Cauchy sequences, Dedekind cuts, base-10 expansions and continued fractions are examples of well-known representations of irrational numbers. But there exist others, not so popular, which can be defined using various kinds of sum approximations and best approximations. In this paper we investigate the complexity of a number of such representations.

For any fast-growing computable function f, we define an irrational number ${\alpha }_f$ by using a series of reciprocals of powers of all primes. We prove that certain representations of ${\alpha }_f$ are of low computational complexity (which does not depend on f), whereas others, apparently similar representations, can be of arbitrarily high computational complexity (which depends on f). The existence of computable numbers like ${\alpha }_f$ allows us to prove new and non-trivial theorems on the computational complexity of representations without resorting to the standard computability-theoretic machinery involving enumerations and diagonalizations.

In the paper we also show how to construct irrational numbers γ whose representations by a Cauchy sequences are of low computational complexity, but whose base-b expansion may be of arbitrarily high computational complexity for all bases b. Moreover, for any ${\mathcal{E}}^2$-irrational number α, there will be an ${\mathcal{E}}^2$-irrational number β, such that $\alpha +\beta$ has the complexity of γ. As a consequence, two numbers which have, let us say, base-10 expansions of low computational complexity, may add up to a number whose base-10 expansion is of arbitrarily high computational complexity. The same goes for representations by base-2 expansions, base-17 expansions, Dedekind cuts, continued fractions, and so on.

Cauchy序列，Dedekind割，以10为底的展开和连续分数是无理数的众所周知表示的示例。但是，还有一些不是很流行的，可以使用各种总和和最佳近似来定义。在本文中，我们研究了许多此类表示的复杂性。

对于任何快速增长的可计算函数f，我们定义一个无理数${\alpha }_F$通过使用一系列所有素数幂的倒数。我们证明${\alpha }_F$具有较低的计算复杂度（不依赖于f），而其他表面上相似的表示形式可以具有任意高的计算复杂度（取决于f）。可计算数字的存在，例如${\alpha }_F$ 使我们能够证明表示的计算复杂度新的且很重要的定理，而无需求助于涉及枚举和对角化的标准可计算性理论机制。

在本文中，我们还展示了如何构造无理数γ，该无理数γ的柯西序列表示具有较低的计算复杂度，但对于所有基数b，其base- b扩展可能具有任意高的计算复杂度。而且，对于任何${\mathcal{Ë}}^2$-无理数α，将有一个${\mathcal{Ë}}^2$-无理数β，使得$\alpha +\beta$具有γ的复杂度。结果，两个具有低计算复杂度的以10为底的扩展数，可以加起来一个以10为基础扩展具有任意高的计算复杂度的数。基数为2的扩展，基数为17的扩展，Dedekind削减，连续分数等的表示方式也是如此。