摘要利用有理数对实数逼近的表示方式，给出黎曼函数处处不可导的一种证明，给出单位圆周上的有理点在单位圆上稠密的证明。

Rational number can approximate to real number use the notation of approximate one can prove riemann function isn't differentiable anywhere that the rational points are dense in unit circle .

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利用有理数对实数逼近的表示方式，给出黎曼函数处处不可导的一种证明，给出单位圆周上的有理点在单位圆上稠密的证明。

Rational number can approximate to real number, use the notation of approximate one can prove Riemann function isn t differentiable anywhere, that the Rational points are dense in unit circle.

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在数集的基础上，在整数域上建立了一个新的交换半群，并在有理数域、实数域和复数域上进行了推广；作为应用，讨论了其元素的表示形式。

Based on the number set, a new commutative semi-group is established in the integer number and extended in number fields of rational number, real number and the complex number.

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