1.745 459 324 169 999 826 281 691 83 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 281 691 83(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
1.745 459 324 169 999 826 281 691 83(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: 0.745 459 324 169 999 826 281 691 83.

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) 0.745 459 324 169 999 826 281 691 83 × 2 = 1 + 0.490 918 648 339 999 652 563 383 66;
  • 2) 0.490 918 648 339 999 652 563 383 66 × 2 = 0 + 0.981 837 296 679 999 305 126 767 32;
  • 3) 0.981 837 296 679 999 305 126 767 32 × 2 = 1 + 0.963 674 593 359 998 610 253 534 64;
  • 4) 0.963 674 593 359 998 610 253 534 64 × 2 = 1 + 0.927 349 186 719 997 220 507 069 28;
  • 5) 0.927 349 186 719 997 220 507 069 28 × 2 = 1 + 0.854 698 373 439 994 441 014 138 56;
  • 6) 0.854 698 373 439 994 441 014 138 56 × 2 = 1 + 0.709 396 746 879 988 882 028 277 12;
  • 7) 0.709 396 746 879 988 882 028 277 12 × 2 = 1 + 0.418 793 493 759 977 764 056 554 24;
  • 8) 0.418 793 493 759 977 764 056 554 24 × 2 = 0 + 0.837 586 987 519 955 528 113 108 48;
  • 9) 0.837 586 987 519 955 528 113 108 48 × 2 = 1 + 0.675 173 975 039 911 056 226 216 96;
  • 10) 0.675 173 975 039 911 056 226 216 96 × 2 = 1 + 0.350 347 950 079 822 112 452 433 92;
  • 11) 0.350 347 950 079 822 112 452 433 92 × 2 = 0 + 0.700 695 900 159 644 224 904 867 84;
  • 12) 0.700 695 900 159 644 224 904 867 84 × 2 = 1 + 0.401 391 800 319 288 449 809 735 68;
  • 13) 0.401 391 800 319 288 449 809 735 68 × 2 = 0 + 0.802 783 600 638 576 899 619 471 36;
  • 14) 0.802 783 600 638 576 899 619 471 36 × 2 = 1 + 0.605 567 201 277 153 799 238 942 72;
  • 15) 0.605 567 201 277 153 799 238 942 72 × 2 = 1 + 0.211 134 402 554 307 598 477 885 44;
  • 16) 0.211 134 402 554 307 598 477 885 44 × 2 = 0 + 0.422 268 805 108 615 196 955 770 88;
  • 17) 0.422 268 805 108 615 196 955 770 88 × 2 = 0 + 0.844 537 610 217 230 393 911 541 76;
  • 18) 0.844 537 610 217 230 393 911 541 76 × 2 = 1 + 0.689 075 220 434 460 787 823 083 52;
  • 19) 0.689 075 220 434 460 787 823 083 52 × 2 = 1 + 0.378 150 440 868 921 575 646 167 04;
  • 20) 0.378 150 440 868 921 575 646 167 04 × 2 = 0 + 0.756 300 881 737 843 151 292 334 08;
  • 21) 0.756 300 881 737 843 151 292 334 08 × 2 = 1 + 0.512 601 763 475 686 302 584 668 16;
  • 22) 0.512 601 763 475 686 302 584 668 16 × 2 = 1 + 0.025 203 526 951 372 605 169 336 32;
  • 23) 0.025 203 526 951 372 605 169 336 32 × 2 = 0 + 0.050 407 053 902 745 210 338 672 64;
  • 24) 0.050 407 053 902 745 210 338 672 64 × 2 = 0 + 0.100 814 107 805 490 420 677 345 28;
  • 25) 0.100 814 107 805 490 420 677 345 28 × 2 = 0 + 0.201 628 215 610 980 841 354 690 56;
  • 26) 0.201 628 215 610 980 841 354 690 56 × 2 = 0 + 0.403 256 431 221 961 682 709 381 12;
  • 27) 0.403 256 431 221 961 682 709 381 12 × 2 = 0 + 0.806 512 862 443 923 365 418 762 24;
