-0.000 000 000 742 147 676 646 733 6 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 646 733 6(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
-0.000 000 000 742 147 676 646 733 6(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:

|-0.000 000 000 742 147 676 646 733 6| = 0.000 000 000 742 147 676 646 733 6


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: 0.000 000 000 742 147 676 646 733 6.

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) 0.000 000 000 742 147 676 646 733 6 × 2 = 0 + 0.000 000 001 484 295 353 293 467 2;
  • 2) 0.000 000 001 484 295 353 293 467 2 × 2 = 0 + 0.000 000 002 968 590 706 586 934 4;
  • 3) 0.000 000 002 968 590 706 586 934 4 × 2 = 0 + 0.000 000 005 937 181 413 173 868 8;
  • 4) 0.000 000 005 937 181 413 173 868 8 × 2 = 0 + 0.000 000 011 874 362 826 347 737 6;
  • 5) 0.000 000 011 874 362 826 347 737 6 × 2 = 0 + 0.000 000 023 748 725 652 695 475 2;
  • 6) 0.000 000 023 748 725 652 695 475 2 × 2 = 0 + 0.000 000 047 497 451 305 390 950 4;
  • 7) 0.000 000 047 497 451 305 390 950 4 × 2 = 0 + 0.000 000 094 994 902 610 781 900 8;
  • 8) 0.000 000 094 994 902 610 781 900 8 × 2 = 0 + 0.000 000 189 989 805 221 563 801 6;
  • 9) 0.000 000 189 989 805 221 563 801 6 × 2 = 0 + 0.000 000 379 979 610 443 127 603 2;
  • 10) 0.000 000 379 979 610 443 127 603 2 × 2 = 0 + 0.000 000 759 959 220 886 255 206 4;
  • 11) 0.000 000 759 959 220 886 255 206 4 × 2 = 0 + 0.000 001 519 918 441 772 510 412 8;
  • 12) 0.000 001 519 918 441 772 510 412 8 × 2 = 0 + 0.000 003 039 836 883 545 020 825 6;
  • 13) 0.000 003 039 836 883 545 020 825 6 × 2 = 0 + 0.000 006 079 673 767 090 041 651 2;
  • 14) 0.000 006 079 673 767 090 041 651 2 × 2 = 0 + 0.000 012 159 347 534 180 083 302 4;
  • 15) 0.000 012 159 347 534 180 083 302 4 × 2 = 0 + 0.000 024 318 695 068 360 166 604 8;
  • 16) 0.000 024 318 695 068 360 166 604 8 × 2 = 0 + 0.000 048 637 390 136 720 333 209 6;
  • 17) 0.000 048 637 390 136 720 333 209 6 × 2 = 0 + 0.000 097 274 780 273 440 666 419 2;
  • 18) 0.000 097 274 780 273 440 666 419 2 × 2 = 0 + 0.000 194 549 560 546 881 332 838 4;
  • 19) 0.000 194 549 560 546 881 332 838 4 × 2 = 0 + 0.000 389 099 121 093 762 665 676 8;
  • 20) 0.000 389 099 121 093 762 665 676 8 × 2 = 0 + 0.000 778 198 242 187 525 331 353 6;
  • 21) 0.000 778 198 242 187 525 331 353 6 × 2 = 0 + 0.001 556 396 484 375 050 662 707 2;
  • 22) 0.001 556 396 484 375 050 662 707 2 × 2 = 0 + 0.003 112 792 968 750 101 325 414 4;
  • 23) 0.003 112 792 968 750 101 325 414 4 × 2 = 0 + 0.006 225 585 937 500 202 650 828 8;
  • 24) 0.006 225 585 937 500 202 650 828 8 × 2 = 0 + 0.012 451 171 875 000 405 301 657 6;
  • 25) 0.012 451 171 875 000 405 301 657 6 × 2 = 0 + 0.024 902 343 750 000 810 603 315 2;
  • 26) 0.024 902 343 750 000 810 603 315 2 × 2 = 0 + 0.049 804 687 500 001 621 206 630 4;
  • 27) 0.049 804 687 500 001 621 206 630 4 × 2 = 0 + 0.099 609 375 000 003 242 413 260 8;
  • 28) 0.099 609 375 000 003 242 413 260 8 × 2 = 0 + 0.199 218 750 000 006 484 826 521 6;
  • 29) 0.199 218 750 000 006 484 826 521 6 × 2 = 0 + 0.398 437 500 000 012 969 653 043 2;
  • 30) 0.398 437 500 000 012 969 653 043 2 × 2 = 0 + 0.796 875 000 000 025 939 306 086 4;
