1, algebraic definition: If the square of one number equals the square of another number, then this number is the square root of another number. This definition can be expressed by mathematical symbols: if a? =b, then A is the square root of B. Under this definition, the radical can be used to solve the solution of the equation, such as X? The solution of -4=0 can be expressed as x=√4=2.
2. Geometric definition: On the Euclidean plane, the distance from point P to point O is the distance from point P to the nearest point on a circle with O as the center and R as the radius. This definition can be expressed by mathematical symbols: if OP=r, then P is on a circle with O as the center and R as the radius. This definition can be used to prove Pythagorean theorem and find the area of a circle.
3. logarithmic definition: for any positive real number x, there is a unique real number y, which makes y? This definition can be expressed by mathematical symbols: if y? =x, then y is the square root of x, and this definition can be used to solve equations, such as log(x? ) =2 can be expressed as x=√2.
Functions of radicals:
1, solving the equation: the equation can be solved by radical. For example, quadratic equations and higher-order equations can be solved by roots. By transforming an equation into a radical, a complex equation can be transformed into a form that is easy to solve, so that the solution of the equation can be found quickly.
2. Simplified expressions: radicals can be used to simplify some complex mathematical expressions. For example, some expressions contain a lot of multiplication and division operations. By using the radical formula, these operations can be transformed into simple multiplication operations, thus simplifying the calculation process of expressions.
3. Prove theorems: Radicals can be used to prove some important mathematical theorems and formulas. For example, Pythagorean theorem, arithmetic mean-geometric mean inequality and so on can all be proved by roots. Using radical formula, some abstract mathematical concepts can be transformed into concrete calculation process, which makes these theorems and formulas easier to understand.
4. Calculate the square root: A radical can be used to calculate the square root of a number. In real life, it is sometimes necessary to calculate the square root of a number, and finding the root formula is a common method.