However, E is the "least" transcendental number possible since it has irrationality measure of 2. Euler proved that E is irrational (meaning it cannot be expressed as a ratio of any two integers) and Hermite subsequently established that it is transcendental (meaning it is not the root of any integer polynomial).Expansion and simplification of complicated expressions involving E may require use of functions such as FunctionExpand and FullSimplify. When E is used as a symbol, it is propagated as an exact quantity. The exponential function Exp is converted to E^ x.It appears in many sums, products, integrals, in equations involving the compounding of interest, in growth laws involving exponential growth or decay, and in formulas from a wide range of other mathematical and scientific fields. With the possible exception of Pi, E is the most important constant in mathematics. E has a number of equivalent definitions in mathematics, including as the infinite sum of reciprocal factorials over non-negative integers and as the limiting value. It is also known as Euler's number and can be input as \. E is the symbol representing the base of the natural logarithm Log.The backward Euler method is also a one-step method similar to the forward Euler rule. Frequently a numerical method like Newton's that we consider in the section must be used to solve for y n+1. The backward Euler method is an implicit method: the new approximation y n+1 appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown y n+1. It requires more effort to solve for y n+1 than Euler's rule because y n+1 appears inside f. The backward Euler formula is an implicit one-step numerical method for solving initial value problems for first order differential equations. Return to Part III of the course y(t))",FontSize->12],]]
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