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Floating-Point Representation

Definition:
Floating-point representation is a method of representing real numbers in a computer in the form of a mantissa and exponent, allowing both very large and very small numbers to be expressed efficiently.

General Form:

±m×be\boxed{\pm m \times b^e}±m×be​

Where:

  • mmm = Mantissa (or significand) – represents the significant digits of the number.

  • bbb = Base (commonly 2 in binary systems).

  • eee = Exponent – determines the scaling or magnitude of the number.

  • ±\pm± = Sign of the number (positive or negative).

Explanation of Terms:

  1. Mantissa (m):

    • Contains the actual digits of the number.

    • Typically normalized so that the first digit is non-zero (e.g., 1.2341.2341.234 in binary normalized form).

  2. Exponent (e):

    • Indicates the power of the base by which mantissa is multiplied.

    • Allows the number to “float” across a wide range of values.

Why Floating-Point Representation is Used:

  1. Large Range:

    • Can represent extremely large or extremely small numbers which cannot be stored in fixed-point form.

  2. Precision:

    • Maintains significant digits accurately.

    • Reduces loss of information compared to simple integer or fixed-point storage.

  3. Flexibility:

    • Useful in scientific calculations, simulations, and engineering computations where numbers vary widely in magnitude.

Example:
Decimal number 0.000560.000560.00056 in floating-point binary form:

0.00056=5.6×10−40.00056 = 5.6 \times 10^{-4} 0.00056=5.6×10−4

Here, 5.65.65.6 is the mantissa, and −4-4−4 is the exponent.

Key Point to Remember for Exam:

  • Normalization ensures mantissa starts with non-zero digit.

  • Floating-point gives both range and precision, which is impossible with simple integer representation.

Normalized Floating-Point Numbers

1. Definition of Normalization:
Normalization is the process of adjusting the mantissa and exponent in a floating-point number so that the mantissa falls within a specific range, usually making the first digit non-zero.

Purpose: To maximize precision and maintain a unique representation.

2. Rule of Normalization:

  • In binary systems: The normalized form requires the mantissa to satisfy

1≤∣m∣<21 \le |m| < 21≤∣m∣<2

  • In decimal systems:

1≤∣m∣<101 \le |m| < 101≤∣m∣<10

  • The exponent is adjusted accordingly to keep the value of the number unchanged.

3. Conversion of a Number into Normalized Form:

Example 1 (Decimal):
Convert 0.00560.00560.0056 to normalized form:

0.0056=5.6×10−30.0056 = 5.6 \times 10^{-3} 0.0056=5.6×10−3

  • Mantissa: 5.65.65.6 (between 1 and 10)

  • Exponent: −3-3−3

Example 2 (Binary):
Convert 0.101120.1011_20.10112​ to normalized form:

0.10112=1.011×2−20.1011_2 = 1.011 \times 2^{-2} 0.10112​=1.011×2−2

  • Mantissa: 1.0111.0111.011

  • Exponent: −2-2−2

4. Consequences of Normalization:

  1. Increased Precision:

    • All significant digits are preserved in mantissa, reducing loss of information.

  2. Reduced Rounding Error:

    • Standardized mantissa minimizes errors when performing arithmetic operations.

  3. Uniform Representation:

    • Every number has a unique normalized form, simplifying comparisons and calculations in computers.

Key Point for Exam:

  • Always remember: Normalized → Mantissa starts with non-zero digit.

  • This ensures maximum precision and consistent storage.

Significant Digits

1. Definition:
Significant digits (or significant figures) of a number are the meaningful digits that contribute to its accuracy.

  • They include all non-zero digits, zeros between non-zero digits, and trailing zeros after a decimal point.

  • Leading zeros are not significant.

2. Rules for Counting Significant Digits:

  1. All non-zero digits are significant.

    • Example: 123.45123.45123.45 → 5 significant digits

  2. Zeros between non-zero digits are significant.

    • Example: 100210021002 → 4 significant digits

  3. Leading zeros (before the first non-zero digit) are not significant.

    • Example: 0.00250.00250.0025 → 2 significant digits

  4. Trailing zeros after a decimal point are significant.

    • Example: 25.0025.0025.00 → 4 significant digits

  5. Trailing zeros in whole numbers without decimal point may or may not be significant; use scientific notation to clarify.

    • Example: 500500500 → 1, 2, or 3 significant digits depending on context

3. Relationship with Accuracy:

  • More significant digits → Higher accuracy

  • The accuracy of a measurement or computation is directly related to the number of significant digits it contains.

  • Rounding off reduces significant digits and may decrease precision.

Example:

  • Number: 3.141593.141593.14159 → 6 significant digits → very precise

  • Number: 3.143.143.14 → 3 significant digits → less precise

Key Point for Exam:

  • Significant digits indicate the precision of a number, not its magnitude.

Types of Errors in Numerical Computation

In numerical computation, errors are the differences between the true value and the approximate or computed value. The main types of errors are:

1. Truncation Error:

  • Definition: Error caused when an infinite process is approximated by a finite process.

  • Example: Using first 3 terms of a series to approximate a function instead of the full infinite series.

  • Cause: Ignoring higher-order terms in a formula or series.

2. Rounding Error:

  • Definition: Error introduced when a number is rounded off to a limited number of significant digits.

  • Example: Approximating 3.141592653.141592653.14159265 as 3.1423.1423.142 for calculations.

  • Cause: Limited precision in computer storage or manual rounding.

3. Inherent (or Pre-existing) Error:

  • Definition: Error already present in measured or given data before computation.

  • Example: Measuring a length with a ruler that has a smallest division of 1 mm → true value has small inherent error.

  • Cause: Limitations of instruments, human errors, or imprecise initial data.

Summary Table:

Type of Error Cause Example
Truncation Error Approximating infinite process Using 3 terms of Taylor series
Rounding Error Limiting number of digits 3.14159265 → 3.142
Inherent Error Measurement/data limitation Ruler reads ±1 mm

Key Point for Exam:

  • Understanding these errors is essential for numerical accuracy.

  • Total error in computation may be a combination of these three types.