This paper attempts to popularize the two conclusions in elementary mathematics:the deduction of "2 is one of the irrational numbers" and "the thermo of internal bisector for triangle".

This article,according to the power series expansion and the Leibniz theorem,had proven,using the reduction to absurdity,when n is the non-vanishing integer,sin(l/n) is an irrational number.

A Taylor formula with integral form surplus term is deduced,by means of which e is simply proved to be an irrational number.

Sum of square roots of prime numbers is irrational;

The generalized Fibonacci sequence can prove that square roots of integer is either integer or irrational.

In the paper,I research into the relation between the recurring continued fraction and the quadratic irrational Firstly I prove any recurring continued faction is all the quadratic irrational,and then offer the general method changing the recurring continued fraction into the quadratic irrationa

The integration by second substitution is an important method of calculating indefinite integral,it has a certain application,it usually applies to calculate some integrals of irrational function.

It is difficult to find the solutions to some complex irrational functions and even worse, there is no solutions to them.

By describing states Euler s transformation of indefinite integral of irrational function,the author analyses what is Euler s transformation through Choosing Q(t) ,and reveals the fundamentals of Euler s transformation.

The paper points out the equivalence proprty of six formulas that have relation with irrational number e and proves them.