  • 28) 0.806 512 862 443 923 365 418 762 24 × 2 = 1 + 0.613 025 724 887 846 730 837 524 48;
  • 29) 0.613 025 724 887 846 730 837 524 48 × 2 = 1 + 0.226 051 449 775 693 461 675 048 96;
  • 30) 0.226 051 449 775 693 461 675 048 96 × 2 = 0 + 0.452 102 899 551 386 923 350 097 92;
  • 31) 0.452 102 899 551 386 923 350 097 92 × 2 = 0 + 0.904 205 799 102 773 846 700 195 84;
  • 32) 0.904 205 799 102 773 846 700 195 84 × 2 = 1 + 0.808 411 598 205 547 693 400 391 68;
  • 33) 0.808 411 598 205 547 693 400 391 68 × 2 = 1 + 0.616 823 196 411 095 386 800 783 36;
  • 34) 0.616 823 196 411 095 386 800 783 36 × 2 = 1 + 0.233 646 392 822 190 773 601 566 72;
  • 35) 0.233 646 392 822 190 773 601 566 72 × 2 = 0 + 0.467 292 785 644 381 547 203 133 44;
  • 36) 0.467 292 785 644 381 547 203 133 44 × 2 = 0 + 0.934 585 571 288 763 094 406 266 88;
  • 37) 0.934 585 571 288 763 094 406 266 88 × 2 = 1 + 0.869 171 142 577 526 188 812 533 76;
  • 38) 0.869 171 142 577 526 188 812 533 76 × 2 = 1 + 0.738 342 285 155 052 377 625 067 52;
  • 39) 0.738 342 285 155 052 377 625 067 52 × 2 = 1 + 0.476 684 570 310 104 755 250 135 04;
  • 40) 0.476 684 570 310 104 755 250 135 04 × 2 = 0 + 0.953 369 140 620 209 510 500 270 08;
  • 41) 0.953 369 140 620 209 510 500 270 08 × 2 = 1 + 0.906 738 281 240 419 021 000 540 16;
  • 42) 0.906 738 281 240 419 021 000 540 16 × 2 = 1 + 0.813 476 562 480 838 042 001 080 32;
  • 43) 0.813 476 562 480 838 042 001 080 32 × 2 = 1 + 0.626 953 124 961 676 084 002 160 64;
  • 44) 0.626 953 124 961 676 084 002 160 64 × 2 = 1 + 0.253 906 249 923 352 168 004 321 28;
  • 45) 0.253 906 249 923 352 168 004 321 28 × 2 = 0 + 0.507 812 499 846 704 336 008 642 56;
  • 46) 0.507 812 499 846 704 336 008 642 56 × 2 = 1 + 0.015 624 999 693 408 672 017 285 12;
  • 47) 0.015 624 999 693 408 672 017 285 12 × 2 = 0 + 0.031 249 999 386 817 344 034 570 24;
  • 48) 0.031 249 999 386 817 344 034 570 24 × 2 = 0 + 0.062 499 998 773 634 688 069 140 48;
  • 49) 0.062 499 998 773 634 688 069 140 48 × 2 = 0 + 0.124 999 997 547 269 376 138 280 96;
  • 50) 0.124 999 997 547 269 376 138 280 96 × 2 = 0 + 0.249 999 995 094 538 752 276 561 92;
  • 51) 0.249 999 995 094 538 752 276 561 92 × 2 = 0 + 0.499 999 990 189 077 504 553 123 84;
  • 52) 0.499 999 990 189 077 504 553 123 84 × 2 = 0 + 0.999 999 980 378 155 009 106 247 68;
  • 53) 0.999 999 980 378 155 009 106 247 68 × 2 = 1 + 0.999 999 960 756 310 018 212 495 36;

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:


0.745 459 324 169 999 826 281 691 83(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 826 281 691 83(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:


1.745 459 324 169 999 826 281 691 83(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 1.745 459 324 169 999 826 281 691 83 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