32bit IEEE 754: Decimal ↗ Single Precision Floating Point Binary: -0.000 000 000 000 000 000 176 182 7 Convert the Number to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number

Number -0.000 000 000 000 000 000 176 182 7(10) converted and written in 32 bit single precision IEEE 754 binary floating point representation (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

1. Start with the positive version of the number:

|-0.000 000 000 000 000 000 176 182 7| = 0.000 000 000 000 000 000 176 182 7

2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 000 176 182 7.

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 7 × 2 = 0 + 0.000 000 000 000 000 000 352 365 4;
  • 2) 0.000 000 000 000 000 000 352 365 4 × 2 = 0 + 0.000 000 000 000 000 000 704 730 8;
  • 3) 0.000 000 000 000 000 000 704 730 8 × 2 = 0 + 0.000 000 000 000 000 001 409 461 6;
  • 4) 0.000 000 000 000 000 001 409 461 6 × 2 = 0 + 0.000 000 000 000 000 002 818 923 2;
  • 5) 0.000 000 000 000 000 002 818 923 2 × 2 = 0 + 0.000 000 000 000 000 005 637 846 4;
  • 6) 0.000 000 000 000 000 005 637 846 4 × 2 = 0 + 0.000 000 000 000 000 011 275 692 8;
  • 7) 0.000 000 000 000 000 011 275 692 8 × 2 = 0 + 0.000 000 000 000 000 022 551 385 6;
  • 8) 0.000 000 000 000 000 022 551 385 6 × 2 = 0 + 0.000 000 000 000 000 045 102 771 2;
  • 9) 0.000 000 000 000 000 045 102 771 2 × 2 = 0 + 0.000 000 000 000 000 090 205 542 4;
  • 10) 0.000 000 000 000 000 090 205 542 4 × 2 = 0 + 0.000 000 000 000 000 180 411 084 8;
  • 11) 0.000 000 000 000 000 180 411 084 8 × 2 = 0 + 0.000 000 000 000 000 360 822 169 6;
  • 12) 0.000 000 000 000 000 360 822 169 6 × 2 = 0 + 0.000 000 000 000 000 721 644 339 2;
  • 13) 0.000 000 000 000 000 721 644 339 2 × 2 = 0 + 0.000 000 000 000 001 443 288 678 4;
  • 14) 0.000 000 000 000 001 443 288 678 4 × 2 = 0 + 0.000 000 000 000 002 886 577 356 8;
  • 15) 0.000 000 000 000 002 886 577 356 8 × 2 = 0 + 0.000 000 000 000 005 773 154 713 6;
  • 16) 0.000 000 000 000 005 773 154 713 6 × 2 = 0 + 0.000 000 000 000 011 546 309 427 2;
  • 17) 0.000 000 000 000 011 546 309 427 2 × 2 = 0 + 0.000 000 000 000 023 092 618 854 4;
  • 18) 0.000 000 000 000 023 092 618 854 4 × 2 = 0 + 0.000 000 000 000 046 185 237 708 8;
  • 19) 0.000 000 000 000 046 185 237 708 8 × 2 = 0 + 0.000 000 000 000 092 370 475 417 6;
  • 20) 0.000 000 000 000 092 370 475 417 6 × 2 = 0 + 0.000 000 000 000 184 740 950 835 2;
  • 21) 0.000 000 000 000 184 740 950 835 2 × 2 = 0 + 0.000 000 000 000 369 481 901 670 4;
  • 22) 0.000 000 000 000 369 481 901 670 4 × 2 = 0 + 0.000 000 000 000 738 963 803 340 8;
  • 23) 0.000 000 000 000 738 963 803 340 8 × 2 = 0 + 0.000 000 000 001 477 927 606 681 6;
  • 24) 0.000 000 000 001 477 927 606 681 6 × 2 = 0 + 0.000 000 000 002 955 855 213 363 2;
  • 25) 0.000 000 000 002 955 855 213 363 2 × 2 = 0 + 0.000 000 000 005 911 710 426 726 4;
  • 26) 0.000 000 000 005 911 710 426 726 4 × 2 = 0 + 0.000 000 000 011 823 420 853 452 8;
  • 27) 0.000 000 000 011 823 420 853 452 8 × 2 = 0 + 0.000 000 000 023 646 841 706 905 6;
  • 28) 0.000 000 000 023 646 841 706 905 6 × 2 = 0 + 0.000 000 000 047 293 683 413 811 2;
  • 29) 0.000 000 000 047 293 683 413 811 2 × 2 = 0 + 0.000 000 000 094 587 366 827 622 4;
  • 30) 0.000 000 000 094 587 366 827 622 4 × 2 = 0 + 0.000 000 000 189 174 733 655 244 8;
  • 31) 0.000 000 000 189 174 733 655 244 8 × 2 = 0 + 0.000 000 000 378 349 467 310 489 6;
