0.000 000 000 000 000 000 176 182 853 46 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 176 182 853 46(10) to 32 bit single precision IEEE 754 binary floating point representation standard (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 000 000 000 176 182 853 46(10) to 32 bit single precision IEEE 754 binary floating point representation (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
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;
  • 0 ÷ 2 = 0 + 0;

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.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 000 176 182 853 46.

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) 0.000 000 000 000 000 000 176 182 853 46 × 2 = 0 + 0.000 000 000 000 000 000 352 365 706 92;
  • 2) 0.000 000 000 000 000 000 352 365 706 92 × 2 = 0 + 0.000 000 000 000 000 000 704 731 413 84;
  • 3) 0.000 000 000 000 000 000 704 731 413 84 × 2 = 0 + 0.000 000 000 000 000 001 409 462 827 68;
  • 4) 0.000 000 000 000 000 001 409 462 827 68 × 2 = 0 + 0.000 000 000 000 000 002 818 925 655 36;
  • 5) 0.000 000 000 000 000 002 818 925 655 36 × 2 = 0 + 0.000 000 000 000 000 005 637 851 310 72;
  • 6) 0.000 000 000 000 000 005 637 851 310 72 × 2 = 0 + 0.000 000 000 000 000 011 275 702 621 44;
  • 7) 0.000 000 000 000 000 011 275 702 621 44 × 2 = 0 + 0.000 000 000 000 000 022 551 405 242 88;
  • 8) 0.000 000 000 000 000 022 551 405 242 88 × 2 = 0 + 0.000 000 000 000 000 045 102 810 485 76;
  • 9) 0.000 000 000 000 000 045 102 810 485 76 × 2 = 0 + 0.000 000 000 000 000 090 205 620 971 52;
  • 10) 0.000 000 000 000 000 090 205 620 971 52 × 2 = 0 + 0.000 000 000 000 000 180 411 241 943 04;
  • 11) 0.000 000 000 000 000 180 411 241 943 04 × 2 = 0 + 0.000 000 000 000 000 360 822 483 886 08;
  • 12) 0.000 000 000 000 000 360 822 483 886 08 × 2 = 0 + 0.000 000 000 000 000 721 644 967 772 16;
  • 13) 0.000 000 000 000 000 721 644 967 772 16 × 2 = 0 + 0.000 000 000 000 001 443 289 935 544 32;
  • 14) 0.000 000 000 000 001 443 289 935 544 32 × 2 = 0 + 0.000 000 000 000 002 886 579 871 088 64;
  • 15) 0.000 000 000 000 002 886 579 871 088 64 × 2 = 0 + 0.000 000 000 000 005 773 159 742 177 28;
  • 16) 0.000 000 000 000 005 773 159 742 177 28 × 2 = 0 + 0.000 000 000 000 011 546 319 484 354 56;
  • 17) 0.000 000 000 000 011 546 319 484 354 56 × 2 = 0 + 0.000 000 000 000 023 092 638 968 709 12;
  • 18) 0.000 000 000 000 023 092 638 968 709 12 × 2 = 0 + 0.000 000 000 000 046 185 277 937 418 24;
  • 19) 0.000 000 000 000 046 185 277 937 418 24 × 2 = 0 + 0.000 000 000 000 092 370 555 874 836 48;
  • 20) 0.000 000 000 000 092 370 555 874 836 48 × 2 = 0 + 0.000 000 000 000 184 741 111 749 672 96;
  • 21) 0.000 000 000 000 184 741 111 749 672 96 × 2 = 0 + 0.000 000 000 000 369 482 223 499 345 92;
  • 22) 0.000 000 000 000 369 482 223 499 345 92 × 2 = 0 + 0.000 000 000 000 738 964 446 998 691 84;
  • 23) 0.000 000 000 000 738 964 446 998 691 84 × 2 = 0 + 0.000 000 000 001 477 928 893 997 383 68;
  • 24) 0.000 000 000 001 477 928 893 997 383 68 × 2 = 0 + 0.000 000 000 002 955 857 787 994 767 36;
