es
Mathematical analysis depends on the properties of the set of real numbers, denoted by ℝ, so we should begin with studying it.

Real numbers

There are two familiar ways to represent real numbers. Geometrically, they may be pictured as the points (abstract objects of zero size) on a line, once the two reference points corresponding to 0 and 1 have been picked. For computation, however, we represent a real number as an infinite decimal, consisting of an integer part and the sign (plus or minus), followed by infinitely many decimal places:
\[ 3.1415926\ldots , \qquad -0.333333\ldots , \qquad 121.567800000\ldots . \]
Recall that a rational number is any number that can be expressed as a fraction of integers and the denominator is not zero. Examples include all integers (which can be written as a fraction with a denominator of 1), terminating decimals (like 0.3), and repeating decimals (like 0.3(21) = 0.3212121…). The set of all rational mi,bers is denoted by ℚ. But if the decimal does not terminate or recur, the number is not the ratio of two whole numbers and is said to be irrational.

Dedekind

Elementary algebra is concerned with the application of the arithmetic operations (+, − *, and ÷) to symbols representing real numbers. However, there are difficulties with decimal representation that we need to think about. The first is that two different infinite decimals can represent the same real number, for according to well-known rules, a decimal having only 9's after some place represents the same real nuber as a different decimal ending with all 0's (such decimals are called finite or terminating):

\[ 3.1415999\ldots = 3.1416000\ldots . \]
This ambiguity is a serious inconvenience in working theoretically with decimals. Notice that when we write a finite decimal, in mathematics the infinite string of decimal place zeros is dropped, whereas in scientific work, some zeroes are retained to indicate how accurately the number has been determined.    
Example 1: We consider two decimals 2.718000… and 2.7179999…. In the former one, we can simply drop zeroes and consider 2.718 instead, which is \[ 2.718 = 2 + \frac{718}{1000} = \frac{2718}{1000} . \] The latter decimal can be written as \[ 2.7179999\ldots = \frac{2717}{1000} + \frac{9}{10^4} +\frac{9}{10^5} + \cdots . \] Factoring 10−3), we rewrite it as \[ \frac{2717}{1000} + \frac{9}{10} \left[ 1 + \frac{1}{10} + \frac{1}{100} + \cdots \right] . \] The expression in brackets is just https://en.wikipedia.org/wiki/Geometric_series">geometric series, \[ 1 + q + q^2 + q^3 + \cdots = \frac{1}{1-q} \qquad\mbox{with}\quad 1 = \frac{1}{10} . \] Since the sum of this series is known to be 10/9, we represent the second decimal as \[ 2.7179999\ldots = \frac{2717}{1000} + \frac{9}{10^4} +\frac{10}{9} = \frac{2717}{1000} + \frac{1}{1000} = \frac{2718}{1000} , \] which is the first decimal.
N[Exp[1], 16]
2.718281828459045
   ■
End of Example 1

Another difficulty with infinite decimals is that it is not immediately obvious how to calculate with them. For finite decimals or ratios of integers (that could be represented by infinite decimals with periodic fractional parts) there is no problem; in this case we just follow the usual rules---add/subtract or multiply/divide starting at the right-hand end:

\[ \frac{72307}{51063} + \frac{19745}{28433} = \frac{3064143866}{1451874279} = 2.1104746535701938693825431423598 \ldots \]
72307/51063 + 19745/28433
3064143866/1451874279
and
\[ \begin{array}{rl} &3,7986 \\ +&2,0179 \\ \hline &5.8165 \end{array} \qquad \begin{array}{rl} &3,7986 \\ \times&2,0179 \\ \hline &7.66519 \end{array} \]
However, an infinite decimal has no right-hand end!

To get around this, instead of calculating with the infinite decimal, we use its truncations to finite decimals, viewing these as approximations to the infinite decimals. For instance, the increasing sequence of finite decimals

\[ 2 \quad 2.7 \quad 2.71 \quad 2.718 \quad 2.7182 %81828459045 \]
gives ever closer approximation to infinite decimal e = 2.718281828459045…; we say that e is the limit of this sequence (a definition of "limit" will be given shortly).    
Example 2: To see how these approximations work, let us calculate the sum \[ e + \pi . \] We write the sequences of finite decimals that approximate these two numbers \[ \begin{array}{l} e\quad\mbox{is the limit of} \\ \pi \quad\mbox{is the limit of} \end{array} \quad \begin{array} 2. \\ 3. \end{array}\ \begin{array} 2.7 \\ 3.1 \end{array}\ \begin{array} 2.71 \\ 3.14 \end{array}\ \begin{array} 2.718 \\ 3.141 \\ \end{array}\ \begin{array} 2.7182\ldots \\ 3.1415 \ldots \\ \end{array} \] Then we add together the successive decimal approximations: e + π is the limit of 5, 5.8, 5.85, 5.859, 5.8597, 5.85987, 5.859873, …, obtaining a sequence of numbers that also increases.

