0.000 000 000 000 079 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 079(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
0.000 000 000 000 079(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. 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;

2. 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)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 000 079.

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) 0.000 000 000 000 079 × 2 = 0 + 0.000 000 000 000 158;
  • 2) 0.000 000 000 000 158 × 2 = 0 + 0.000 000 000 000 316;
  • 3) 0.000 000 000 000 316 × 2 = 0 + 0.000 000 000 000 632;
  • 4) 0.000 000 000 000 632 × 2 = 0 + 0.000 000 000 001 264;
  • 5) 0.000 000 000 001 264 × 2 = 0 + 0.000 000 000 002 528;
  • 6) 0.000 000 000 002 528 × 2 = 0 + 0.000 000 000 005 056;
  • 7) 0.000 000 000 005 056 × 2 = 0 + 0.000 000 000 010 112;
  • 8) 0.000 000 000 010 112 × 2 = 0 + 0.000 000 000 020 224;
  • 9) 0.000 000 000 020 224 × 2 = 0 + 0.000 000 000 040 448;
  • 10) 0.000 000 000 040 448 × 2 = 0 + 0.000 000 000 080 896;
  • 11) 0.000 000 000 080 896 × 2 = 0 + 0.000 000 000 161 792;
  • 12) 0.000 000 000 161 792 × 2 = 0 + 0.000 000 000 323 584;
  • 13) 0.000 000 000 323 584 × 2 = 0 + 0.000 000 000 647 168;
  • 14) 0.000 000 000 647 168 × 2 = 0 + 0.000 000 001 294 336;
  • 15) 0.000 000 001 294 336 × 2 = 0 + 0.000 000 002 588 672;
  • 16) 0.000 000 002 588 672 × 2 = 0 + 0.000 000 005 177 344;
  • 17) 0.000 000 005 177 344 × 2 = 0 + 0.000 000 010 354 688;
  • 18) 0.000 000 010 354 688 × 2 = 0 + 0.000 000 020 709 376;
  • 19) 0.000 000 020 709 376 × 2 = 0 + 0.000 000 041 418 752;
  • 20) 0.000 000 041 418 752 × 2 = 0 + 0.000 000 082 837 504;
  • 21) 0.000 000 082 837 504 × 2 = 0 + 0.000 000 165 675 008;
  • 22) 0.000 000 165 675 008 × 2 = 0 + 0.000 000 331 350 016;
  • 23) 0.000 000 331 350 016 × 2 = 0 + 0.000 000 662 700 032;
  • 24) 0.000 000 662 700 032 × 2 = 0 + 0.000 001 325 400 064;
  • 25) 0.000 001 325 400 064 × 2 = 0 + 0.000 002 650 800 128;
  • 26) 0.000 002 650 800 128 × 2 = 0 + 0.000 005 301 600 256;
  • 27) 0.000 005 301 600 256 × 2 = 0 + 0.000 010 603 200 512;
  • 28) 0.000 010 603 200 512 × 2 = 0 + 0.000 021 206 401 024;
  • 29) 0.000 021 206 401 024 × 2 = 0 + 0.000 042 412 802 048;
  • 30) 0.000 042 412 802 048 × 2 = 0 + 0.000 084 825 604 096;
  • 31) 0.000 084 825 604 096 × 2 = 0 + 0.000 169 651 208 192;
  • 32) 0.000 169 651 208 192 × 2 = 0 + 0.000 339 302 416 384;
  • 33) 0.000 339 302 416 384 × 2 = 0 + 0.000 678 604 832 768;
  • 34) 0.000 678 604 832 768 × 2 = 0 + 0.001 357 209 665 536;
  • 35) 0.001 357 209 665 536 × 2 = 0 + 0.002 714 419 331 072;
  • 36) 0.002 714 419 331 072 × 2 = 0 + 0.005 428 838 662 144;
  • 37) 0.005 428 838 662 144 × 2 = 0 + 0.010 857 677 324 288;
  • 38) 0.010 857 677 324 288 × 2 = 0 + 0.021 715 354 648 576;
  • 39) 0.021 715 354 648 576 × 2 = 0 + 0.043 430 709 297 152;
  • 40) 0.043 430 709 297 152 × 2 = 0 + 0.086 861 418 594 304;
  • 41) 0.086 861 418 594 304 × 2 = 0 + 0.173 722 837 188 608;
  • 42) 0.173 722 837 188 608 × 2 = 0 + 0.347 445 674 377 216;
  • 43) 0.347 445 674 377 216 × 2 = 0 + 0.694 891 348 754 432;
  • 44) 0.694 891 348 754 432 × 2 = 1 + 0.389 782 697 508 864;
  • 45) 0.389 782 697 508 864 × 2 = 0 + 0.779 565 395 017 728;