  • 32) 0.000 000 000 378 349 467 310 489 6 × 2 = 0 + 0.000 000 000 756 698 934 620 979 2;
  • 33) 0.000 000 000 756 698 934 620 979 2 × 2 = 0 + 0.000 000 001 513 397 869 241 958 4;
  • 34) 0.000 000 001 513 397 869 241 958 4 × 2 = 0 + 0.000 000 003 026 795 738 483 916 8;
  • 35) 0.000 000 003 026 795 738 483 916 8 × 2 = 0 + 0.000 000 006 053 591 476 967 833 6;
  • 36) 0.000 000 006 053 591 476 967 833 6 × 2 = 0 + 0.000 000 012 107 182 953 935 667 2;
  • 37) 0.000 000 012 107 182 953 935 667 2 × 2 = 0 + 0.000 000 024 214 365 907 871 334 4;
  • 38) 0.000 000 024 214 365 907 871 334 4 × 2 = 0 + 0.000 000 048 428 731 815 742 668 8;
  • 39) 0.000 000 048 428 731 815 742 668 8 × 2 = 0 + 0.000 000 096 857 463 631 485 337 6;
  • 40) 0.000 000 096 857 463 631 485 337 6 × 2 = 0 + 0.000 000 193 714 927 262 970 675 2;
  • 41) 0.000 000 193 714 927 262 970 675 2 × 2 = 0 + 0.000 000 387 429 854 525 941 350 4;
  • 42) 0.000 000 387 429 854 525 941 350 4 × 2 = 0 + 0.000 000 774 859 709 051 882 700 8;
  • 43) 0.000 000 774 859 709 051 882 700 8 × 2 = 0 + 0.000 001 549 719 418 103 765 401 6;
  • 44) 0.000 001 549 719 418 103 765 401 6 × 2 = 0 + 0.000 003 099 438 836 207 530 803 2;
  • 45) 0.000 003 099 438 836 207 530 803 2 × 2 = 0 + 0.000 006 198 877 672 415 061 606 4;
  • 46) 0.000 006 198 877 672 415 061 606 4 × 2 = 0 + 0.000 012 397 755 344 830 123 212 8;
  • 47) 0.000 012 397 755 344 830 123 212 8 × 2 = 0 + 0.000 024 795 510 689 660 246 425 6;
  • 48) 0.000 024 795 510 689 660 246 425 6 × 2 = 0 + 0.000 049 591 021 379 320 492 851 2;
  • 49) 0.000 049 591 021 379 320 492 851 2 × 2 = 0 + 0.000 099 182 042 758 640 985 702 4;
  • 50) 0.000 099 182 042 758 640 985 702 4 × 2 = 0 + 0.000 198 364 085 517 281 971 404 8;
  • 51) 0.000 198 364 085 517 281 971 404 8 × 2 = 0 + 0.000 396 728 171 034 563 942 809 6;
  • 52) 0.000 396 728 171 034 563 942 809 6 × 2 = 0 + 0.000 793 456 342 069 127 885 619 2;
  • 53) 0.000 793 456 342 069 127 885 619 2 × 2 = 0 + 0.001 586 912 684 138 255 771 238 4;
  • 54) 0.001 586 912 684 138 255 771 238 4 × 2 = 0 + 0.003 173 825 368 276 511 542 476 8;
  • 55) 0.003 173 825 368 276 511 542 476 8 × 2 = 0 + 0.006 347 650 736 553 023 084 953 6;
  • 56) 0.006 347 650 736 553 023 084 953 6 × 2 = 0 + 0.012 695 301 473 106 046 169 907 2;
  • 57) 0.012 695 301 473 106 046 169 907 2 × 2 = 0 + 0.025 390 602 946 212 092 339 814 4;
  • 58) 0.025 390 602 946 212 092 339 814 4 × 2 = 0 + 0.050 781 205 892 424 184 679 628 8;
  • 59) 0.050 781 205 892 424 184 679 628 8 × 2 = 0 + 0.101 562 411 784 848 369 359 257 6;
  • 60) 0.101 562 411 784 848 369 359 257 6 × 2 = 0 + 0.203 124 823 569 696 738 718 515 2;
  • 61) 0.203 124 823 569 696 738 718 515 2 × 2 = 0 + 0.406 249 647 139 393 477 437 030 4;
  • 62) 0.406 249 647 139 393 477 437 030 4 × 2 = 0 + 0.812 499 294 278 786 954 874 060 8;
  • 63) 0.812 499 294 278 786 954 874 060 8 × 2 = 1 + 0.624 998 588 557 573 909 748 121 6;
  • 64) 0.624 998 588 557 573 909 748 121 6 × 2 = 1 + 0.249 997 177 115 147 819 496 243 2;
  • 65) 0.249 997 177 115 147 819 496 243 2 × 2 = 0 + 0.499 994 354 230 295 638 992 486 4;
  • 66) 0.499 994 354 230 295 638 992 486 4 × 2 = 0 + 0.999 988 708 460 591 277 984 972 8;
  • 67) 0.999 988 708 460 591 277 984 972 8 × 2 = 1 + 0.999 977 416 921 182 555 969 945 6;
  • 68) 0.999 977 416 921 182 555 969 945 6 × 2 = 1 + 0.999 954 833 842 365 111 939 891 2;
