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Let P be the set of primes, and let Q_p and Z_p(t) be the fields of p-adic numbers and formal power series over Z_p=(0,1,...,p-1). Further, suppose that D is a "nonprincipal ...

The points on a line can be put into a one-to-one correspondence with the real numbers.

A proof which can be accomplished using only real numbers (i.e., real analysis instead of complex analysis; Hoffman 1998, pp. 92-93).

The field of rationals is the set of rational numbers, which form a field. This field is commonly denoted Q (doublestruck Q).

If x_1<x_2<...<x_n denote the zeros of p_n(x), there exist real numbers lambda_1,lambda_2,...,lambda_n such that ...

A line in the complex plane with slope +/-i. An isotropic line passes through either of the circular points at infinity. Isotropic lines are perpendicular to themselves.

It is possible to construct simple functions which produce growing patterns. For example, the Baxter-Hickerson function f(n)=1/3(2·10^(5n)-10^(4n)+2·10^(3n)+10^(2n)+10^n+1) ...

A set of numbers obeying a pattern like the following: 91·37 = 3367 (1) 9901·3367 = 33336667 (2) 999001·333667 = 333333666667 (3) 99990001·33336667 = 3333333366666667 (4) 4^2 ...

The only linear associative algebra in which the coordinates are real numbers and products vanish only if one factor is zero are the field of real numbers, the field of ...

The complex plane C with the origin removed, i.e., C-{0}. The punctured plane is sometimes denoted C^* (although this notation conflicts with that for the Riemann sphere C-*, ...