  • 25) 0.000 000 000 002 955 857 787 994 767 36 × 2 = 0 + 0.000 000 000 005 911 715 575 989 534 72;
  • 26) 0.000 000 000 005 911 715 575 989 534 72 × 2 = 0 + 0.000 000 000 011 823 431 151 979 069 44;
  • 27) 0.000 000 000 011 823 431 151 979 069 44 × 2 = 0 + 0.000 000 000 023 646 862 303 958 138 88;
  • 28) 0.000 000 000 023 646 862 303 958 138 88 × 2 = 0 + 0.000 000 000 047 293 724 607 916 277 76;
  • 29) 0.000 000 000 047 293 724 607 916 277 76 × 2 = 0 + 0.000 000 000 094 587 449 215 832 555 52;
  • 30) 0.000 000 000 094 587 449 215 832 555 52 × 2 = 0 + 0.000 000 000 189 174 898 431 665 111 04;
  • 31) 0.000 000 000 189 174 898 431 665 111 04 × 2 = 0 + 0.000 000 000 378 349 796 863 330 222 08;
  • 32) 0.000 000 000 378 349 796 863 330 222 08 × 2 = 0 + 0.000 000 000 756 699 593 726 660 444 16;
  • 33) 0.000 000 000 756 699 593 726 660 444 16 × 2 = 0 + 0.000 000 001 513 399 187 453 320 888 32;
  • 34) 0.000 000 001 513 399 187 453 320 888 32 × 2 = 0 + 0.000 000 003 026 798 374 906 641 776 64;
  • 35) 0.000 000 003 026 798 374 906 641 776 64 × 2 = 0 + 0.000 000 006 053 596 749 813 283 553 28;
  • 36) 0.000 000 006 053 596 749 813 283 553 28 × 2 = 0 + 0.000 000 012 107 193 499 626 567 106 56;
  • 37) 0.000 000 012 107 193 499 626 567 106 56 × 2 = 0 + 0.000 000 024 214 386 999 253 134 213 12;
  • 38) 0.000 000 024 214 386 999 253 134 213 12 × 2 = 0 + 0.000 000 048 428 773 998 506 268 426 24;
  • 39) 0.000 000 048 428 773 998 506 268 426 24 × 2 = 0 + 0.000 000 096 857 547 997 012 536 852 48;
  • 40) 0.000 000 096 857 547 997 012 536 852 48 × 2 = 0 + 0.000 000 193 715 095 994 025 073 704 96;
  • 41) 0.000 000 193 715 095 994 025 073 704 96 × 2 = 0 + 0.000 000 387 430 191 988 050 147 409 92;
  • 42) 0.000 000 387 430 191 988 050 147 409 92 × 2 = 0 + 0.000 000 774 860 383 976 100 294 819 84;
  • 43) 0.000 000 774 860 383 976 100 294 819 84 × 2 = 0 + 0.000 001 549 720 767 952 200 589 639 68;
  • 44) 0.000 001 549 720 767 952 200 589 639 68 × 2 = 0 + 0.000 003 099 441 535 904 401 179 279 36;
  • 45) 0.000 003 099 441 535 904 401 179 279 36 × 2 = 0 + 0.000 006 198 883 071 808 802 358 558 72;
  • 46) 0.000 006 198 883 071 808 802 358 558 72 × 2 = 0 + 0.000 012 397 766 143 617 604 717 117 44;
  • 47) 0.000 012 397 766 143 617 604 717 117 44 × 2 = 0 + 0.000 024 795 532 287 235 209 434 234 88;
  • 48) 0.000 024 795 532 287 235 209 434 234 88 × 2 = 0 + 0.000 049 591 064 574 470 418 868 469 76;
  • 49) 0.000 049 591 064 574 470 418 868 469 76 × 2 = 0 + 0.000 099 182 129 148 940 837 736 939 52;
  • 50) 0.000 099 182 129 148 940 837 736 939 52 × 2 = 0 + 0.000 198 364 258 297 881 675 473 879 04;
  • 51) 0.000 198 364 258 297 881 675 473 879 04 × 2 = 0 + 0.000 396 728 516 595 763 350 947 758 08;
  • 52) 0.000 396 728 516 595 763 350 947 758 08 × 2 = 0 + 0.000 793 457 033 191 526 701 895 516 16;
  • 53) 0.000 793 457 033 191 526 701 895 516 16 × 2 = 0 + 0.001 586 914 066 383 053 403 791 032 32;
  • 54) 0.001 586 914 066 383 053 403 791 032 32 × 2 = 0 + 0.003 173 828 132 766 106 807 582 064 64;
  • 55) 0.003 173 828 132 766 106 807 582 064 64 × 2 = 0 + 0.006 347 656 265 532 213 615 164 129 28;
