"Approximate formula" means that in mathematics, we often use some approximate methods to calculate or estimate the result of a mathematical expression. These approximate formulas are usually based on some mathematical principles or empirical formulas, and approximate answers can be given quickly without accurate calculation.
Explanation:
Approximate formulas are widely used in mathematics. They are usually used to simplify complex mathematical expressions or calculations, or in some cases, we may not get accurate answers, but we can get a rough result by using approximate formulas.
The following are some examples of common approximate formulas:
Approximation of pi: π is an irrational number, and we can't express it with a finite decimal. However, many mathematicians in history tried to approximate the value of π in different ways. For example, Archimedes used the following approximate formula: π≈ 10/3. This formula is very close to the true value under the calculation conditions at that time.
Approximation of natural constant: e is also an irrational number, but we can use some approximate formulas to estimate its value. Perhaps the most famous is Taylor series expansion: e ≈1+11! + 1/2! + 1/3! + 1/n！ . Although this series is infinite, we only need to calculate the first few terms to get a fairly accurate approximation.
Approximation of square root: If we find the square root of a number, but we don't know its exact value, we can use some approximate formulas to estimate it. For example, for non-negative real number x, √x≈x/2. This formula can give a fairly accurate answer when x is relatively small.
Besides these examples, there are many other approximate formulas used in different mathematical problems. These approximate formulas are usually based on some mathematical principles or empirical formulas, such as Taylor series expansion and power series expansion. When using these approximate formulas, we need to pay attention to their application range and error range to ensure that the obtained results are reliable.