1.745 459 324 169 999 826 281 889 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 281 889(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
1.745 459 324 169 999 826 281 889(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: 0.745 459 324 169 999 826 281 889.

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) 0.745 459 324 169 999 826 281 889 × 2 = 1 + 0.490 918 648 339 999 652 563 778;
  • 2) 0.490 918 648 339 999 652 563 778 × 2 = 0 + 0.981 837 296 679 999 305 127 556;
  • 3) 0.981 837 296 679 999 305 127 556 × 2 = 1 + 0.963 674 593 359 998 610 255 112;
  • 4) 0.963 674 593 359 998 610 255 112 × 2 = 1 + 0.927 349 186 719 997 220 510 224;
  • 5) 0.927 349 186 719 997 220 510 224 × 2 = 1 + 0.854 698 373 439 994 441 020 448;
  • 6) 0.854 698 373 439 994 441 020 448 × 2 = 1 + 0.709 396 746 879 988 882 040 896;
  • 7) 0.709 396 746 879 988 882 040 896 × 2 = 1 + 0.418 793 493 759 977 764 081 792;
  • 8) 0.418 793 493 759 977 764 081 792 × 2 = 0 + 0.837 586 987 519 955 528 163 584;
  • 9) 0.837 586 987 519 955 528 163 584 × 2 = 1 + 0.675 173 975 039 911 056 327 168;
  • 10) 0.675 173 975 039 911 056 327 168 × 2 = 1 + 0.350 347 950 079 822 112 654 336;
  • 11) 0.350 347 950 079 822 112 654 336 × 2 = 0 + 0.700 695 900 159 644 225 308 672;
  • 12) 0.700 695 900 159 644 225 308 672 × 2 = 1 + 0.401 391 800 319 288 450 617 344;
  • 13) 0.401 391 800 319 288 450 617 344 × 2 = 0 + 0.802 783 600 638 576 901 234 688;
  • 14) 0.802 783 600 638 576 901 234 688 × 2 = 1 + 0.605 567 201 277 153 802 469 376;
  • 15) 0.605 567 201 277 153 802 469 376 × 2 = 1 + 0.211 134 402 554 307 604 938 752;
  • 16) 0.211 134 402 554 307 604 938 752 × 2 = 0 + 0.422 268 805 108 615 209 877 504;
  • 17) 0.422 268 805 108 615 209 877 504 × 2 = 0 + 0.844 537 610 217 230 419 755 008;
  • 18) 0.844 537 610 217 230 419 755 008 × 2 = 1 + 0.689 075 220 434 460 839 510 016;
  • 19) 0.689 075 220 434 460 839 510 016 × 2 = 1 + 0.378 150 440 868 921 679 020 032;
  • 20) 0.378 150 440 868 921 679 020 032 × 2 = 0 + 0.756 300 881 737 843 358 040 064;
  • 21) 0.756 300 881 737 843 358 040 064 × 2 = 1 + 0.512 601 763 475 686 716 080 128;
  • 22) 0.512 601 763 475 686 716 080 128 × 2 = 1 + 0.025 203 526 951 373 432 160 256;
  • 23) 0.025 203 526 951 373 432 160 256 × 2 = 0 + 0.050 407 053 902 746 864 320 512;
  • 24) 0.050 407 053 902 746 864 320 512 × 2 = 0 + 0.100 814 107 805 493 728 641 024;
  • 25) 0.100 814 107 805 493 728 641 024 × 2 = 0 + 0.201 628 215 610 987 457 282 048;
  • 26) 0.201 628 215 610 987 457 282 048 × 2 = 0 + 0.403 256 431 221 974 914 564 096;
  • 27) 0.403 256 431 221 974 914 564 096 × 2 = 0 + 0.806 512 862 443 949 829 128 192;
  • 28) 0.806 512 862 443 949 829 128 192 × 2 = 1 + 0.613 025 724 887 899 658 256 384;
  • 29) 0.613 025 724 887 899 658 256 384 × 2 = 1 + 0.226 051 449 775 799 316 512 768;
  • 30) 0.226 051 449 775 799 316 512 768 × 2 = 0 + 0.452 102 899 551 598 633 025 536;
  • 31) 0.452 102 899 551 598 633 025 536 × 2 = 0 + 0.904 205 799 103 197 266 051 072;
  • 32) 0.904 205 799 103 197 266 051 072 × 2 = 1 + 0.808 411 598 206 394 532 102 144;
  • 33) 0.808 411 598 206 394 532 102 144 × 2 = 1 + 0.616 823 196 412 789 064 204 288;
  • 34) 0.616 823 196 412 789 064 204 288 × 2 = 1 + 0.233 646 392 825 578 128 408 576;
  • 35) 0.233 646 392 825 578 128 408 576 × 2 = 0 + 0.467 292 785 651 156 256 817 152;
  • 36) 0.467 292 785 651 156 256 817 152 × 2 = 0 + 0.934 585 571 302 312 513 634 304;
  • 37) 0.934 585 571 302 312 513 634 304 × 2 = 1 + 0.869 171 142 604 625 027 268 608;
  • 38) 0.869 171 142 604 625 027 268 608 × 2 = 1 + 0.738 342 285 209 250 054 537 216;
  • 39) 0.738 342 285 209 250 054 537 216 × 2 = 1 + 0.476 684 570 418 500 109 074 432;
  • 40) 0.476 684 570 418 500 109 074 432 × 2 = 0 + 0.953 369 140 837 000 218 148 864;
  • 41) 0.953 369 140 837 000 218 148 864 × 2 = 1 + 0.906 738 281 674 000 436 297 728;
  • 42) 0.906 738 281 674 000 436 297 728 × 2 = 1 + 0.813 476 563 348 000 872 595 456;
  • 43) 0.813 476 563 348 000 872 595 456 × 2 = 1 + 0.626 953 126 696 001 745 190 912;
  • 44) 0.626 953 126 696 001 745 190 912 × 2 = 1 + 0.253 906 253 392 003 490 381 824;
  • 45) 0.253 906 253 392 003 490 381 824 × 2 = 0 + 0.507 812 506 784 006 980 763 648;
  • 46) 0.507 812 506 784 006 980 763 648 × 2 = 1 + 0.015 625 013 568 013 961 527 296;
  • 47) 0.015 625 013 568 013 961 527 296 × 2 = 0 + 0.031 250 027 136 027 923 054 592;
  • 48) 0.031 250 027 136 027 923 054 592 × 2 = 0 + 0.062 500 054 272 055 846 109 184;
  • 49) 0.062 500 054 272 055 846 109 184 × 2 = 0 + 0.125 000 108 544 111 692 218 368;
  • 50) 0.125 000 108 544 111 692 218 368 × 2 = 0 + 0.250 000 217 088 223 384 436 736;
  • 51) 0.250 000 217 088 223 384 436 736 × 2 = 0 + 0.500 000 434 176 446 768 873 472;
  • 52) 0.500 000 434 176 446 768 873 472 × 2 = 1 + 0.000 000 868 352 893 537 746 944;
  • 53) 0.000 000 868 352 893 537 746 944 × 2 = 0 + 0.000 001 736 705 787 075 493 888;

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:


0.745 459 324 169 999 826 281 889(10) =

0.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0001 0(2)

5. Positive number before normalization:

1.745 459 324 169 999 826 281 889(10) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0001 0(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:


1.745 459 324 169 999 826 281 889(10) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0001 0(2) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0001 0(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 0001 0


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 0001 0 =

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


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 0001


Decimal number 1.745 459 324 169 999 826 281 889 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 0001

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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