The decimal representation of this sequence is not as simple as it was for the sequence representing e or π where each new decimal digit is added on. The sequence representing e + π may contain changes in two decimal laces. For instance, in the fifth step of the last row, the first decimal place changes from 7 to 8. Nevertheless, as we compute more and more places, the earlier part of the decimals in this sequence ultimately does not change any more, and in this way we get the decimal expansion of a new number; we then define the sum e + π to be this number.    ■

End of Example 2

As this example shows, even the simplest arithmetic operations with real numbers require an understanding of sequences and their limits. So you get an answer not at once, but rather by making closer and closer approximations to it.

ℝ as a field

We discuss properties of arithmetic operations (addition, subtraction, multiplication, and division) from formal algebraic prospective.

A commutative monoid is a set S equipped with a binary operation S × S → S, which we will denote •, that satisfies the following axioms (using multiplication):
  1. Closure: For any two elements 𝑎, b in semigroup S, the result of the operation, 𝑎 • b, is also an element in S.
  2. Associativity: For all 𝑎, b, and c in group S, the equation (𝑎 • b) • c = 𝑎 • (b • c) holds.
  3. Identity element: There exists an element e in S, such that for all elements 𝑎 in S, the equation e • 𝑎 = 𝑎 • e = 𝑎 holds. This identity element is usually denoted by 1.
  4. Commutativity: For all 𝑎, b in S, 𝑎 • b = b • 𝑎.
A monoid in which each element has an inverse is a group. All three sets (ℝ, •) (real numbers), (ℚ, •) (rational numbers) and (ℤ, •) (integers) are commutative monoids with respect to multiplication.

An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two its elements does not depend on the order in which they are written. That is, the group operations satisfy the following axioms (using addition):
  1. Closure: For any two elements 𝑎, b in A, the result of the operation, 𝑎 + b, is also an element in A.
  2. Associativity: For all 𝑎, b, and c in group A, the equation (𝑎 + b) + c = 𝑎 + (b + c) holds.
  3. Identity element: There exists an element e in A, such that for all elements 𝑎 in A, the equation e + 𝑎 = 𝑎 + e = 𝑎 holds. This element is usually denoted by zero.
  4. Inverse element: For each 𝑎 in A there exists an element b in A such that 𝑎 + b = b + 𝑎 = e (= 0), where e is the identity element (zero).
  5. Commutativity: For all 𝑎, b in A, 𝑎 + b = b + 𝑎.
A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group." The set of real numbers (ℝ, +) as well as its two subsets (ℚ, +) (rational numbers) and (ℤ, +) (integers) are Abelian groups with respect to addition.

The set ℝ, equipped with the usual addition "+" and multiplication "·" (also denoted by "•"), is a field. This means it satisfies the field axioms: commutative groups under addition, commutative monoid under multiplication with multiplicative inverses for nonzero elements, and distributively of multiplication over addition.

A field is a set on which addition, subtraction, multiplication, and division are defined, satisfying the field axioms that are generally written in additive and multiplicative pairs.
Field axioms
name     addition                      multiplication                 
associativity:     (𝑎 + b) + c = 𝑎 + (b + c)     (𝑎 • b) • c = 𝑎 • (b • c)
commutativity:     𝑎 + b = b + 𝑎     𝑎 • b = b • 𝑎
distributivity:     𝑎 • (b + c) = 𝑎•b + 𝑎•c     (𝑎 + b) • c = 𝑎•c + b•c
identity:     𝑎 + 0 = 𝑎 = 0 + 𝑎     𝑎•1 = 1•𝑎 = 𝑎
inverses:     𝑎 + (−𝑎) = 0 = (−a) + 𝑎     𝑎•𝑎−1 = 1 = 𝑎−1•𝑎 if 𝑎 ≠ 0

 

 

  1. Apostol, T.M., Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability, Wiley; 2nd edition, 1991; ISBN-13: ‎ 978-0471000075.