 187 328;
  • 43) 0.000 136 718 749 999 999 999 560 803 657 187 328 × 2 = 0 + 0.000 273 437 499 999 999 999 121 607 314 374 656;
  • 44) 0.000 273 437 499 999 999 999 121 607 314 374 656 × 2 = 0 + 0.000 546 874 999 999 999 998 243 214 628 749 312;
  • 45) 0.000 546 874 999 999 999 998 243 214 628 749 312 × 2 = 0 + 0.001 093 749 999 999 999 996 486 429 257 498 624;
  • 46) 0.001 093 749 999 999 999 996 486 429 257 498 624 × 2 = 0 + 0.002 187 499 999 999 999 992 972 858 514 997 248;
  • 47) 0.002 187 499 999 999 999 992 972 858 514 997 248 × 2 = 0 + 0.004 374 999 999 999 999 985 945 717 029 994 496;
  • 48) 0.004 374 999 999 999 999 985 945 717 029 994 496 × 2 = 0 + 0.008 749 999 999 999 999 971 891 434 059 988 992;
  • 49) 0.008 749 999 999 999 999 971 891 434 059 988 992 × 2 = 0 + 0.017 499 999 999 999 999 943 782 868 119 977 984;
  • 50) 0.017 499 999 999 999 999 943 782 868 119 977 984 × 2 = 0 + 0.034 999 999 999 999 999 887 565 736 239 955 968;
  • 51) 0.034 999 999 999 999 999 887 565 736 239 955 968 × 2 = 0 + 0.069 999 999 999 999 999 775 131 472 479 911 936;
  • 52) 0.069 999 999 999 999 999 775 131 472 479 911 936 × 2 = 0 + 0.139 999 999 999 999 999 550 262 944 959 823 872;
  • 53) 0.139 999 999 999 999 999 550 262 944 959 823 872 × 2 = 0 + 0.279 999 999 999 999 999 100 525 889 919 647 744;
  • 54) 0.279 999 999 999 999 999 100 525 889 919 647 744 × 2 = 0 + 0.559 999 999 999 999 998 201 051 779 839 295 488;
  • 55) 0.559 999 999 999 999 998 201 051 779 839 295 488 × 2 = 1 + 0.119 999 999 999 999 996 402 103 559 678 590 976;
  • 56) 0.119 999 999 999 999 996 402 103 559 678 590 976 × 2 = 0 + 0.239 999 999 999 999 992 804 207 119 357 181 952;
  • 57) 0.239 999 999 999 999 992 804 207 119 357 181 952 × 2 = 0 + 0.479 999 999 999 999 985 608 414 238 714 363 904;
  • 58) 0.479 999 999 999 999 985 608 414 238 714 363 904 × 2 = 0 + 0.959 999 999 999 999 971 216 828 477 428 727 808;

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:


0.016 738 891 601 562 531 086 244 689 504 383 032(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 531 086 244 689 504 383 032(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:


0.016 738 891 601 562 531 086 244 689 504 383 032(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 -0.016 738 891 601 562 531 086 244 689 504 383 032 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