1.745 459 324 169 999 826 281 696 083 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 281 696 083(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
1.745 459 324 169 999 826 281 696 083(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: 0.745 459 324 169 999 826 281 696 083.

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) 0.745 459 324 169 999 826 281 696 083 × 2 = 1 + 0.490 918 648 339 999 652 563 392 166;
  • 2) 0.490 918 648 339 999 652 563 392 166 × 2 = 0 + 0.981 837 296 679 999 305 126 784 332;
  • 3) 0.981 837 296 679 999 305 126 784 332 × 2 = 1 + 0.963 674 593 359 998 610 253 568 664;
  • 4) 0.963 674 593 359 998 610 253 568 664 × 2 = 1 + 0.927 349 186 719 997 220 507 137 328;
  • 5) 0.927 349 186 719 997 220 507 137 328 × 2 = 1 + 0.854 698 373 439 994 441 014 274 656;
  • 6) 0.854 698 373 439 994 441 014 274 656 × 2 = 1 + 0.709 396 746 879 988 882 028 549 312;
  • 7) 0.709 396 746 879 988 882 028 549 312 × 2 = 1 + 0.418 793 493 759 977 764 057 098 624;
  • 8) 0.418 793 493 759 977 764 057 098 624 × 2 = 0 + 0.837 586 987 519 955 528 114 197 248;
  • 9) 0.837 586 987 519 955 528 114 197 248 × 2 = 1 + 0.675 173 975 039 911 056 228 394 496;
  • 10) 0.675 173 975 039 911 056 228 394 496 × 2 = 1 + 0.350 347 950 079 822 112 456 788 992;
  • 11) 0.350 347 950 079 822 112 456 788 992 × 2 = 0 + 0.700 695 900 159 644 224 913 577 984;
  • 12) 0.700 695 900 159 644 224 913 577 984 × 2 = 1 + 0.401 391 800 319 288 449 827 155 968;
  • 13) 0.401 391 800 319 288 449 827 155 968 × 2 = 0 + 0.802 783 600 638 576 899 654 311 936;
  • 14) 0.802 783 600 638 576 899 654 311 936 × 2 = 1 + 0.605 567 201 277 153 799 308 623 872;
  • 15) 0.605 567 201 277 153 799 308 623 872 × 2 = 1 + 0.211 134 402 554 307 598 617 247 744;
  • 16) 0.211 134 402 554 307 598 617 247 744 × 2 = 0 + 0.422 268 805 108 615 197 234 495 488;
  • 17) 0.422 268 805 108 615 197 234 495 488 × 2 = 0 + 0.844 537 610 217 230 394 468 990 976;
  • 18) 0.844 537 610 217 230 394 468 990 976 × 2 = 1 + 0.689 075 220 434 460 788 937 981 952;
  • 19) 0.689 075 220 434 460 788 937 981 952 × 2 = 1 + 0.378 150 440 868 921 577 875 963 904;
  • 20) 0.378 150 440 868 921 577 875 963 904 × 2 = 0 + 0.756 300 881 737 843 155 751 927 808;
  • 21) 0.756 300 881 737 843 155 751 927 808 × 2 = 1 + 0.512 601 763 475 686 311 503 855 616;
  • 22) 0.512 601 763 475 686 311 503 855 616 × 2 = 1 + 0.025 203 526 951 372 623 007 711 232;
  • 23) 0.025 203 526 951 372 623 007 711 232 × 2 = 0 + 0.050 407 053 902 745 246 015 422 464;
  • 24) 0.050 407 053 902 745 246 015 422 464 × 2 = 0 + 0.100 814 107 805 490 492 030 844 928;
  • 25) 0.100 814 107 805 490 492 030 844 928 × 2 = 0 + 0.201 628 215 610 980 984 061 689 856;
  • 26) 0.201 628 215 610 980 984 061 689 856 × 2 = 0 + 0.403 256 431 221 961 968 123 379 712;
  • 27) 0.403 256 431 221 961 968 123 379 712 × 2 = 0 + 0.806 512 862 443 923 936 246 759 424;
