1.745 459 324 169 999 826 281 698 67 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 281 698 67(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
1.745 459 324 169 999 826 281 698 67(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: 0.745 459 324 169 999 826 281 698 67.

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) 0.745 459 324 169 999 826 281 698 67 × 2 = 1 + 0.490 918 648 339 999 652 563 397 34;
  • 2) 0.490 918 648 339 999 652 563 397 34 × 2 = 0 + 0.981 837 296 679 999 305 126 794 68;
  • 3) 0.981 837 296 679 999 305 126 794 68 × 2 = 1 + 0.963 674 593 359 998 610 253 589 36;
  • 4) 0.963 674 593 359 998 610 253 589 36 × 2 = 1 + 0.927 349 186 719 997 220 507 178 72;
  • 5) 0.927 349 186 719 997 220 507 178 72 × 2 = 1 + 0.854 698 373 439 994 441 014 357 44;
  • 6) 0.854 698 373 439 994 441 014 357 44 × 2 = 1 + 0.709 396 746 879 988 882 028 714 88;
  • 7) 0.709 396 746 879 988 882 028 714 88 × 2 = 1 + 0.418 793 493 759 977 764 057 429 76;
  • 8) 0.418 793 493 759 977 764 057 429 76 × 2 = 0 + 0.837 586 987 519 955 528 114 859 52;
  • 9) 0.837 586 987 519 955 528 114 859 52 × 2 = 1 + 0.675 173 975 039 911 056 229 719 04;
  • 10) 0.675 173 975 039 911 056 229 719 04 × 2 = 1 + 0.350 347 950 079 822 112 459 438 08;
  • 11) 0.350 347 950 079 822 112 459 438 08 × 2 = 0 + 0.700 695 900 159 644 224 918 876 16;
  • 12) 0.700 695 900 159 644 224 918 876 16 × 2 = 1 + 0.401 391 800 319 288 449 837 752 32;
  • 13) 0.401 391 800 319 288 449 837 752 32 × 2 = 0 + 0.802 783 600 638 576 899 675 504 64;
  • 14) 0.802 783 600 638 576 899 675 504 64 × 2 = 1 + 0.605 567 201 277 153 799 351 009 28;
  • 15) 0.605 567 201 277 153 799 351 009 28 × 2 = 1 + 0.211 134 402 554 307 598 702 018 56;
  • 16) 0.211 134 402 554 307 598 702 018 56 × 2 = 0 + 0.422 268 805 108 615 197 404 037 12;
  • 17) 0.422 268 805 108 615 197 404 037 12 × 2 = 0 + 0.844 537 610 217 230 394 808 074 24;
  • 18) 0.844 537 610 217 230 394 808 074 24 × 2 = 1 + 0.689 075 220 434 460 789 616 148 48;
  • 19) 0.689 075 220 434 460 789 616 148 48 × 2 = 1 + 0.378 150 440 868 921 579 232 296 96;
  • 20) 0.378 150 440 868 921 579 232 296 96 × 2 = 0 + 0.756 300 881 737 843 158 464 593 92;
  • 21) 0.756 300 881 737 843 158 464 593 92 × 2 = 1 + 0.512 601 763 475 686 316 929 187 84;
  • 22) 0.512 601 763 475 686 316 929 187 84 × 2 = 1 + 0.025 203 526 951 372 633 858 375 68;
  • 23) 0.025 203 526 951 372 633 858 375 68 × 2 = 0 + 0.050 407 053 902 745 267 716 751 36;
  • 24) 0.050 407 053 902 745 267 716 751 36 × 2 = 0 + 0.100 814 107 805 490 535 433 502 72;
  • 25) 0.100 814 107 805 490 535 433 502 72 × 2 = 0 + 0.201 628 215 610 981 070 867 005 44;
  • 26) 0.201 628 215 610 981 070 867 005 44 × 2 = 0 + 0.403 256 431 221 962 141 734 010 88;
  • 27) 0.403 256 431 221 962 141 734 010 88 × 2 = 0 + 0.806 512 862 443 924 283 468 021 76;
