1.3. Floating-Point Representation

Computers use floating-point representation to handle real numbers. However, due to limited memory and precision, floating-point numbers are only an approximation of real numbers. Understanding how numbers are represented and the potential errors introduced in this process is critical for numerical methods.

Floating-point representation is a way to store real numbers in computers. Instead of holding every possible real number, computers approximate these numbers because of memory and precision constraints. It is important for students to grasp how floating-point representation works and the limits it imposes in calculations, as this knowledge is essential when performing numerical methods.

Think of floating-point representation like a painter mixing colors. Just like a painter may not have every shade available (only mixing what's closest), computers also cannot represent every number exactly and must approximate them.

In floating-point representation, numbers are stored in scientific notation, which is typically represented as:

x=(-1)^s⋅m⋅2^e Where: ● s is the sign bit (0 for positive, 1 for negative). ● m is the mantissa or significand, a fractional number. ● e is the exponent that scales the mantissa by a power of 2.

For example, in single precision (32 bits): ● 1 bit for the sign. ● 8 bits for the exponent (with a bias of 127). ● 23 bits for the mantissa. Double precision (64 bits) uses more bits for the exponent and mantissa, offering higher precision and range.

The floating-point format organizes the way numbers are stored in computers. It uses three components: the sign bit, which indicates whether the number is positive or negative; the mantissa, which contains the significant digits of the number; and the exponent, which scales the mantissa to represent very large or very small values. In single precision, these bits are divided as specified, and this combination allows for a wide range of real numbers to be represented.

Imagine a digital clock. The numbers represent time (like the mantissa), and the hours (like the exponent) determine the scale—without the correct scaling, even with the right time details, you'd have the wrong display.

● Precision: The number of significant digits that a floating-point number can represent. ○ For single precision, you have about 7 decimal digits of precision. ○ For double precision, you have about 15 decimal digits of precision. ● Accuracy: The degree to which the floating-point representation approximates the real value. Accuracy is limited by machine epsilon, which is the smallest difference between two representable numbers.

Precision refers to how many digits a floating-point representation can maintain accurately. Single precision typically keeps about 7 digits, while double precision allows for around 15 digits. Accuracy is about how close the represented number is to the actual number, governed by a value known as machine epsilon. This underscores the limitations in numerical methods, as certain values may not be accurately represented.

Consider as a measurement tool like a ruler. A ruler that measures only in centimeters (similar to single precision) can only show a limited level of detail compared to a finer measuring tape that shows millimeters (analogous to double precision).

● Rounding Errors: Because floating-point numbers cannot represent all real numbers exactly, rounding errors occur when numbers are approximated to the nearest representable value. ● Overflow and Underflow: Overflow occurs when the number exceeds the largest representable value, and underflow occurs when the number is too small to be represented, leading to loss of precision or infinity. ● Loss of Significance: This occurs when subtracting two nearly equal numbers, which can result in large relative errors in the result.

Floating-point representation has inherent limitations. Rounding errors are common as a floating-point number must round to a nearby representable value. Overflow might occur when a computed number exceeds what can be stored, while underflow happens when a number is too small, often producing zeros or undefined results. Also, when two nearly equal numbers are subtracted, the result can become imprecise, leading to significant errors in calculations due to the loss of significance.

Think of pouring water into glasses. If you have a glass that can only hold a certain amount of water (like a variable limit in floating-point), pouring too much will cause it to overflow. Conversely, if you have very little remaining and you try to pour it into a tiny shot glass (too small), you may lose the actual volume due to overflow or underflow issues.

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