  • 28) 0.806 512 862 443 923 936 246 759 424 × 2 = 1 + 0.613 025 724 887 847 872 493 518 848;
  • 29) 0.613 025 724 887 847 872 493 518 848 × 2 = 1 + 0.226 051 449 775 695 744 987 037 696;
  • 30) 0.226 051 449 775 695 744 987 037 696 × 2 = 0 + 0.452 102 899 551 391 489 974 075 392;
  • 31) 0.452 102 899 551 391 489 974 075 392 × 2 = 0 + 0.904 205 799 102 782 979 948 150 784;
  • 32) 0.904 205 799 102 782 979 948 150 784 × 2 = 1 + 0.808 411 598 205 565 959 896 301 568;
  • 33) 0.808 411 598 205 565 959 896 301 568 × 2 = 1 + 0.616 823 196 411 131 919 792 603 136;
  • 34) 0.616 823 196 411 131 919 792 603 136 × 2 = 1 + 0.233 646 392 822 263 839 585 206 272;
  • 35) 0.233 646 392 822 263 839 585 206 272 × 2 = 0 + 0.467 292 785 644 527 679 170 412 544;
  • 36) 0.467 292 785 644 527 679 170 412 544 × 2 = 0 + 0.934 585 571 289 055 358 340 825 088;
  • 37) 0.934 585 571 289 055 358 340 825 088 × 2 = 1 + 0.869 171 142 578 110 716 681 650 176;
  • 38) 0.869 171 142 578 110 716 681 650 176 × 2 = 1 + 0.738 342 285 156 221 433 363 300 352;
  • 39) 0.738 342 285 156 221 433 363 300 352 × 2 = 1 + 0.476 684 570 312 442 866 726 600 704;
  • 40) 0.476 684 570 312 442 866 726 600 704 × 2 = 0 + 0.953 369 140 624 885 733 453 201 408;
  • 41) 0.953 369 140 624 885 733 453 201 408 × 2 = 1 + 0.906 738 281 249 771 466 906 402 816;
  • 42) 0.906 738 281 249 771 466 906 402 816 × 2 = 1 + 0.813 476 562 499 542 933 812 805 632;
  • 43) 0.813 476 562 499 542 933 812 805 632 × 2 = 1 + 0.626 953 124 999 085 867 625 611 264;
  • 44) 0.626 953 124 999 085 867 625 611 264 × 2 = 1 + 0.253 906 249 998 171 735 251 222 528;
  • 45) 0.253 906 249 998 171 735 251 222 528 × 2 = 0 + 0.507 812 499 996 343 470 502 445 056;
  • 46) 0.507 812 499 996 343 470 502 445 056 × 2 = 1 + 0.015 624 999 992 686 941 004 890 112;
  • 47) 0.015 624 999 992 686 941 004 890 112 × 2 = 0 + 0.031 249 999 985 373 882 009 780 224;
  • 48) 0.031 249 999 985 373 882 009 780 224 × 2 = 0 + 0.062 499 999 970 747 764 019 560 448;
  • 49) 0.062 499 999 970 747 764 019 560 448 × 2 = 0 + 0.124 999 999 941 495 528 039 120 896;
  • 50) 0.124 999 999 941 495 528 039 120 896 × 2 = 0 + 0.249 999 999 882 991 056 078 241 792;
  • 51) 0.249 999 999 882 991 056 078 241 792 × 2 = 0 + 0.499 999 999 765 982 112 156 483 584;
  • 52) 0.499 999 999 765 982 112 156 483 584 × 2 = 0 + 0.999 999 999 531 964 224 312 967 168;
  • 53) 0.999 999 999 531 964 224 312 967 168 × 2 = 1 + 0.999 999 999 063 928 448 625 934 336;

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).


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:


0.745 459 324 169 999 826 281 696 083(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 826 281 696 083(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:


1.745 459 324 169 999 826 281 696 083(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 1.745 459 324 169 999 826 281 696 083 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