  • 56) 0.006 347 656 265 532 213 615 164 129 28 × 2 = 0 + 0.012 695 312 531 064 427 230 328 258 56;
  • 57) 0.012 695 312 531 064 427 230 328 258 56 × 2 = 0 + 0.025 390 625 062 128 854 460 656 517 12;
  • 58) 0.025 390 625 062 128 854 460 656 517 12 × 2 = 0 + 0.050 781 250 124 257 708 921 313 034 24;
  • 59) 0.050 781 250 124 257 708 921 313 034 24 × 2 = 0 + 0.101 562 500 248 515 417 842 626 068 48;
  • 60) 0.101 562 500 248 515 417 842 626 068 48 × 2 = 0 + 0.203 125 000 497 030 835 685 252 136 96;
  • 61) 0.203 125 000 497 030 835 685 252 136 96 × 2 = 0 + 0.406 250 000 994 061 671 370 504 273 92;
  • 62) 0.406 250 000 994 061 671 370 504 273 92 × 2 = 0 + 0.812 500 001 988 123 342 741 008 547 84;
  • 63) 0.812 500 001 988 123 342 741 008 547 84 × 2 = 1 + 0.625 000 003 976 246 685 482 017 095 68;
  • 64) 0.625 000 003 976 246 685 482 017 095 68 × 2 = 1 + 0.250 000 007 952 493 370 964 034 191 36;
  • 65) 0.250 000 007 952 493 370 964 034 191 36 × 2 = 0 + 0.500 000 015 904 986 741 928 068 382 72;
  • 66) 0.500 000 015 904 986 741 928 068 382 72 × 2 = 1 + 0.000 000 031 809 973 483 856 136 765 44;
  • 67) 0.000 000 031 809 973 483 856 136 765 44 × 2 = 0 + 0.000 000 063 619 946 967 712 273 530 88;
  • 68) 0.000 000 063 619 946 967 712 273 530 88 × 2 = 0 + 0.000 000 127 239 893 935 424 547 061 76;
  • 69) 0.000 000 127 239 893 935 424 547 061 76 × 2 = 0 + 0.000 000 254 479 787 870 849 094 123 52;
  • 70) 0.000 000 254 479 787 870 849 094 123 52 × 2 = 0 + 0.000 000 508 959 575 741 698 188 247 04;
  • 71) 0.000 000 508 959 575 741 698 188 247 04 × 2 = 0 + 0.000 001 017 919 151 483 396 376 494 08;
  • 72) 0.000 001 017 919 151 483 396 376 494 08 × 2 = 0 + 0.000 002 035 838 302 966 792 752 988 16;
  • 73) 0.000 002 035 838 302 966 792 752 988 16 × 2 = 0 + 0.000 004 071 676 605 933 585 505 976 32;
  • 74) 0.000 004 071 676 605 933 585 505 976 32 × 2 = 0 + 0.000 008 143 353 211 867 171 011 952 64;
  • 75) 0.000 008 143 353 211 867 171 011 952 64 × 2 = 0 + 0.000 016 286 706 423 734 342 023 905 28;
  • 76) 0.000 016 286 706 423 734 342 023 905 28 × 2 = 0 + 0.000 032 573 412 847 468 684 047 810 56;
  • 77) 0.000 032 573 412 847 468 684 047 810 56 × 2 = 0 + 0.000 065 146 825 694 937 368 095 621 12;
  • 78) 0.000 065 146 825 694 937 368 095 621 12 × 2 = 0 + 0.000 130 293 651 389 874 736 191 242 24;
  • 79) 0.000 130 293 651 389 874 736 191 242 24 × 2 = 0 + 0.000 260 587 302 779 749 472 382 484 48;
  • 80) 0.000 260 587 302 779 749 472 382 484 48 × 2 = 0 + 0.000 521 174 605 559 498 944 764 968 96;
  • 81) 0.000 521 174 605 559 498 944 764 968 96 × 2 = 0 + 0.001 042 349 211 118 997 889 529 937 92;
  • 82) 0.001 042 349 211 118 997 889 529 937 92 × 2 = 0 + 0.002 084 698 422 237 995 779 059 875 84;
  • 83) 0.002 084 698 422 237 995 779 059 875 84 × 2 = 0 + 0.004 169 396 844 475 991 558 119 751 68;
  • 84) 0.004 169 396 844 475 991 558 119 751 68 × 2 = 0 + 0.008 338 793 688 951 983 116 239 503 36;
  • 85) 0.008 338 793 688 951 983 116 239 503 36 × 2 = 0 + 0.016 677 587 377 903 966 232 479 006 72;
  • 86) 0.016 677 587 377 903 966 232 479 006 72 × 2 = 0 + 0.033 355 174 755 807 932 464 958 013 44;

