1.745 459 324 169 999 826 267 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 267(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
1.745 459 324 169 999 826 267(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: 1.
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;
  • 1 ÷ 2 = 0 + 1;

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.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.745 459 324 169 999 826 267.

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) 0.745 459 324 169 999 826 267 × 2 = 1 + 0.490 918 648 339 999 652 534;
  • 2) 0.490 918 648 339 999 652 534 × 2 = 0 + 0.981 837 296 679 999 305 068;
  • 3) 0.981 837 296 679 999 305 068 × 2 = 1 + 0.963 674 593 359 998 610 136;
  • 4) 0.963 674 593 359 998 610 136 × 2 = 1 + 0.927 349 186 719 997 220 272;
  • 5) 0.927 349 186 719 997 220 272 × 2 = 1 + 0.854 698 373 439 994 440 544;
  • 6) 0.854 698 373 439 994 440 544 × 2 = 1 + 0.709 396 746 879 988 881 088;
  • 7) 0.709 396 746 879 988 881 088 × 2 = 1 + 0.418 793 493 759 977 762 176;
  • 8) 0.418 793 493 759 977 762 176 × 2 = 0 + 0.837 586 987 519 955 524 352;
  • 9) 0.837 586 987 519 955 524 352 × 2 = 1 + 0.675 173 975 039 911 048 704;
  • 10) 0.675 173 975 039 911 048 704 × 2 = 1 + 0.350 347 950 079 822 097 408;
  • 11) 0.350 347 950 079 822 097 408 × 2 = 0 + 0.700 695 900 159 644 194 816;
  • 12) 0.700 695 900 159 644 194 816 × 2 = 1 + 0.401 391 800 319 288 389 632;
  • 13) 0.401 391 800 319 288 389 632 × 2 = 0 + 0.802 783 600 638 576 779 264;
  • 14) 0.802 783 600 638 576 779 264 × 2 = 1 + 0.605 567 201 277 153 558 528;
  • 15) 0.605 567 201 277 153 558 528 × 2 = 1 + 0.211 134 402 554 307 117 056;
  • 16) 0.211 134 402 554 307 117 056 × 2 = 0 + 0.422 268 805 108 614 234 112;
  • 17) 0.422 268 805 108 614 234 112 × 2 = 0 + 0.844 537 610 217 228 468 224;
  • 18) 0.844 537 610 217 228 468 224 × 2 = 1 + 0.689 075 220 434 456 936 448;
  • 19) 0.689 075 220 434 456 936 448 × 2 = 1 + 0.378 150 440 868 913 872 896;
  • 20) 0.378 150 440 868 913 872 896 × 2 = 0 + 0.756 300 881 737 827 745 792;
  • 21) 0.756 300 881 737 827 745 792 × 2 = 1 + 0.512 601 763 475 655 491 584;
  • 22) 0.512 601 763 475 655 491 584 × 2 = 1 + 0.025 203 526 951 310 983 168;
  • 23) 0.025 203 526 951 310 983 168 × 2 = 0 + 0.050 407 053 902 621 966 336;
  • 24) 0.050 407 053 902 621 966 336 × 2 = 0 + 0.100 814 107 805 243 932 672;
  • 25) 0.100 814 107 805 243 932 672 × 2 = 0 + 0.201 628 215 610 487 865 344;
  • 26) 0.201 628 215 610 487 865 344 × 2 = 0 + 0.403 256 431 220 975 730 688;
  • 27) 0.403 256 431 220 975 730 688 × 2 = 0 + 0.806 512 862 441 951 461 376;
  • 28) 0.806 512 862 441 951 461 376 × 2 = 1 + 0.613 025 724 883 902 922 752;
  • 29) 0.613 025 724 883 902 922 752 × 2 = 1 + 0.226 051 449 767 805 845 504;
  • 30) 0.226 051 449 767 805 845 504 × 2 = 0 + 0.452 102 899 535 611 691 008;
  • 31) 0.452 102 899 535 611 691 008 × 2 = 0 + 0.904 205 799 071 223 382 016;
  • 32) 0.904 205 799 071 223 382 016 × 2 = 1 + 0.808 411 598 142 446 764 032;
  • 33) 0.808 411 598 142 446 764 032 × 2 = 1 + 0.616 823 196 284 893 528 064;
  • 34) 0.616 823 196 284 893 528 064 × 2 = 1 + 0.233 646 392 569 787 056 128;
  • 35) 0.233 646 392 569 787 056 128 × 2 = 0 + 0.467 292 785 139 574 112 256;
  • 36) 0.467 292 785 139 574 112 256 × 2 = 0 + 0.934 585 570 279 148 224 512;
  • 37) 0.934 585 570 279 148 224 512 × 2 = 1 + 0.869 171 140 558 296 449 024;
  • 38) 0.869 171 140 558 296 449 024 × 2 = 1 + 0.738 342 281 116 592 898 048;
  • 39) 0.738 342 281 116 592 898 048 × 2 = 1 + 0.476 684 562 233 185 796 096;
  • 40) 0.476 684 562 233 185 796 096 × 2 = 0 + 0.953 369 124 466 371 592 192;
  • 41) 0.953 369 124 466 371 592 192 × 2 = 1 + 0.906 738 248 932 743 184 384;
  • 42) 0.906 738 248 932 743 184 384 × 2 = 1 + 0.813 476 497 865 486 368 768;
  • 43) 0.813 476 497 865 486 368 768 × 2 = 1 + 0.626 952 995 730 972 737 536;
  • 44) 0.626 952 995 730 972 737 536 × 2 = 1 + 0.253 905 991 461 945 475 072;
  • 45) 0.253 905 991 461 945 475 072 × 2 = 0 + 0.507 811 982 923 890 950 144;
  • 46) 0.507 811 982 923 890 950 144 × 2 = 1 + 0.015 623 965 847 781 900 288;
  • 47) 0.015 623 965 847 781 900 288 × 2 = 0 + 0.031 247 931 695 563 800 576;
  • 48) 0.031 247 931 695 563 800 576 × 2 = 0 + 0.062 495 863 391 127 601 152;
  • 49) 0.062 495 863 391 127 601 152 × 2 = 0 + 0.124 991 726 782 255 202 304;
  • 50) 0.124 991 726 782 255 202 304 × 2 = 0 + 0.249 983 453 564 510 404 608;
  • 51) 0.249 983 453 564 510 404 608 × 2 = 0 + 0.499 966 907 129 020 809 216;
  • 52) 0.499 966 907 129 020 809 216 × 2 = 0 + 0.999 933 814 258 041 618 432;
  • 53) 0.999 933 814 258 041 618 432 × 2 = 1 + 0.999 867 628 516 083 236 864;

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:


0.745 459 324 169 999 826 267(10) =

0.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1(2)

5. Positive number before normalization:

1.745 459 324 169 999 826 267(10) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.745 459 324 169 999 826 267(10) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1(2) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1(2) × 20


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): 0

Mantissa (not normalized):
1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

0 + 2(11-1) - 1 =

(0 + 1 023)(10) =

1 023(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;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 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;

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) =

1023(10) =

011 1111 1111(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1 =

1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000


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 1111 1111

Mantissa (52 bits) =
1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000


Decimal number 1.745 459 324 169 999 826 267 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000

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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