  • 31) 0.796 875 000 000 025 939 306 086 4 × 2 = 1 + 0.593 750 000 000 051 878 612 172 8;
  • 32) 0.593 750 000 000 051 878 612 172 8 × 2 = 1 + 0.187 500 000 000 103 757 224 345 6;
  • 33) 0.187 500 000 000 103 757 224 345 6 × 2 = 0 + 0.375 000 000 000 207 514 448 691 2;
  • 34) 0.375 000 000 000 207 514 448 691 2 × 2 = 0 + 0.750 000 000 000 415 028 897 382 4;
  • 35) 0.750 000 000 000 415 028 897 382 4 × 2 = 1 + 0.500 000 000 000 830 057 794 764 8;
  • 36) 0.500 000 000 000 830 057 794 764 8 × 2 = 1 + 0.000 000 000 001 660 115 589 529 6;
  • 37) 0.000 000 000 001 660 115 589 529 6 × 2 = 0 + 0.000 000 000 003 320 231 179 059 2;
  • 38) 0.000 000 000 003 320 231 179 059 2 × 2 = 0 + 0.000 000 000 006 640 462 358 118 4;
  • 39) 0.000 000 000 006 640 462 358 118 4 × 2 = 0 + 0.000 000 000 013 280 924 716 236 8;
  • 40) 0.000 000 000 013 280 924 716 236 8 × 2 = 0 + 0.000 000 000 026 561 849 432 473 6;
  • 41) 0.000 000 000 026 561 849 432 473 6 × 2 = 0 + 0.000 000 000 053 123 698 864 947 2;
  • 42) 0.000 000 000 053 123 698 864 947 2 × 2 = 0 + 0.000 000 000 106 247 397 729 894 4;
  • 43) 0.000 000 000 106 247 397 729 894 4 × 2 = 0 + 0.000 000 000 212 494 795 459 788 8;
  • 44) 0.000 000 000 212 494 795 459 788 8 × 2 = 0 + 0.000 000 000 424 989 590 919 577 6;
  • 45) 0.000 000 000 424 989 590 919 577 6 × 2 = 0 + 0.000 000 000 849 979 181 839 155 2;
  • 46) 0.000 000 000 849 979 181 839 155 2 × 2 = 0 + 0.000 000 001 699 958 363 678 310 4;
  • 47) 0.000 000 001 699 958 363 678 310 4 × 2 = 0 + 0.000 000 003 399 916 727 356 620 8;
  • 48) 0.000 000 003 399 916 727 356 620 8 × 2 = 0 + 0.000 000 006 799 833 454 713 241 6;
  • 49) 0.000 000 006 799 833 454 713 241 6 × 2 = 0 + 0.000 000 013 599 666 909 426 483 2;
  • 50) 0.000 000 013 599 666 909 426 483 2 × 2 = 0 + 0.000 000 027 199 333 818 852 966 4;
  • 51) 0.000 000 027 199 333 818 852 966 4 × 2 = 0 + 0.000 000 054 398 667 637 705 932 8;
  • 52) 0.000 000 054 398 667 637 705 932 8 × 2 = 0 + 0.000 000 108 797 335 275 411 865 6;
  • 53) 0.000 000 108 797 335 275 411 865 6 × 2 = 0 + 0.000 000 217 594 670 550 823 731 2;
  • 54) 0.000 000 217 594 670 550 823 731 2 × 2 = 0 + 0.000 000 435 189 341 101 647 462 4;

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:


0.000 000 000 742 147 676 646 733 6(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 646 733 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0011 0000 0000 0000 0000 00(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:


0.000 000 000 742 147 676 646 733 6(10) =

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

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

1.1001 1000 0000 0000 0000 000(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 1000 0000 0000 0000 000


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 1100 0000 0000 0000 0000 =

100 1100 0000 0000 0000 0000


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 1100 0000 0000 0000 0000


Decimal number -0.000 000 000 742 147 676 646 733 6 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0110 0000 - 100 1100 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