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Download e-book for iPad: Algebraic numbers and Diophantine approximation by Kenneth B Stolarsky

By Kenneth B Stolarsky

ISBN-10: 0824761022

ISBN-13: 9780824761028

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Example text

For example, the function max can be applied to two arguments in a function application: max 4 7. If you prefer, you can use it as an operator, like this: 4 ‘max‘ 7. 4 Function Definitions You can define new functions by giving the type declaration followed by the defining equation. The type declaration has the form: function name :: argType1 → argType2 → . . → argTypen → resultType The arrows are written as → in mathematical notation, and they are written as -> in Haskell programs. The defining equation has the form: function name arg1 arg2 .

Higher order functions make possible a variety of powerful programming techniques. The twice function takes another function f as its first argument, and it applies f two times to its second argument x: twice :: (a->a) -> a -> a twice f x = f (f x) We can work out an application using equational reasoning. For example, twice sqrt 81 is evaluated as follows: twice sqrt 81 = sqrt (sqrt 81) = sqrt 9 = 3 Let’s examine the type of twice in detail. Assuming the second argument has type a, then the argument function has to accept an argument of type a and also return a value of the same type (because the result of the inner application becomes the argument to the outer application).

6. 6 21 Local Variables: let Expressions There are many times when we need to use computed values more than once. Instead of repeating the expression several times, it is better to give it a local name that can be reused. This can be done with a let expression. The general form is: let equation equation .. equation in expression This entire construct is just one big expression, and it can be used anywhere an expression would be valid. When it is evaluated, the local equations give temporary values to the variables in their left-hand sides; the final expression after in is the value of the entire let expression.

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Algebraic numbers and Diophantine approximation by Kenneth B Stolarsky


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