1.745 459 324 169 999 826 281 786 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 281 786(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
1.745 459 324 169 999 826 281 786(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: 0.745 459 324 169 999 826 281 786.

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) 0.745 459 324 169 999 826 281 786 × 2 = 1 + 0.490 918 648 339 999 652 563 572;
  • 2) 0.490 918 648 339 999 652 563 572 × 2 = 0 + 0.981 837 296 679 999 305 127 144;
  • 3) 0.981 837 296 679 999 305 127 144 × 2 = 1 + 0.963 674 593 359 998 610 254 288;
  • 4) 0.963 674 593 359 998 610 254 288 × 2 = 1 + 0.927 349 186 719 997 220 508 576;
  • 5) 0.927 349 186 719 997 220 508 576 × 2 = 1 + 0.854 698 373 439 994 441 017 152;
  • 6) 0.854 698 373 439 994 441 017 152 × 2 = 1 + 0.709 396 746 879 988 882 034 304;
  • 7) 0.709 396 746 879 988 882 034 304 × 2 = 1 + 0.418 793 493 759 977 764 068 608;
  • 8) 0.418 793 493 759 977 764 068 608 × 2 = 0 + 0.837 586 987 519 955 528 137 216;
  • 9) 0.837 586 987 519 955 528 137 216 × 2 = 1 + 0.675 173 975 039 911 056 274 432;
  • 10) 0.675 173 975 039 911 056 274 432 × 2 = 1 + 0.350 347 950 079 822 112 548 864;
  • 11) 0.350 347 950 079 822 112 548 864 × 2 = 0 + 0.700 695 900 159 644 225 097 728;
  • 12) 0.700 695 900 159 644 225 097 728 × 2 = 1 + 0.401 391 800 319 288 450 195 456;
  • 13) 0.401 391 800 319 288 450 195 456 × 2 = 0 + 0.802 783 600 638 576 900 390 912;
  • 14) 0.802 783 600 638 576 900 390 912 × 2 = 1 + 0.605 567 201 277 153 800 781 824;
  • 15) 0.605 567 201 277 153 800 781 824 × 2 = 1 + 0.211 134 402 554 307 601 563 648;
  • 16) 0.211 134 402 554 307 601 563 648 × 2 = 0 + 0.422 268 805 108 615 203 127 296;
  • 17) 0.422 268 805 108 615 203 127 296 × 2 = 0 + 0.844 537 610 217 230 406 254 592;
  • 18) 0.844 537 610 217 230 406 254 592 × 2 = 1 + 0.689 075 220 434 460 812 509 184;
  • 19) 0.689 075 220 434 460 812 509 184 × 2 = 1 + 0.378 150 440 868 921 625 018 368;
  • 20) 0.378 150 440 868 921 625 018 368 × 2 = 0 + 0.756 300 881 737 843 250 036 736;
  • 21) 0.756 300 881 737 843 250 036 736 × 2 = 1 + 0.512 601 763 475 686 500 073 472;
  • 22) 0.512 601 763 475 686 500 073 472 × 2 = 1 + 0.025 203 526 951 373 000 146 944;
  • 23) 0.025 203 526 951 373 000 146 944 × 2 = 0 + 0.050 407 053 902 746 000 293 888;
  • 24) 0.050 407 053 902 746 000 293 888 × 2 = 0 + 0.100 814 107 805 492 000 587 776;
  • 25) 0.100 814 107 805 492 000 587 776 × 2 = 0 + 0.201 628 215 610 984 001 175 552;
  • 26) 0.201 628 215 610 984 001 175 552 × 2 = 0 + 0.403 256 431 221 968 002 351 104;
  • 27) 0.403 256 431 221 968 002 351 104 × 2 = 0 + 0.806 512 862 443 936 004 702 208;
  • 28) 0.806 512 862 443 936 004 702 208 × 2 = 1 + 0.613 025 724 887 872 009 404 416;
  • 29) 0.613 025 724 887 872 009 404 416 × 2 = 1 + 0.226 051 449 775 744 018 808 832;
  • 30) 0.226 051 449 775 744 018 808 832 × 2 = 0 + 0.452 102 899 551 488 037 617 664;
  • 31) 0.452 102 899 551 488 037 617 664 × 2 = 0 + 0.904 205 799 102 976 075 235 328;
  • 32) 0.904 205 799 102 976 075 235 328 × 2 = 1 + 0.808 411 598 205 952 150 470 656;
  • 33) 0.808 411 598 205 952 150 470 656 × 2 = 1 + 0.616 823 196 411 904 300 941 312;
  • 34) 0.616 823 196 411 904 300 941 312 × 2 = 1 + 0.233 646 392 823 808 601 882 624;
  • 35) 0.233 646 392 823 808 601 882 624 × 2 = 0 + 0.467 292 785 647 617 203 765 248;
  • 36) 0.467 292 785 647 617 203 765 248 × 2 = 0 + 0.934 585 571 295 234 407 530 496;
  • 37) 0.934 585 571 295 234 407 530 496 × 2 = 1 + 0.869 171 142 590 468 815 060 992;
  • 38) 0.869 171 142 590 468 815 060 992 × 2 = 1 + 0.738 342 285 180 937 630 121 984;
  • 39) 0.738 342 285 180 937 630 121 984 × 2 = 1 + 0.476 684 570 361 875 260 243 968;
  • 40) 0.476 684 570 361 875 260 243 968 × 2 = 0 + 0.953 369 140 723 750 520 487 936;
  • 41) 0.953 369 140 723 750 520 487 936 × 2 = 1 + 0.906 738 281 447 501 040 975 872;
  • 42) 0.906 738 281 447 501 040 975 872 × 2 = 1 + 0.813 476 562 895 002 081 951 744;
  • 43) 0.813 476 562 895 002 081 951 744 × 2 = 1 + 0.626 953 125 790 004 163 903 488;
  • 44) 0.626 953 125 790 004 163 903 488 × 2 = 1 + 0.253 906 251 580 008 327 806 976;
  • 45) 0.253 906 251 580 008 327 806 976 × 2 = 0 + 0.507 812 503 160 016 655 613 952;
  • 46) 0.507 812 503 160 016 655 613 952 × 2 = 1 + 0.015 625 006 320 033 311 227 904;
  • 47) 0.015 625 006 320 033 311 227 904 × 2 = 0 + 0.031 250 012 640 066 622 455 808;
  • 48) 0.031 250 012 640 066 622 455 808 × 2 = 0 + 0.062 500 025 280 133 244 911 616;
  • 49) 0.062 500 025 280 133 244 911 616 × 2 = 0 + 0.125 000 050 560 266 489 823 232;
  • 50) 0.125 000 050 560 266 489 823 232 × 2 = 0 + 0.250 000 101 120 532 979 646 464;
  • 51) 0.250 000 101 120 532 979 646 464 × 2 = 0 + 0.500 000 202 241 065 959 292 928;
  • 52) 0.500 000 202 241 065 959 292 928 × 2 = 1 + 0.000 000 404 482 131 918 585 856;
  • 53) 0.000 000 404 482 131 918 585 856 × 2 = 0 + 0.000 000 808 964 263 837 171 712;

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:


0.745 459 324 169 999 826 281 786(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 826 281 786(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:


1.745 459 324 169 999 826 281 786(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 1.745 459 324 169 999 826 281 786 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