  • 28) 0.806 512 862 443 924 283 468 021 76 × 2 = 1 + 0.613 025 724 887 848 566 936 043 52;
  • 29) 0.613 025 724 887 848 566 936 043 52 × 2 = 1 + 0.226 051 449 775 697 133 872 087 04;
  • 30) 0.226 051 449 775 697 133 872 087 04 × 2 = 0 + 0.452 102 899 551 394 267 744 174 08;
  • 31) 0.452 102 899 551 394 267 744 174 08 × 2 = 0 + 0.904 205 799 102 788 535 488 348 16;
  • 32) 0.904 205 799 102 788 535 488 348 16 × 2 = 1 + 0.808 411 598 205 577 070 976 696 32;
  • 33) 0.808 411 598 205 577 070 976 696 32 × 2 = 1 + 0.616 823 196 411 154 141 953 392 64;
  • 34) 0.616 823 196 411 154 141 953 392 64 × 2 = 1 + 0.233 646 392 822 308 283 906 785 28;
  • 35) 0.233 646 392 822 308 283 906 785 28 × 2 = 0 + 0.467 292 785 644 616 567 813 570 56;
  • 36) 0.467 292 785 644 616 567 813 570 56 × 2 = 0 + 0.934 585 571 289 233 135 627 141 12;
  • 37) 0.934 585 571 289 233 135 627 141 12 × 2 = 1 + 0.869 171 142 578 466 271 254 282 24;
  • 38) 0.869 171 142 578 466 271 254 282 24 × 2 = 1 + 0.738 342 285 156 932 542 508 564 48;
  • 39) 0.738 342 285 156 932 542 508 564 48 × 2 = 1 + 0.476 684 570 313 865 085 017 128 96;
  • 40) 0.476 684 570 313 865 085 017 128 96 × 2 = 0 + 0.953 369 140 627 730 170 034 257 92;
  • 41) 0.953 369 140 627 730 170 034 257 92 × 2 = 1 + 0.906 738 281 255 460 340 068 515 84;
  • 42) 0.906 738 281 255 460 340 068 515 84 × 2 = 1 + 0.813 476 562 510 920 680 137 031 68;
  • 43) 0.813 476 562 510 920 680 137 031 68 × 2 = 1 + 0.626 953 125 021 841 360 274 063 36;
  • 44) 0.626 953 125 021 841 360 274 063 36 × 2 = 1 + 0.253 906 250 043 682 720 548 126 72;
  • 45) 0.253 906 250 043 682 720 548 126 72 × 2 = 0 + 0.507 812 500 087 365 441 096 253 44;
  • 46) 0.507 812 500 087 365 441 096 253 44 × 2 = 1 + 0.015 625 000 174 730 882 192 506 88;
  • 47) 0.015 625 000 174 730 882 192 506 88 × 2 = 0 + 0.031 250 000 349 461 764 385 013 76;
  • 48) 0.031 250 000 349 461 764 385 013 76 × 2 = 0 + 0.062 500 000 698 923 528 770 027 52;
  • 49) 0.062 500 000 698 923 528 770 027 52 × 2 = 0 + 0.125 000 001 397 847 057 540 055 04;
  • 50) 0.125 000 001 397 847 057 540 055 04 × 2 = 0 + 0.250 000 002 795 694 115 080 110 08;
  • 51) 0.250 000 002 795 694 115 080 110 08 × 2 = 0 + 0.500 000 005 591 388 230 160 220 16;
  • 52) 0.500 000 005 591 388 230 160 220 16 × 2 = 1 + 0.000 000 011 182 776 460 320 440 32;
  • 53) 0.000 000 011 182 776 460 320 440 32 × 2 = 0 + 0.000 000 022 365 552 920 640 880 64;

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:


0.745 459 324 169 999 826 281 698 67(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 826 281 698 67(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:


1.745 459 324 169 999 826 281 698 67(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 1.745 459 324 169 999 826 281 698 67 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