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Representation of Decimal Numbers

Decimal Numbers

• A decimal number contains a decimal point. (e.g., 125.55 is a decimal number or a real number.)

• In computer memory, real numbers are stored according to the IEEE Standard 754 floating-point representation.

• IEEE754 floating-point representation is an approximate representation of real numbers.

• All computer systems and microcontrollers nowadays use this standard to store real numbers in memory.

• If you are working with numbers that have a fractional part or in case you are using integers that don't fit into a long data type, then you can use the floating-point representation.

• In C, too small or too big numbers or numbers having a fractional part are represented using floating-point representation data types such as float and double.

  • Too small real number

    e.g., Charge of an electron = -1.60217662 x 19^-19 coulombs

  • Too big real number

    e.g., Distance between the earth and the Andromeda galaxy = 2.3651826 x 10¹⁹ Km

• In summary, floating-point representation is needed:

  • To store too-small numbers in memory

  • To store too-big numbers in memory

  • To store decimal numbers

IEEE-754 Floating-Point Standard

• The IEEE-754 is a standard for representing and manipulating floating-point quantities that are followed by all modern computer systems and microcontrollers.

• Let's say you are to store +5.432 x 10⁴⁸ in memory. You can't just store it as is in binary format because it will take up too much space. The IEEE-754 suggests, approximate the number and store only the necessary information in memory.

• You cannot represent decimal numbers like 246.55 in a program using integer data types such as int, char, or long. When these data types are used, the fractional part will be lost. Following two data types are used to represent decimal numbers.

  • Float (32-bit floating-point representation, single precision)

  • Double (64-bit floating-point representation, double precision)

• Range of float

  Storage size: 4 bytes

  Precision: Up to 6 decimal places

  Value range: 1.2 x 10^-38 to 3.4 x 10³⁸

• Range of double

  Storage size: 8 bytes

  Precision: Up to 15 decimal places

  Value range: 2.3 x 10^-308 to 1.7 x 10³⁰⁸

Format Specifiers for float and double Data Types

• Use %lf format specifier to read or write double type variable.

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  double number = 12.123456789;
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  printf("number = %0.14lf\n", number);   /* 12.12345678900000 */
  

  Why is plain %lf format specifier for printing double not working in the CubeIDE?

• Use %f format specifier to read or write float type variable.

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  float number = 12.123456789;
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  printf("number = %f\n", number);    /* 12.123457 (up to 6 decimal places) */
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  printf("number = %0.9f\n", number); /* 12.123456955 (loosing precision after 6 decimal places) */
  

• Use %e, %le format specifier to read or write real numbers in scientific notation.

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  float number = 12.123456789;
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  printf("number = %0.2f\n", number); /* 12.12 */
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  printf("number = %0.2e\n", number); /* 1.21e+01 */
  4
  
  5
  float chargeE = -1.60217662e-19;    /* storing a decimal number in scientific notation */
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  printf("number = %f\n", number);    /* -0.000000 (approximation of very small number) */
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  printf("number = %e\n", number);    /* -1.602177e-19 (precise up to 6 decimal places) */
  

  Note that you cannot use float for storing very small (or very large) numbers due to the precision loss. Use double instead.

• All constants with a decimal point are considered as double by default.

References

Nayak, K. (2022). Microcontroller Embedded C Programming: Absolute Beginners [Video file]. Retrieved from https://www.udemy.com/course/microcontroller-embedded-c-programming/