Arithmetic Groups and Their Generalizations (Ams/Ip Studies by Lizhen Ji

By Lizhen Ji

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In elementary algebra a technique known as proof by induction is used. As an example, let us prove the formula with its help. First, we shall abbreviate the formula we are to prove. It is a statement involving the symbol n; let us denote it by T(n). Then T(3), for example, denotes the formula 1 + 2 + 3 = 6; and T(n + 1) denotes We want to prove that the formula T(n) holds for all whole numbers n. Let us try to prove it step by step, proving T( 1), T(2), T(3) and so on. After a little we find that it is natural to use the previous step in proving any given step.

But for other calculations they are inadequate: we want perhaps to weigh parts of a pound or to measure fractions of a foot. What number is suitable for measuring the weight of half of a five-pound cake? If x is the number of pounds in the half-cake, we must have 2·x = 5. But there is no whole number for which this is true. The equation a·x = b is not always solvable. We must therefore invent extra numbers—fractions—to make this equation solvable as far as possible. Without going into the logical development of mathematics, every schoolboy learns about fractions.

F( x), in fact, makes a discontinuous change of amount 2g where x = 1. e. define ϕ by the formula then f is discontinuous where x = 1. A function f is continuous at x = a if it does not suddenly change as x changes from value to value near a. This means that if a1, a2, a3, … is any succession of values of x whose limit is a, then the succession must have f( a) for its limit. We take this, then, as our definition of continuous function. An important property of a continuous function f which we can prove immediately is that if f( x) is equal to a for some value of x and to b for some other value, then it must take all values between a and b as x varies from one value to the other.

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