1.745 459 324 169 999 826 141 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 141(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
1.745 459 324 169 999 826 141(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: 0.745 459 324 169 999 826 141.

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) 0.745 459 324 169 999 826 141 × 2 = 1 + 0.490 918 648 339 999 652 282;
  • 2) 0.490 918 648 339 999 652 282 × 2 = 0 + 0.981 837 296 679 999 304 564;
  • 3) 0.981 837 296 679 999 304 564 × 2 = 1 + 0.963 674 593 359 998 609 128;
  • 4) 0.963 674 593 359 998 609 128 × 2 = 1 + 0.927 349 186 719 997 218 256;
  • 5) 0.927 349 186 719 997 218 256 × 2 = 1 + 0.854 698 373 439 994 436 512;
  • 6) 0.854 698 373 439 994 436 512 × 2 = 1 + 0.709 396 746 879 988 873 024;
  • 7) 0.709 396 746 879 988 873 024 × 2 = 1 + 0.418 793 493 759 977 746 048;
  • 8) 0.418 793 493 759 977 746 048 × 2 = 0 + 0.837 586 987 519 955 492 096;
  • 9) 0.837 586 987 519 955 492 096 × 2 = 1 + 0.675 173 975 039 910 984 192;
  • 10) 0.675 173 975 039 910 984 192 × 2 = 1 + 0.350 347 950 079 821 968 384;
  • 11) 0.350 347 950 079 821 968 384 × 2 = 0 + 0.700 695 900 159 643 936 768;
  • 12) 0.700 695 900 159 643 936 768 × 2 = 1 + 0.401 391 800 319 287 873 536;
  • 13) 0.401 391 800 319 287 873 536 × 2 = 0 + 0.802 783 600 638 575 747 072;
  • 14) 0.802 783 600 638 575 747 072 × 2 = 1 + 0.605 567 201 277 151 494 144;
  • 15) 0.605 567 201 277 151 494 144 × 2 = 1 + 0.211 134 402 554 302 988 288;
  • 16) 0.211 134 402 554 302 988 288 × 2 = 0 + 0.422 268 805 108 605 976 576;
  • 17) 0.422 268 805 108 605 976 576 × 2 = 0 + 0.844 537 610 217 211 953 152;
  • 18) 0.844 537 610 217 211 953 152 × 2 = 1 + 0.689 075 220 434 423 906 304;
  • 19) 0.689 075 220 434 423 906 304 × 2 = 1 + 0.378 150 440 868 847 812 608;
  • 20) 0.378 150 440 868 847 812 608 × 2 = 0 + 0.756 300 881 737 695 625 216;
  • 21) 0.756 300 881 737 695 625 216 × 2 = 1 + 0.512 601 763 475 391 250 432;
  • 22) 0.512 601 763 475 391 250 432 × 2 = 1 + 0.025 203 526 950 782 500 864;
  • 23) 0.025 203 526 950 782 500 864 × 2 = 0 + 0.050 407 053 901 565 001 728;
  • 24) 0.050 407 053 901 565 001 728 × 2 = 0 + 0.100 814 107 803 130 003 456;
  • 25) 0.100 814 107 803 130 003 456 × 2 = 0 + 0.201 628 215 606 260 006 912;
  • 26) 0.201 628 215 606 260 006 912 × 2 = 0 + 0.403 256 431 212 520 013 824;
  • 27) 0.403 256 431 212 520 013 824 × 2 = 0 + 0.806 512 862 425 040 027 648;
  • 28) 0.806 512 862 425 040 027 648 × 2 = 1 + 0.613 025 724 850 080 055 296;
  • 29) 0.613 025 724 850 080 055 296 × 2 = 1 + 0.226 051 449 700 160 110 592;
  • 30) 0.226 051 449 700 160 110 592 × 2 = 0 + 0.452 102 899 400 320 221 184;
  • 31) 0.452 102 899 400 320 221 184 × 2 = 0 + 0.904 205 798 800 640 442 368;
  • 32) 0.904 205 798 800 640 442 368 × 2 = 1 + 0.808 411 597 601 280 884 736;
  • 33) 0.808 411 597 601 280 884 736 × 2 = 1 + 0.616 823 195 202 561 769 472;
  • 34) 0.616 823 195 202 561 769 472 × 2 = 1 + 0.233 646 390 405 123 538 944;
  • 35) 0.233 646 390 405 123 538 944 × 2 = 0 + 0.467 292 780 810 247 077 888;
  • 36) 0.467 292 780 810 247 077 888 × 2 = 0 + 0.934 585 561 620 494 155 776;
  • 37) 0.934 585 561 620 494 155 776 × 2 = 1 + 0.869 171 123 240 988 311 552;
  • 38) 0.869 171 123 240 988 311 552 × 2 = 1 + 0.738 342 246 481 976 623 104;
  • 39) 0.738 342 246 481 976 623 104 × 2 = 1 + 0.476 684 492 963 953 246 208;
  • 40) 0.476 684 492 963 953 246 208 × 2 = 0 + 0.953 368 985 927 906 492 416;
  • 41) 0.953 368 985 927 906 492 416 × 2 = 1 + 0.906 737 971 855 812 984 832;
  • 42) 0.906 737 971 855 812 984 832 × 2 = 1 + 0.813 475 943 711 625 969 664;
  • 43) 0.813 475 943 711 625 969 664 × 2 = 1 + 0.626 951 887 423 251 939 328;
  • 44) 0.626 951 887 423 251 939 328 × 2 = 1 + 0.253 903 774 846 503 878 656;
  • 45) 0.253 903 774 846 503 878 656 × 2 = 0 + 0.507 807 549 693 007 757 312;
  • 46) 0.507 807 549 693 007 757 312 × 2 = 1 + 0.015 615 099 386 015 514 624;
  • 47) 0.015 615 099 386 015 514 624 × 2 = 0 + 0.031 230 198 772 031 029 248;
  • 48) 0.031 230 198 772 031 029 248 × 2 = 0 + 0.062 460 397 544 062 058 496;
  • 49) 0.062 460 397 544 062 058 496 × 2 = 0 + 0.124 920 795 088 124 116 992;
  • 50) 0.124 920 795 088 124 116 992 × 2 = 0 + 0.249 841 590 176 248 233 984;
  • 51) 0.249 841 590 176 248 233 984 × 2 = 0 + 0.499 683 180 352 496 467 968;
  • 52) 0.499 683 180 352 496 467 968 × 2 = 0 + 0.999 366 360 704 992 935 936;
  • 53) 0.999 366 360 704 992 935 936 × 2 = 1 + 0.998 732 721 409 985 871 872;

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:


0.745 459 324 169 999 826 141(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 826 141(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:


1.745 459 324 169 999 826 141(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 1.745 459 324 169 999 826 141 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