Real Numbers

Understanding Real Numbers

• Real numbers encompass all the values that can be represented on a number line, including integers, fractions and irrational numbers.
• They can take on any value within a range, unlike integers, and so are needed for high-precision tasks.
• Real numbers include positive, negative values and zero. They also include infinitesimally small and infinitely large values.
• Real numbers are crucial for tasks in computing that involve measurements or approximations, like calculating distances or processing graphics.

Binary Representation of Real Numbers

• The binary representation of real numbers often involves the use of floating point representation.
• A floating point number is divided into two parts - the mantissa and the exponent. The mantissa carries the main digits of the number, while the exponent determines the position of the decimal.
• There are standards like IEEE 754 that outline the way real numbers ought to be represented and manipulated within computers.

Encoding and Decoding Real Numbers

• Encoding a real number often involves converting from a base-10 number system to the binary system.
• During this representation, precision may be lost due to limitations in the length of the mantissa. This is a truncation error.
• Decoding a real number involves converting it back from its binary representation to the base-10 number system.

Importance of Real Numbers in Computing

• Real numbers are crucial in graphic design software and animation to render realistic, smooth graphics.
• They are also important in scientific computing, for tasks like modelling real-world scenarios or simulating natural phenomena.
• Understanding real number representation in computing is fundamental to understanding errors and accuracy of calculations in a program.
• Having a robust understanding of how to manipulate real numbers is important when working within fields that rely heavily on decimal precision, like machine learning and data analysis.