-0.000 000 000 742 147 676 391 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 391(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
-0.000 000 000 742 147 676 391(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:

|-0.000 000 000 742 147 676 391| = 0.000 000 000 742 147 676 391


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: 0.000 000 000 742 147 676 391.

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) 0.000 000 000 742 147 676 391 × 2 = 0 + 0.000 000 001 484 295 352 782;
  • 2) 0.000 000 001 484 295 352 782 × 2 = 0 + 0.000 000 002 968 590 705 564;
  • 3) 0.000 000 002 968 590 705 564 × 2 = 0 + 0.000 000 005 937 181 411 128;
  • 4) 0.000 000 005 937 181 411 128 × 2 = 0 + 0.000 000 011 874 362 822 256;
  • 5) 0.000 000 011 874 362 822 256 × 2 = 0 + 0.000 000 023 748 725 644 512;
  • 6) 0.000 000 023 748 725 644 512 × 2 = 0 + 0.000 000 047 497 451 289 024;
  • 7) 0.000 000 047 497 451 289 024 × 2 = 0 + 0.000 000 094 994 902 578 048;
  • 8) 0.000 000 094 994 902 578 048 × 2 = 0 + 0.000 000 189 989 805 156 096;
  • 9) 0.000 000 189 989 805 156 096 × 2 = 0 + 0.000 000 379 979 610 312 192;
  • 10) 0.000 000 379 979 610 312 192 × 2 = 0 + 0.000 000 759 959 220 624 384;
  • 11) 0.000 000 759 959 220 624 384 × 2 = 0 + 0.000 001 519 918 441 248 768;
  • 12) 0.000 001 519 918 441 248 768 × 2 = 0 + 0.000 003 039 836 882 497 536;
  • 13) 0.000 003 039 836 882 497 536 × 2 = 0 + 0.000 006 079 673 764 995 072;
  • 14) 0.000 006 079 673 764 995 072 × 2 = 0 + 0.000 012 159 347 529 990 144;
  • 15) 0.000 012 159 347 529 990 144 × 2 = 0 + 0.000 024 318 695 059 980 288;
  • 16) 0.000 024 318 695 059 980 288 × 2 = 0 + 0.000 048 637 390 119 960 576;
  • 17) 0.000 048 637 390 119 960 576 × 2 = 0 + 0.000 097 274 780 239 921 152;
  • 18) 0.000 097 274 780 239 921 152 × 2 = 0 + 0.000 194 549 560 479 842 304;
  • 19) 0.000 194 549 560 479 842 304 × 2 = 0 + 0.000 389 099 120 959 684 608;
  • 20) 0.000 389 099 120 959 684 608 × 2 = 0 + 0.000 778 198 241 919 369 216;
  • 21) 0.000 778 198 241 919 369 216 × 2 = 0 + 0.001 556 396 483 838 738 432;
  • 22) 0.001 556 396 483 838 738 432 × 2 = 0 + 0.003 112 792 967 677 476 864;
  • 23) 0.003 112 792 967 677 476 864 × 2 = 0 + 0.006 225 585 935 354 953 728;
  • 24) 0.006 225 585 935 354 953 728 × 2 = 0 + 0.012 451 171 870 709 907 456;
  • 25) 0.012 451 171 870 709 907 456 × 2 = 0 + 0.024 902 343 741 419 814 912;
  • 26) 0.024 902 343 741 419 814 912 × 2 = 0 + 0.049 804 687 482 839 629 824;
  • 27) 0.049 804 687 482 839 629 824 × 2 = 0 + 0.099 609 374 965 679 259 648;
  • 28) 0.099 609 374 965 679 259 648 × 2 = 0 + 0.199 218 749 931 358 519 296;
  • 29) 0.199 218 749 931 358 519 296 × 2 = 0 + 0.398 437 499 862 717 038 592;
  • 30) 0.398 437 499 862 717 038 592 × 2 = 0 + 0.796 874 999 725 434 077 184;
  • 31) 0.796 874 999 725 434 077 184 × 2 = 1 + 0.593 749 999 450 868 154 368;
  • 32) 0.593 749 999 450 868 154 368 × 2 = 1 + 0.187 499 998 901 736 308 736;
  • 33) 0.187 499 998 901 736 308 736 × 2 = 0 + 0.374 999 997 803 472 617 472;
  • 34) 0.374 999 997 803 472 617 472 × 2 = 0 + 0.749 999 995 606 945 234 944;
  • 35) 0.749 999 995 606 945 234 944 × 2 = 1 + 0.499 999 991 213 890 469 888;
  • 36) 0.499 999 991 213 890 469 888 × 2 = 0 + 0.999 999 982 427 780 939 776;
  • 37) 0.999 999 982 427 780 939 776 × 2 = 1 + 0.999 999 964 855 561 879 552;
  • 38) 0.999 999 964 855 561 879 552 × 2 = 1 + 0.999 999 929 711 123 759 104;
  • 39) 0.999 999 929 711 123 759 104 × 2 = 1 + 0.999 999 859 422 247 518 208;
  • 40) 0.999 999 859 422 247 518 208 × 2 = 1 + 0.999 999 718 844 495 036 416;
  • 41) 0.999 999 718 844 495 036 416 × 2 = 1 + 0.999 999 437 688 990 072 832;
  • 42) 0.999 999 437 688 990 072 832 × 2 = 1 + 0.999 998 875 377 980 145 664;
  • 43) 0.999 998 875 377 980 145 664 × 2 = 1 + 0.999 997 750 755 960 291 328;
  • 44) 0.999 997 750 755 960 291 328 × 2 = 1 + 0.999 995 501 511 920 582 656;
  • 45) 0.999 995 501 511 920 582 656 × 2 = 1 + 0.999 991 003 023 841 165 312;
  • 46) 0.999 991 003 023 841 165 312 × 2 = 1 + 0.999 982 006 047 682 330 624;
  • 47) 0.999 982 006 047 682 330 624 × 2 = 1 + 0.999 964 012 095 364 661 248;
  • 48) 0.999 964 012 095 364 661 248 × 2 = 1 + 0.999 928 024 190 729 322 496;
  • 49) 0.999 928 024 190 729 322 496 × 2 = 1 + 0.999 856 048 381 458 644 992;
  • 50) 0.999 856 048 381 458 644 992 × 2 = 1 + 0.999 712 096 762 917 289 984;
  • 51) 0.999 712 096 762 917 289 984 × 2 = 1 + 0.999 424 193 525 834 579 968;
  • 52) 0.999 424 193 525 834 579 968 × 2 = 1 + 0.998 848 387 051 669 159 936;
  • 53) 0.998 848 387 051 669 159 936 × 2 = 1 + 0.997 696 774 103 338 319 872;
  • 54) 0.997 696 774 103 338 319 872 × 2 = 1 + 0.995 393 548 206 676 639 744;

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:


0.000 000 000 742 147 676 391(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 391(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:


0.000 000 000 742 147 676 391(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 -0.000 000 000 742 147 676 391 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