-0.000 000 000 742 147 676 401 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 401(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
-0.000 000 000 742 147 676 401(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. Start with the positive version of the number:

|-0.000 000 000 742 147 676 401| = 0.000 000 000 742 147 676 401


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: 0.000 000 000 742 147 676 401.

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) 0.000 000 000 742 147 676 401 × 2 = 0 + 0.000 000 001 484 295 352 802;
  • 2) 0.000 000 001 484 295 352 802 × 2 = 0 + 0.000 000 002 968 590 705 604;
  • 3) 0.000 000 002 968 590 705 604 × 2 = 0 + 0.000 000 005 937 181 411 208;
  • 4) 0.000 000 005 937 181 411 208 × 2 = 0 + 0.000 000 011 874 362 822 416;
  • 5) 0.000 000 011 874 362 822 416 × 2 = 0 + 0.000 000 023 748 725 644 832;
  • 6) 0.000 000 023 748 725 644 832 × 2 = 0 + 0.000 000 047 497 451 289 664;
  • 7) 0.000 000 047 497 451 289 664 × 2 = 0 + 0.000 000 094 994 902 579 328;
  • 8) 0.000 000 094 994 902 579 328 × 2 = 0 + 0.000 000 189 989 805 158 656;
  • 9) 0.000 000 189 989 805 158 656 × 2 = 0 + 0.000 000 379 979 610 317 312;
  • 10) 0.000 000 379 979 610 317 312 × 2 = 0 + 0.000 000 759 959 220 634 624;
  • 11) 0.000 000 759 959 220 634 624 × 2 = 0 + 0.000 001 519 918 441 269 248;
  • 12) 0.000 001 519 918 441 269 248 × 2 = 0 + 0.000 003 039 836 882 538 496;
  • 13) 0.000 003 039 836 882 538 496 × 2 = 0 + 0.000 006 079 673 765 076 992;
  • 14) 0.000 006 079 673 765 076 992 × 2 = 0 + 0.000 012 159 347 530 153 984;
  • 15) 0.000 012 159 347 530 153 984 × 2 = 0 + 0.000 024 318 695 060 307 968;
  • 16) 0.000 024 318 695 060 307 968 × 2 = 0 + 0.000 048 637 390 120 615 936;
  • 17) 0.000 048 637 390 120 615 936 × 2 = 0 + 0.000 097 274 780 241 231 872;
  • 18) 0.000 097 274 780 241 231 872 × 2 = 0 + 0.000 194 549 560 482 463 744;
  • 19) 0.000 194 549 560 482 463 744 × 2 = 0 + 0.000 389 099 120 964 927 488;
  • 20) 0.000 389 099 120 964 927 488 × 2 = 0 + 0.000 778 198 241 929 854 976;
  • 21) 0.000 778 198 241 929 854 976 × 2 = 0 + 0.001 556 396 483 859 709 952;
  • 22) 0.001 556 396 483 859 709 952 × 2 = 0 + 0.003 112 792 967 719 419 904;
  • 23) 0.003 112 792 967 719 419 904 × 2 = 0 + 0.006 225 585 935 438 839 808;
  • 24) 0.006 225 585 935 438 839 808 × 2 = 0 + 0.012 451 171 870 877 679 616;
  • 25) 0.012 451 171 870 877 679 616 × 2 = 0 + 0.024 902 343 741 755 359 232;
  • 26) 0.024 902 343 741 755 359 232 × 2 = 0 + 0.049 804 687 483 510 718 464;
  • 27) 0.049 804 687 483 510 718 464 × 2 = 0 + 0.099 609 374 967 021 436 928;
  • 28) 0.099 609 374 967 021 436 928 × 2 = 0 + 0.199 218 749 934 042 873 856;
  • 29) 0.199 218 749 934 042 873 856 × 2 = 0 + 0.398 437 499 868 085 747 712;
  • 30) 0.398 437 499 868 085 747 712 × 2 = 0 + 0.796 874 999 736 171 495 424;
  • 31) 0.796 874 999 736 171 495 424 × 2 = 1 + 0.593 749 999 472 342 990 848;
  • 32) 0.593 749 999 472 342 990 848 × 2 = 1 + 0.187 499 998 944 685 981 696;
  • 33) 0.187 499 998 944 685 981 696 × 2 = 0 + 0.374 999 997 889 371 963 392;