  • 46) 0.779 565 395 017 728 × 2 = 1 + 0.559 130 790 035 456;
  • 47) 0.559 130 790 035 456 × 2 = 1 + 0.118 261 580 070 912;
  • 48) 0.118 261 580 070 912 × 2 = 0 + 0.236 523 160 141 824;
  • 49) 0.236 523 160 141 824 × 2 = 0 + 0.473 046 320 283 648;
  • 50) 0.473 046 320 283 648 × 2 = 0 + 0.946 092 640 567 296;
  • 51) 0.946 092 640 567 296 × 2 = 1 + 0.892 185 281 134 592;
  • 52) 0.892 185 281 134 592 × 2 = 1 + 0.784 370 562 269 184;
  • 53) 0.784 370 562 269 184 × 2 = 1 + 0.568 741 124 538 368;
  • 54) 0.568 741 124 538 368 × 2 = 1 + 0.137 482 249 076 736;
  • 55) 0.137 482 249 076 736 × 2 = 0 + 0.274 964 498 153 472;
  • 56) 0.274 964 498 153 472 × 2 = 0 + 0.549 928 996 306 944;
  • 57) 0.549 928 996 306 944 × 2 = 1 + 0.099 857 992 613 888;
  • 58) 0.099 857 992 613 888 × 2 = 0 + 0.199 715 985 227 776;
  • 59) 0.199 715 985 227 776 × 2 = 0 + 0.399 431 970 455 552;
  • 60) 0.399 431 970 455 552 × 2 = 0 + 0.798 863 940 911 104;
  • 61) 0.798 863 940 911 104 × 2 = 1 + 0.597 727 881 822 208;
  • 62) 0.597 727 881 822 208 × 2 = 1 + 0.195 455 763 644 416;
  • 63) 0.195 455 763 644 416 × 2 = 0 + 0.390 911 527 288 832;
  • 64) 0.390 911 527 288 832 × 2 = 0 + 0.781 823 054 577 664;
  • 65) 0.781 823 054 577 664 × 2 = 1 + 0.563 646 109 155 328;
  • 66) 0.563 646 109 155 328 × 2 = 1 + 0.127 292 218 310 656;
  • 67) 0.127 292 218 310 656 × 2 = 0 + 0.254 584 436 621 312;
  • 68) 0.254 584 436 621 312 × 2 = 0 + 0.509 168 873 242 624;
  • 69) 0.509 168 873 242 624 × 2 = 1 + 0.018 337 746 485 248;
  • 70) 0.018 337 746 485 248 × 2 = 0 + 0.036 675 492 970 496;
  • 71) 0.036 675 492 970 496 × 2 = 0 + 0.073 350 985 940 992;
  • 72) 0.073 350 985 940 992 × 2 = 0 + 0.146 701 971 881 984;
  • 73) 0.146 701 971 881 984 × 2 = 0 + 0.293 403 943 763 968;
  • 74) 0.293 403 943 763 968 × 2 = 0 + 0.586 807 887 527 936;
  • 75) 0.586 807 887 527 936 × 2 = 1 + 0.173 615 775 055 872;
  • 76) 0.173 615 775 055 872 × 2 = 0 + 0.347 231 550 111 744;
  • 77) 0.347 231 550 111 744 × 2 = 0 + 0.694 463 100 223 488;
  • 78) 0.694 463 100 223 488 × 2 = 1 + 0.388 926 200 446 976;
  • 79) 0.388 926 200 446 976 × 2 = 0 + 0.777 852 400 893 952;
  • 80) 0.777 852 400 893 952 × 2 = 1 + 0.555 704 801 787 904;
  • 81) 0.555 704 801 787 904 × 2 = 1 + 0.111 409 603 575 808;
  • 82) 0.111 409 603 575 808 × 2 = 0 + 0.222 819 207 151 616;
  • 83) 0.222 819 207 151 616 × 2 = 0 + 0.445 638 414 303 232;
  • 84) 0.445 638 414 303 232 × 2 = 0 + 0.891 276 828 606 464;
  • 85) 0.891 276 828 606 464 × 2 = 1 + 0.782 553 657 212 928;
  • 86) 0.782 553 657 212 928 × 2 = 1 + 0.565 107 314 425 856;
  • 87) 0.565 107 314 425 856 × 2 = 1 + 0.130 214 628 851 712;
  • 88) 0.130 214 628 851 712 × 2 = 0 + 0.260 429 257 703 424;
  • 89) 0.260 429 257 703 424 × 2 = 0 + 0.520 858 515 406 848;
  • 90) 0.520 858 515 406 848 × 2 = 1 + 0.041 717 030 813 696;
  • 91) 0.041 717 030 813 696 × 2 = 0 + 0.083 434 061 627 392;
  • 92) 0.083 434 061 627 392 × 2 = 0 + 0.166 868 123 254 784;
  • 93) 0.166 868 123 254 784 × 2 = 0 + 0.333 736 246 509 568;
  • 94) 0.333 736 246 509 568 × 2 = 0 + 0.667 472 493 019 136;
  • 95) 0.667 472 493 019 136 × 2 = 1 + 0.334 944 986 038 272;
  • 96) 0.334 944 986 038 272 × 2 = 0 + 0.669 889 972 076 544;

