-0.000 000 000 742 147 676 599 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 599(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
-0.000 000 000 742 147 676 599(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:

|-0.000 000 000 742 147 676 599| = 0.000 000 000 742 147 676 599


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: 0.000 000 000 742 147 676 599.

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) 0.000 000 000 742 147 676 599 × 2 = 0 + 0.000 000 001 484 295 353 198;
  • 2) 0.000 000 001 484 295 353 198 × 2 = 0 + 0.000 000 002 968 590 706 396;
  • 3) 0.000 000 002 968 590 706 396 × 2 = 0 + 0.000 000 005 937 181 412 792;
  • 4) 0.000 000 005 937 181 412 792 × 2 = 0 + 0.000 000 011 874 362 825 584;
  • 5) 0.000 000 011 874 362 825 584 × 2 = 0 + 0.000 000 023 748 725 651 168;
  • 6) 0.000 000 023 748 725 651 168 × 2 = 0 + 0.000 000 047 497 451 302 336;
  • 7) 0.000 000 047 497 451 302 336 × 2 = 0 + 0.000 000 094 994 902 604 672;
  • 8) 0.000 000 094 994 902 604 672 × 2 = 0 + 0.000 000 189 989 805 209 344;
  • 9) 0.000 000 189 989 805 209 344 × 2 = 0 + 0.000 000 379 979 610 418 688;
  • 10) 0.000 000 379 979 610 418 688 × 2 = 0 + 0.000 000 759 959 220 837 376;
  • 11) 0.000 000 759 959 220 837 376 × 2 = 0 + 0.000 001 519 918 441 674 752;
  • 12) 0.000 001 519 918 441 674 752 × 2 = 0 + 0.000 003 039 836 883 349 504;
  • 13) 0.000 003 039 836 883 349 504 × 2 = 0 + 0.000 006 079 673 766 699 008;
  • 14) 0.000 006 079 673 766 699 008 × 2 = 0 + 0.000 012 159 347 533 398 016;
  • 15) 0.000 012 159 347 533 398 016 × 2 = 0 + 0.000 024 318 695 066 796 032;
  • 16) 0.000 024 318 695 066 796 032 × 2 = 0 + 0.000 048 637 390 133 592 064;
  • 17) 0.000 048 637 390 133 592 064 × 2 = 0 + 0.000 097 274 780 267 184 128;
  • 18) 0.000 097 274 780 267 184 128 × 2 = 0 + 0.000 194 549 560 534 368 256;
  • 19) 0.000 194 549 560 534 368 256 × 2 = 0 + 0.000 389 099 121 068 736 512;
  • 20) 0.000 389 099 121 068 736 512 × 2 = 0 + 0.000 778 198 242 137 473 024;
  • 21) 0.000 778 198 242 137 473 024 × 2 = 0 + 0.001 556 396 484 274 946 048;
  • 22) 0.001 556 396 484 274 946 048 × 2 = 0 + 0.003 112 792 968 549 892 096;
  • 23) 0.003 112 792 968 549 892 096 × 2 = 0 + 0.006 225 585 937 099 784 192;
  • 24) 0.006 225 585 937 099 784 192 × 2 = 0 + 0.012 451 171 874 199 568 384;
  • 25) 0.012 451 171 874 199 568 384 × 2 = 0 + 0.024 902 343 748 399 136 768;
  • 26) 0.024 902 343 748 399 136 768 × 2 = 0 + 0.049 804 687 496 798 273 536;
  • 27) 0.049 804 687 496 798 273 536 × 2 = 0 + 0.099 609 374 993 596 547 072;
  • 28) 0.099 609 374 993 596 547 072 × 2 = 0 + 0.199 218 749 987 193 094 144;
  • 29) 0.199 218 749 987 193 094 144 × 2 = 0 + 0.398 437 499 974 386 188 288;
  • 30) 0.398 437 499 974 386 188 288 × 2 = 0 + 0.796 874 999 948 772 376 576;
  • 31) 0.796 874 999 948 772 376 576 × 2 = 1 + 0.593 749 999 897 544 753 152;
  • 32) 0.593 749 999 897 544 753 152 × 2 = 1 + 0.187 499 999 795 089 506 304;
  • 33) 0.187 499 999 795 089 506 304 × 2 = 0 + 0.374 999 999 590 179 012 608;
  • 34) 0.374 999 999 590 179 012 608 × 2 = 0 + 0.749 999 999 180 358 025 216;
  • 35) 0.749 999 999 180 358 025 216 × 2 = 1 + 0.499 999 998 360 716 050 432;
  • 36) 0.499 999 998 360 716 050 432 × 2 = 0 + 0.999 999 996 721 432 100 864;
  • 37) 0.999 999 996 721 432 100 864 × 2 = 1 + 0.999 999 993 442 864 201 728;
  • 38) 0.999 999 993 442 864 201 728 × 2 = 1 + 0.999 999 986 885 728 403 456;
  • 39) 0.999 999 986 885 728 403 456 × 2 = 1 + 0.999 999 973 771 456 806 912;
  • 40) 0.999 999 973 771 456 806 912 × 2 = 1 + 0.999 999 947 542 913 613 824;
  • 41) 0.999 999 947 542 913 613 824 × 2 = 1 + 0.999 999 895 085 827 227 648;
  • 42) 0.999 999 895 085 827 227 648 × 2 = 1 + 0.999 999 790 171 654 455 296;
  • 43) 0.999 999 790 171 654 455 296 × 2 = 1 + 0.999 999 580 343 308 910 592;
  • 44) 0.999 999 580 343 308 910 592 × 2 = 1 + 0.999 999 160 686 617 821 184;
  • 45) 0.999 999 160 686 617 821 184 × 2 = 1 + 0.999 998 321 373 235 642 368;
  • 46) 0.999 998 321 373 235 642 368 × 2 = 1 + 0.999 996 642 746 471 284 736;
  • 47) 0.999 996 642 746 471 284 736 × 2 = 1 + 0.999 993 285 492 942 569 472;
  • 48) 0.999 993 285 492 942 569 472 × 2 = 1 + 0.999 986 570 985 885 138 944;
  • 49) 0.999 986 570 985 885 138 944 × 2 = 1 + 0.999 973 141 971 770 277 888;
  • 50) 0.999 973 141 971 770 277 888 × 2 = 1 + 0.999 946 283 943 540 555 776;
  • 51) 0.999 946 283 943 540 555 776 × 2 = 1 + 0.999 892 567 887 081 111 552;
  • 52) 0.999 892 567 887 081 111 552 × 2 = 1 + 0.999 785 135 774 162 223 104;
  • 53) 0.999 785 135 774 162 223 104 × 2 = 1 + 0.999 570 271 548 324 446 208;
  • 54) 0.999 570 271 548 324 446 208 × 2 = 1 + 0.999 140 543 096 648 892 416;

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:


0.000 000 000 742 147 676 599(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 599(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:


0.000 000 000 742 147 676 599(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 -0.000 000 000 742 147 676 599 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