-0.000 000 000 742 147 676 701 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 701(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
-0.000 000 000 742 147 676 701(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:

|-0.000 000 000 742 147 676 701| = 0.000 000 000 742 147 676 701


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: 0.000 000 000 742 147 676 701.

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) 0.000 000 000 742 147 676 701 × 2 = 0 + 0.000 000 001 484 295 353 402;
  • 2) 0.000 000 001 484 295 353 402 × 2 = 0 + 0.000 000 002 968 590 706 804;
  • 3) 0.000 000 002 968 590 706 804 × 2 = 0 + 0.000 000 005 937 181 413 608;
  • 4) 0.000 000 005 937 181 413 608 × 2 = 0 + 0.000 000 011 874 362 827 216;
  • 5) 0.000 000 011 874 362 827 216 × 2 = 0 + 0.000 000 023 748 725 654 432;
  • 6) 0.000 000 023 748 725 654 432 × 2 = 0 + 0.000 000 047 497 451 308 864;
  • 7) 0.000 000 047 497 451 308 864 × 2 = 0 + 0.000 000 094 994 902 617 728;
  • 8) 0.000 000 094 994 902 617 728 × 2 = 0 + 0.000 000 189 989 805 235 456;
  • 9) 0.000 000 189 989 805 235 456 × 2 = 0 + 0.000 000 379 979 610 470 912;
  • 10) 0.000 000 379 979 610 470 912 × 2 = 0 + 0.000 000 759 959 220 941 824;
  • 11) 0.000 000 759 959 220 941 824 × 2 = 0 + 0.000 001 519 918 441 883 648;
  • 12) 0.000 001 519 918 441 883 648 × 2 = 0 + 0.000 003 039 836 883 767 296;
  • 13) 0.000 003 039 836 883 767 296 × 2 = 0 + 0.000 006 079 673 767 534 592;
  • 14) 0.000 006 079 673 767 534 592 × 2 = 0 + 0.000 012 159 347 535 069 184;
  • 15) 0.000 012 159 347 535 069 184 × 2 = 0 + 0.000 024 318 695 070 138 368;
  • 16) 0.000 024 318 695 070 138 368 × 2 = 0 + 0.000 048 637 390 140 276 736;
  • 17) 0.000 048 637 390 140 276 736 × 2 = 0 + 0.000 097 274 780 280 553 472;
  • 18) 0.000 097 274 780 280 553 472 × 2 = 0 + 0.000 194 549 560 561 106 944;
  • 19) 0.000 194 549 560 561 106 944 × 2 = 0 + 0.000 389 099 121 122 213 888;
  • 20) 0.000 389 099 121 122 213 888 × 2 = 0 + 0.000 778 198 242 244 427 776;
  • 21) 0.000 778 198 242 244 427 776 × 2 = 0 + 0.001 556 396 484 488 855 552;
  • 22) 0.001 556 396 484 488 855 552 × 2 = 0 + 0.003 112 792 968 977 711 104;
  • 23) 0.003 112 792 968 977 711 104 × 2 = 0 + 0.006 225 585 937 955 422 208;
  • 24) 0.006 225 585 937 955 422 208 × 2 = 0 + 0.012 451 171 875 910 844 416;
  • 25) 0.012 451 171 875 910 844 416 × 2 = 0 + 0.024 902 343 751 821 688 832;
  • 26) 0.024 902 343 751 821 688 832 × 2 = 0 + 0.049 804 687 503 643 377 664;
  • 27) 0.049 804 687 503 643 377 664 × 2 = 0 + 0.099 609 375 007 286 755 328;
  • 28) 0.099 609 375 007 286 755 328 × 2 = 0 + 0.199 218 750 014 573 510 656;
  • 29) 0.199 218 750 014 573 510 656 × 2 = 0 + 0.398 437 500 029 147 021 312;
  • 30) 0.398 437 500 029 147 021 312 × 2 = 0 + 0.796 875 000 058 294 042 624;
  • 31) 0.796 875 000 058 294 042 624 × 2 = 1 + 0.593 750 000 116 588 085 248;
  • 32) 0.593 750 000 116 588 085 248 × 2 = 1 + 0.187 500 000 233 176 170 496;
  • 33) 0.187 500 000 233 176 170 496 × 2 = 0 + 0.375 000 000 466 352 340 992;
  • 34) 0.375 000 000 466 352 340 992 × 2 = 0 + 0.750 000 000 932 704 681 984;
  • 35) 0.750 000 000 932 704 681 984 × 2 = 1 + 0.500 000 001 865 409 363 968;
  • 36) 0.500 000 001 865 409 363 968 × 2 = 1 + 0.000 000 003 730 818 727 936;
  • 37) 0.000 000 003 730 818 727 936 × 2 = 0 + 0.000 000 007 461 637 455 872;
  • 38) 0.000 000 007 461 637 455 872 × 2 = 0 + 0.000 000 014 923 274 911 744;
  • 39) 0.000 000 014 923 274 911 744 × 2 = 0 + 0.000 000 029 846 549 823 488;
  • 40) 0.000 000 029 846 549 823 488 × 2 = 0 + 0.000 000 059 693 099 646 976;
  • 41) 0.000 000 059 693 099 646 976 × 2 = 0 + 0.000 000 119 386 199 293 952;
  • 42) 0.000 000 119 386 199 293 952 × 2 = 0 + 0.000 000 238 772 398 587 904;
  • 43) 0.000 000 238 772 398 587 904 × 2 = 0 + 0.000 000 477 544 797 175 808;
  • 44) 0.000 000 477 544 797 175 808 × 2 = 0 + 0.000 000 955 089 594 351 616;
  • 45) 0.000 000 955 089 594 351 616 × 2 = 0 + 0.000 001 910 179 188 703 232;
  • 46) 0.000 001 910 179 188 703 232 × 2 = 0 + 0.000 003 820 358 377 406 464;
  • 47) 0.000 003 820 358 377 406 464 × 2 = 0 + 0.000 007 640 716 754 812 928;
  • 48) 0.000 007 640 716 754 812 928 × 2 = 0 + 0.000 015 281 433 509 625 856;
  • 49) 0.000 015 281 433 509 625 856 × 2 = 0 + 0.000 030 562 867 019 251 712;
  • 50) 0.000 030 562 867 019 251 712 × 2 = 0 + 0.000 061 125 734 038 503 424;
  • 51) 0.000 061 125 734 038 503 424 × 2 = 0 + 0.000 122 251 468 077 006 848;
  • 52) 0.000 122 251 468 077 006 848 × 2 = 0 + 0.000 244 502 936 154 013 696;
  • 53) 0.000 244 502 936 154 013 696 × 2 = 0 + 0.000 489 005 872 308 027 392;
  • 54) 0.000 489 005 872 308 027 392 × 2 = 0 + 0.000 978 011 744 616 054 784;

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:


0.000 000 000 742 147 676 701(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 701(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:


0.000 000 000 742 147 676 701(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 -0.000 000 000 742 147 676 701 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