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).


4. 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:


0.000 000 000 000 079(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0110 0011 1100 1000 1100 1100 1000 0010 0101 1000 1110 0100 0010(2)

5. Positive number before normalization:

0.000 000 000 000 079(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0110 0011 1100 1000 1100 1100 1000 0010 0101 1000 1110 0100 0010(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 44 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 079(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0110 0011 1100 1000 1100 1100 1000 0010 0101 1000 1110 0100 0010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0110 0011 1100 1000 1100 1100 1000 0010 0101 1000 1110 0100 0010(2) × 20 =

1.0110 0011 1100 1000 1100 1100 1000 0010 0101 1000 1110 0100 0010(2) × 2-44


7. 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 0 (a positive number)

Exponent (unadjusted): -44

Mantissa (not normalized):
1.0110 0011 1100 1000 1100 1100 1000 0010 0101 1000 1110 0100 0010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-44 + 2(11-1) - 1 =

(-44 + 1 023)(10) =

979(10)


9. 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;
  • 979 ÷ 2 = 489 + 1;
  • 489 ÷ 2 = 244 + 1;
  • 244 ÷ 2 = 122 + 0;
  • 122 ÷ 2 = 61 + 0;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

979(10) =

011 1101 0011(2)


11. 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. 0110 0011 1100 1000 1100 1100 1000 0010 0101 1000 1110 0100 0010 =

0110 0011 1100 1000 1100 1100 1000 0010 0101 1000 1110 0100 0010


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1101 0011

Mantissa (52 bits) =
0110 0011 1100 1000 1100 1100 1000 0010 0101 1000 1110 0100 0010


Decimal number 0.000 000 000 000 079 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 0011 - 0110 0011 1100 1000 1100 1100 1000 0010 0101 1000 1110 0100 0010

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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