  • 69) 0.999 954 833 842 365 111 939 891 2 × 2 = 1 + 0.999 909 667 684 730 223 879 782 4;
  • 70) 0.999 909 667 684 730 223 879 782 4 × 2 = 1 + 0.999 819 335 369 460 447 759 564 8;
  • 71) 0.999 819 335 369 460 447 759 564 8 × 2 = 1 + 0.999 638 670 738 920 895 519 129 6;
  • 72) 0.999 638 670 738 920 895 519 129 6 × 2 = 1 + 0.999 277 341 477 841 791 038 259 2;
  • 73) 0.999 277 341 477 841 791 038 259 2 × 2 = 1 + 0.998 554 682 955 683 582 076 518 4;
  • 74) 0.998 554 682 955 683 582 076 518 4 × 2 = 1 + 0.997 109 365 911 367 164 153 036 8;
  • 75) 0.997 109 365 911 367 164 153 036 8 × 2 = 1 + 0.994 218 731 822 734 328 306 073 6;
  • 76) 0.994 218 731 822 734 328 306 073 6 × 2 = 1 + 0.988 437 463 645 468 656 612 147 2;
  • 77) 0.988 437 463 645 468 656 612 147 2 × 2 = 1 + 0.976 874 927 290 937 313 224 294 4;
  • 78) 0.976 874 927 290 937 313 224 294 4 × 2 = 1 + 0.953 749 854 581 874 626 448 588 8;
  • 79) 0.953 749 854 581 874 626 448 588 8 × 2 = 1 + 0.907 499 709 163 749 252 897 177 6;
  • 80) 0.907 499 709 163 749 252 897 177 6 × 2 = 1 + 0.814 999 418 327 498 505 794 355 2;
  • 81) 0.814 999 418 327 498 505 794 355 2 × 2 = 1 + 0.629 998 836 654 997 011 588 710 4;
  • 82) 0.629 998 836 654 997 011 588 710 4 × 2 = 1 + 0.259 997 673 309 994 023 177 420 8;
  • 83) 0.259 997 673 309 994 023 177 420 8 × 2 = 0 + 0.519 995 346 619 988 046 354 841 6;
  • 84) 0.519 995 346 619 988 046 354 841 6 × 2 = 1 + 0.039 990 693 239 976 092 709 683 2;
  • 85) 0.039 990 693 239 976 092 709 683 2 × 2 = 0 + 0.079 981 386 479 952 185 419 366 4;
  • 86) 0.079 981 386 479 952 185 419 366 4 × 2 = 0 + 0.159 962 772 959 904 370 838 732 8;

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...)


5. 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 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0011 1111 1111 1111 1101 00(2)


6. Positive number before normalization:

0.000 000 000 000 000 000 176 182 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0011 1111 1111 1111 1101 00(2)

7. 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 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0011 1111 1111 1111 1101 00(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0011 1111 1111 1111 1101 00(2) × 20 =

1.1001 1111 1111 1111 1110 100(2) × 2-63


8. 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 1 (a negative number)

Exponent (unadjusted): -63

Mantissa (not normalized):
1.1001 1111 1111 1111 1110 100


9. 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)


10. 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;

11. 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)


12. 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. 100 1111 1111 1111 1111 0100 =

100 1111 1111 1111 1111 0100


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

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

Exponent (8 bits) =
0100 0000

Mantissa (23 bits) =
100 1111 1111 1111 1111 0100


The base ten decimal number -0.000 000 000 000 000 000 176 182 7 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
1 - 0100 0000 - 100 1111 1111 1111 1111 0100

More operations with decimal numbers converted to 32 bit single precision IEEE 754 binary floating point representation:

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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