Why float is accurate

The reasons why float is accurate: 1. The accuracy of floating point numbers is limited by computer hardware. Computers use binary to represent numbers, and real numbers have infinite precision. Therefore, when a real number is converted to a floating point number, Its accuracy will be limited; 2. The accuracy of floating point numbers is also affected by rounding errors. In computers, floating point number operations are performed through approximations. Since the representation of floating point numbers is limited, so in computers Rounding errors will occur when performing operations; 3. The accuracy of floating point numbers is also affected by the algorithm and calculation order, so minimize the loss of accuracy.

Operating system for this tutorial: Windows 10 system, Dell G3 computer.

In computer science, floating point (float) is a data type used to represent real numbers. It consists of two parts: mantissa and exponent. The mantissa represents the number of significant digits of the real number, while the exponent represents the magnitude of the real number. However, although floating point numbers are widely used in computer science, they are not completely accurate.

First of all, the accuracy of floating point numbers is limited by computer hardware. Computers use binary to represent numbers, while real numbers have infinite precision. Therefore, when a real number is converted to a floating point number, its accuracy is limited. For example, consider an irrational number such as π, whose decimal part is infinite. When π is converted to a floating point number, only a limited number of bits can be represented, so its accuracy is limited.

Secondly, the accuracy of floating point numbers is also affected by rounding errors. In computers, operations with floating point numbers are performed through approximations. Since the representation of floating point numbers is finite, rounding errors will occur when performing operations. For example, when two floating point numbers are added, their mantissas and exponents may not match exactly, causing rounding errors.

In addition, the accuracy of floating point numbers is also affected by the algorithm and calculation order. In some specific calculations, the accuracy of floating point numbers may be affected by the choice of algorithm and the order of calculations. For example, in the calculation of consecutive multiplications, if large numbers are multiplied first and then small numbers are multiplied, there may be a loss of accuracy. Therefore, when writing computer programs, one needs to carefully choose the algorithm and order of calculations to minimize the loss of accuracy.

Despite the accuracy limitations of floating point numbers, they are still very useful in computer science. Floating point numbers can be used to represent and calculate various practical problems, such as scientific computing, financial analysis, and graphics processing. Additionally, the precision of floating-point numbers can be improved by increasing the number of digits in the mantissa or using a higher-precision data type. For example, double precision floating point numbers (double) have higher precision and can represent a larger range and higher precision of real numbers.

In short, despite the accuracy limitations of floating point numbers, they are still very useful in computer science. By understanding the precision limits of floating point numbers and the associated sources of error, we can better understand and use floating point numbers and avoid precision losses when writing computer programs.

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