  • 34) 0.374 999 997 889 371 963 392 × 2 = 0 + 0.749 999 995 778 743 926 784;
  • 35) 0.749 999 995 778 743 926 784 × 2 = 1 + 0.499 999 991 557 487 853 568;
  • 36) 0.499 999 991 557 487 853 568 × 2 = 0 + 0.999 999 983 114 975 707 136;
  • 37) 0.999 999 983 114 975 707 136 × 2 = 1 + 0.999 999 966 229 951 414 272;
  • 38) 0.999 999 966 229 951 414 272 × 2 = 1 + 0.999 999 932 459 902 828 544;
  • 39) 0.999 999 932 459 902 828 544 × 2 = 1 + 0.999 999 864 919 805 657 088;
  • 40) 0.999 999 864 919 805 657 088 × 2 = 1 + 0.999 999 729 839 611 314 176;
  • 41) 0.999 999 729 839 611 314 176 × 2 = 1 + 0.999 999 459 679 222 628 352;
  • 42) 0.999 999 459 679 222 628 352 × 2 = 1 + 0.999 998 919 358 445 256 704;
  • 43) 0.999 998 919 358 445 256 704 × 2 = 1 + 0.999 997 838 716 890 513 408;
  • 44) 0.999 997 838 716 890 513 408 × 2 = 1 + 0.999 995 677 433 781 026 816;
  • 45) 0.999 995 677 433 781 026 816 × 2 = 1 + 0.999 991 354 867 562 053 632;
  • 46) 0.999 991 354 867 562 053 632 × 2 = 1 + 0.999 982 709 735 124 107 264;
  • 47) 0.999 982 709 735 124 107 264 × 2 = 1 + 0.999 965 419 470 248 214 528;
  • 48) 0.999 965 419 470 248 214 528 × 2 = 1 + 0.999 930 838 940 496 429 056;
  • 49) 0.999 930 838 940 496 429 056 × 2 = 1 + 0.999 861 677 880 992 858 112;
  • 50) 0.999 861 677 880 992 858 112 × 2 = 1 + 0.999 723 355 761 985 716 224;
  • 51) 0.999 723 355 761 985 716 224 × 2 = 1 + 0.999 446 711 523 971 432 448;
  • 52) 0.999 446 711 523 971 432 448 × 2 = 1 + 0.998 893 423 047 942 864 896;
  • 53) 0.998 893 423 047 942 864 896 × 2 = 1 + 0.997 786 846 095 885 729 792;
  • 54) 0.997 786 846 095 885 729 792 × 2 = 1 + 0.995 573 692 191 771 459 584;

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).


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:


0.000 000 000 742 147 676 401(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2)

6. Positive number before normalization:

0.000 000 000 742 147 676 401(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 31 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 742 147 676 401(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2) × 20 =

1.1001 0111 1111 1111 1111 111(2) × 2-31


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): -31

Mantissa (not normalized):
1.1001 0111 1111 1111 1111 111


9. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-31 + 2(8-1) - 1 =

(-31 + 127)(10) =

96(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;
  • 96 ÷ 2 = 48 + 0;
  • 48 ÷ 2 = 24 + 0;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 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) =

96(10) =

0110 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 1011 1111 1111 1111 1111 =

100 1011 1111 1111 1111 1111


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) =
0110 0000

Mantissa (23 bits) =
100 1011 1111 1111 1111 1111


Decimal number -0.000 000 000 742 147 676 401 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0110 0000 - 100 1011 1111 1111 1111 1111

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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