# Real Numbers: Constructive Mathematics

To keep track of position and time and quantities of different sorts, such as mass, volume, temperature, velocity,…, we use *numbers:*

- denotes the set of
*natural numbers* - denotes the set of
*integers*, - denotes the set of
*rational numbers*, where with .

A rational number has a *decimal representation* or *expansion* with a finite number of non-zero decimals or a neverending periodically repeating decimal representation. For example, and

A neverending decimal representation which is not periodic , is said to be an* irrational number.* The set of rational and irrational numbers with finite or infinite decimal representations forms the set of* real numbers* denoted by *.*

In *constructive mathematics* the decimals of decimal expansions, typically of roots to given equations, are computed successively by iterative methods such as *fixed point iteration* with *Newton’s method* as special case, which generate sequences of numbers as successively more accurate root approximations.

We say that a *sequence of real numbers* , also denoted or for short, is *convergent* if for a positive constant and a positive constant :

- for (1)

A convergent sequence of real numbers specifies a unique decimal expansion which defines a real number called the *limit* of the sequence, satisfying

- for (2)

To see this we note that for :

- ,

since

- ,

where we used the following formula for the sum of a finite geometric series with :

For example, if and , then it follows from (2) that the first decimals of are specified by the first decimals of , for .

We shall see that a convergent sequence is typically constructed by *time stepping* or *fixed point iteration* (with *Newton’s method* as an important special case) with each iteration determining a new decimal of the limit if (and binary digit if ) and as a root to an equation for a given function .

We shall see that for Newton’s method the number of correct decimals doubles with each iteration which gives a very fast convergence.

In short, we will construct convergent sequences by fixed point iteration with the decimals of the limit succesively being determined by the decimals of for increasing .

**Example 1: **Consider the sequence with , that is the sequence, . We see that

- ,

showing that the sequence satisfies (1) with and and thus converges, and we see that since .

**Example 2: **We shall discover (MST Chapter 8 and 190) that the sequence generated by the successive iteration for with , which is Newton’s method for computing a root of the equation , that is the sequence , converges and .

**Example 3**: Below we will meet a typical case of time stepping, where we will compare the result computed with time step and the result computed with half the time step , and find that with a constant independent of , which leads to a sequence satisfying (1) with , so that one binary digit of precision is gained in each step.

**Rema****rk:** We will use the concepts of convergence and limit only with (1) as criterion, and not in the more general but more vague epsilon-delta form typically used in standard versions of Calculus (to define continuity and derivate in terms of limits):

- if for all there is such that for .

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