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:


0.000 000 000 000 000 000 176 182 853 46(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0100 0000 0000 0000 0000 00(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 176 182 853 46(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0100 0000 0000 0000 0000 00(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 63 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 176 182 853 46(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0100 0000 0000 0000 0000 00(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0100 0000 0000 0000 0000 00(2) × 20 =

1.1010 0000 0000 0000 0000 000(2) × 2-63


7. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -63

Mantissa (not normalized):
1.1010 0000 0000 0000 0000 000


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(8-1) - 1 =

-63 + 2(8-1) - 1 =

(-63 + 127)(10) =

64(10)


9. Convert the adjusted exponent from the decimal (base 10) to 8 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

64(10) =

0100 0000(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 23 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 101 0000 0000 0000 0000 0000 =

101 0000 0000 0000 0000 0000


12. The three elements that make up the number's 32 bit single precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (8 bits) =
0100 0000

Mantissa (23 bits) =
101 0000 0000 0000 0000 0000


Decimal number 0.000 000 000 000 000 000 176 182 853 46 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0100 0000 - 101 0000 0000 0000 0000 0000

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How to convert decimal numbers from base ten to 32 bit single precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 32 bit single 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 base ten 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 of the previous dividing 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 previous multiplying operations, starting from the top of the constructed list 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, by shifting the decimal point (or if you prefer, the decimal mark) "n" positions either to the left or to the right, so that only one non zero digit remains to the left of the decimal point.
  • 7. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(8-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign if the case) and adjust its length to 23 bits, either by removing the excess bits from the right (losing precision...) or by adding extra '0' bits 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 -25.347 from decimal system (base ten) to 32 bit single precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-25.347| = 25.347

  • 2. First convert the integer part, 25. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 25 ÷ 2 = 12 + 1;
    • 12 ÷ 2 = 6 + 0;
    • 6 ÷ 2 = 3 + 0;
    • 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:

    25(10) = 1 1001(2)

  • 4. Then convert the fractional part, 0.347. 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) 0.347 × 2 = 0 + 0.694;
    • 2) 0.694 × 2 = 1 + 0.388;
    • 3) 0.388 × 2 = 0 + 0.776;
    • 4) 0.776 × 2 = 1 + 0.552;
    • 5) 0.552 × 2 = 1 + 0.104;
    • 6) 0.104 × 2 = 0 + 0.208;
    • 7) 0.208 × 2 = 0 + 0.416;
    • 8) 0.416 × 2 = 0 + 0.832;
    • 9) 0.832 × 2 = 1 + 0.664;
    • 10) 0.664 × 2 = 1 + 0.328;
    • 11) 0.328 × 2 = 0 + 0.656;
    • 12) 0.656 × 2 = 1 + 0.312;
    • 13) 0.312 × 2 = 0 + 0.624;
    • 14) 0.624 × 2 = 1 + 0.248;
    • 15) 0.248 × 2 = 0 + 0.496;
    • 16) 0.496 × 2 = 0 + 0.992;
    • 17) 0.992 × 2 = 1 + 0.984;
    • 18) 0.984 × 2 = 1 + 0.968;
    • 19) 0.968 × 2 = 1 + 0.936;
    • 20) 0.936 × 2 = 1 + 0.872;
    • 21) 0.872 × 2 = 1 + 0.744;
    • 22) 0.744 × 2 = 1 + 0.488;
    • 23) 0.488 × 2 = 0 + 0.976;
    • 24) 0.976 × 2 = 1 + 0.952;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 23) 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:

    0.347(10) = 0.0101 1000 1101 0100 1111 1101(2)

  • 6. Summarizing - the positive number before normalization:

    25.347(10) = 1 1001.0101 1000 1101 0100 1111 1101(2)

  • 7. Normalize the binary representation of the number, shifting the decimal point 4 positions to the left so that only one non-zero digit stays to the left of the decimal point:

    25.347(10) =
    1 1001.0101 1000 1101 0100 1111 1101(2) =
    1 1001.0101 1000 1101 0100 1111 1101(2) × 20 =
    1.1001 0101 1000 1101 0100 1111 1101(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1001 0101 1000 1101 0100 1111 1101

  • 9. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as already demonstrated above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(8-1) - 1 = (4 + 127)(10) = 131(10) =
    1000 0011(2)

  • 10. Normalize the mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal point) and adjust its length to 23 bits, by removing the excess bits from the right (losing precision...):

    Mantissa (not-normalized): 1.1001 0101 1000 1101 0100 1111 1101

    Mantissa (normalized): 100 1010 1100 0110 1010 0111

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 1000 0011

    Mantissa (23 bits) = 100 1010 1100 0110 1010 0111

  • Number -25.347, converted from the decimal system (base 10) to 32 bit single precision IEEE 754 binary floating point =
    1 - 1000 0011 - 100 1010 1100 0110 1010 0111