-0.000 000 000 742 147 676 626 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 626(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
-0.000 000 000 742 147 676 626(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:

|-0.000 000 000 742 147 676 626| = 0.000 000 000 742 147 676 626


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: 0.000 000 000 742 147 676 626.

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) 0.000 000 000 742 147 676 626 × 2 = 0 + 0.000 000 001 484 295 353 252;
  • 2) 0.000 000 001 484 295 353 252 × 2 = 0 + 0.000 000 002 968 590 706 504;
  • 3) 0.000 000 002 968 590 706 504 × 2 = 0 + 0.000 000 005 937 181 413 008;
  • 4) 0.000 000 005 937 181 413 008 × 2 = 0 + 0.000 000 011 874 362 826 016;
  • 5) 0.000 000 011 874 362 826 016 × 2 = 0 + 0.000 000 023 748 725 652 032;
  • 6) 0.000 000 023 748 725 652 032 × 2 = 0 + 0.000 000 047 497 451 304 064;
  • 7) 0.000 000 047 497 451 304 064 × 2 = 0 + 0.000 000 094 994 902 608 128;
  • 8) 0.000 000 094 994 902 608 128 × 2 = 0 + 0.000 000 189 989 805 216 256;
  • 9) 0.000 000 189 989 805 216 256 × 2 = 0 + 0.000 000 379 979 610 432 512;
  • 10) 0.000 000 379 979 610 432 512 × 2 = 0 + 0.000 000 759 959 220 865 024;
  • 11) 0.000 000 759 959 220 865 024 × 2 = 0 + 0.000 001 519 918 441 730 048;
  • 12) 0.000 001 519 918 441 730 048 × 2 = 0 + 0.000 003 039 836 883 460 096;
  • 13) 0.000 003 039 836 883 460 096 × 2 = 0 + 0.000 006 079 673 766 920 192;
  • 14) 0.000 006 079 673 766 920 192 × 2 = 0 + 0.000 012 159 347 533 840 384;
  • 15) 0.000 012 159 347 533 840 384 × 2 = 0 + 0.000 024 318 695 067 680 768;
  • 16) 0.000 024 318 695 067 680 768 × 2 = 0 + 0.000 048 637 390 135 361 536;
  • 17) 0.000 048 637 390 135 361 536 × 2 = 0 + 0.000 097 274 780 270 723 072;
  • 18) 0.000 097 274 780 270 723 072 × 2 = 0 + 0.000 194 549 560 541 446 144;
  • 19) 0.000 194 549 560 541 446 144 × 2 = 0 + 0.000 389 099 121 082 892 288;
  • 20) 0.000 389 099 121 082 892 288 × 2 = 0 + 0.000 778 198 242 165 784 576;
  • 21) 0.000 778 198 242 165 784 576 × 2 = 0 + 0.001 556 396 484 331 569 152;
  • 22) 0.001 556 396 484 331 569 152 × 2 = 0 + 0.003 112 792 968 663 138 304;
  • 23) 0.003 112 792 968 663 138 304 × 2 = 0 + 0.006 225 585 937 326 276 608;
  • 24) 0.006 225 585 937 326 276 608 × 2 = 0 + 0.012 451 171 874 652 553 216;
  • 25) 0.012 451 171 874 652 553 216 × 2 = 0 + 0.024 902 343 749 305 106 432;
  • 26) 0.024 902 343 749 305 106 432 × 2 = 0 + 0.049 804 687 498 610 212 864;
  • 27) 0.049 804 687 498 610 212 864 × 2 = 0 + 0.099 609 374 997 220 425 728;
  • 28) 0.099 609 374 997 220 425 728 × 2 = 0 + 0.199 218 749 994 440 851 456;
  • 29) 0.199 218 749 994 440 851 456 × 2 = 0 + 0.398 437 499 988 881 702 912;
  • 30) 0.398 437 499 988 881 702 912 × 2 = 0 + 0.796 874 999 977 763 405 824;
  • 31) 0.796 874 999 977 763 405 824 × 2 = 1 + 0.593 749 999 955 526 811 648;
  • 32) 0.593 749 999 955 526 811 648 × 2 = 1 + 0.187 499 999 911 053 623 296;
  • 33) 0.187 499 999 911 053 623 296 × 2 = 0 + 0.374 999 999 822 107 246 592;
  • 34) 0.374 999 999 822 107 246 592 × 2 = 0 + 0.749 999 999 644 214 493 184;
  • 35) 0.749 999 999 644 214 493 184 × 2 = 1 + 0.499 999 999 288 428 986 368;
  • 36) 0.499 999 999 288 428 986 368 × 2 = 0 + 0.999 999 998 576 857 972 736;
  • 37) 0.999 999 998 576 857 972 736 × 2 = 1 + 0.999 999 997 153 715 945 472;
  • 38) 0.999 999 997 153 715 945 472 × 2 = 1 + 0.999 999 994 307 431 890 944;
  • 39) 0.999 999 994 307 431 890 944 × 2 = 1 + 0.999 999 988 614 863 781 888;
  • 40) 0.999 999 988 614 863 781 888 × 2 = 1 + 0.999 999 977 229 727 563 776;
  • 41) 0.999 999 977 229 727 563 776 × 2 = 1 + 0.999 999 954 459 455 127 552;
  • 42) 0.999 999 954 459 455 127 552 × 2 = 1 + 0.999 999 908 918 910 255 104;
  • 43) 0.999 999 908 918 910 255 104 × 2 = 1 + 0.999 999 817 837 820 510 208;
  • 44) 0.999 999 817 837 820 510 208 × 2 = 1 + 0.999 999 635 675 641 020 416;
  • 45) 0.999 999 635 675 641 020 416 × 2 = 1 + 0.999 999 271 351 282 040 832;
  • 46) 0.999 999 271 351 282 040 832 × 2 = 1 + 0.999 998 542 702 564 081 664;
  • 47) 0.999 998 542 702 564 081 664 × 2 = 1 + 0.999 997 085 405 128 163 328;
  • 48) 0.999 997 085 405 128 163 328 × 2 = 1 + 0.999 994 170 810 256 326 656;
  • 49) 0.999 994 170 810 256 326 656 × 2 = 1 + 0.999 988 341 620 512 653 312;
  • 50) 0.999 988 341 620 512 653 312 × 2 = 1 + 0.999 976 683 241 025 306 624;
  • 51) 0.999 976 683 241 025 306 624 × 2 = 1 + 0.999 953 366 482 050 613 248;
  • 52) 0.999 953 366 482 050 613 248 × 2 = 1 + 0.999 906 732 964 101 226 496;
  • 53) 0.999 906 732 964 101 226 496 × 2 = 1 + 0.999 813 465 928 202 452 992;
  • 54) 0.999 813 465 928 202 452 992 × 2 = 1 + 0.999 626 931 856 404 905 984;

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:


0.000 000 000 742 147 676 626(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 626(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:


0.000 000 000 742 147 676 626(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 -0.000 000 000 742 147 676 626 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