Chapter 4
The Irrational

26. The System of Rational Numbers Inadequate. The system of rational numbers, while it suffices for the four fundamental operations of arithmetic and finite combinations of these operations, does not fully meet the needs of algebra.

The great central problem of algebra is the equation, and that only is an adequate number-system for algebra which supplies the means of expressing the roots of all possible equations. The system of rational numbers, however, is equal to the requirements of equations of the first degree only; it contains symbols not even for the roots of such elementary equations of higher degrees as $x^2=2$, $x^2=-1$.

But how is the system of rational numbers to be enlarged into an algebraic system which shall be adequate and at the same time sufficiently simple?

The roots of the equation

The most general characteristics of the roots must be discovered and defined and embodied in symbols—by a method which does not depend on processes for solving equations. These symbols, of course, however characterized otherwise, must stand in consistent relations with the system of rational numbers and their operations.

An investigation shows that the forms of number necessary to complete the algebraic system may be reduced to two: the symbol $\sqrt{-1}$, called the imaginary (an indicated root of the equation $x^2+1=0$), and the class of symbols called irrational, to which the roots of the equation $x^2-2=0$ belong.

27. Numbers Defined by Regular Sequences. The Irrational. On applying to 2 the ordinary method for extracting the square root of a number, there is obtained the following sequence of numbers, the results of carrying the reckoning out to 0, 1, 2, 3, 4, …places of decimals, viz.:

These numbers are rational; the first of them differs from each that follows it by less than 1, the second by less than $\frac{1}{10}$, the third by less than $\frac{1}{100}$, …the $n$th by less than $\frac{1}{10^{n-1}}$. And $\frac{1}{10^{n-1}}$ is a fraction which may be made less than any assignable number whatsoever by taking $n$ great enough.

This sequence may be regarded as a definition of the square root of 2. It is such in the sense that a term may be found in it the square of which, as well as of each following term, differs from 2 by less than any assignable number.

Any sequence of rational numbers

The entire class of operations which lead to regular sequences may be called regular sequence-building. Evolution is only one of many operations belonging to this class.

Any regular sequence is said to “define a number,”—this “number” being merely the symbolic, ideal, result of the operation which led to the sequence. It will sometimes be convenient to represent numbers thus defined by the single letters $a$, $b$, $c$, etc., which have heretofore represented positive integers only.

After some particular term all terms of the sequence ${\alpha }_1$, ${\alpha }_2,\cdots$ may be the same, say $\alpha$. The number defined by the sequence is then $\alpha$ itself. A place is thus provided for rational numbers in the general scheme of numbers which the definition contemplates.

When not a rational, the number defined by a regular sequence is called irrational.

The regular sequence .3, .33, …, has a limiting value, viz., $\frac{1}{3}$; which is to say that a term can be found in this sequence which itself, as well as each term which follows it, differs from $\frac{1}{3}$ by less than any assignable number. In other words, the difference between $\frac{1}{3}$ and the $\mu$th term of the sequence may be made less than any assignable number whatsoever by taking $\mu$ great enough. It will be shown presently that the number defined by any regular sequence, ${\alpha }_1$, ${\alpha }_2,\cdots$ stands in this same relation to its term ${\alpha }_{\mu }$.

28. Zero, Positive, Negative. In any regular sequence ${\alpha }_1,{\alpha }_2,\cdots$ a term ${\alpha }_{\mu }$ may always be found which itself, as well as each term which follows it, is either

(1) numerically less than any assignable number,
or (2) greater than some definite positive rational number,
or (3) less than some definite negative rational number.

In the first case the number $a$, which the sequence defines, is said to be zero, in the second positive, in the third negative.

29. The Four Fundamental Operations. Of the numbers defined by the two sequences:

(1) The sum is the number defined by the sequence:

(2) The difference is the number defined by the sequence:

(3) The product is the number defined by the sequence:

(4) The quotient is the number defined by the sequence:

For these definitions are consistent with the corresponding definitions for rational numbers; they reduce to these elementary definitions, in fact, whenever the sequences ${\alpha }_1$, ${\alpha }_2,\dots$; ${\beta }_1$, ${\beta }_2,\dots$ either reduce to the forms $\alpha$, $\alpha ,\dots$; $\beta$, $\beta ,\dots$ or have rational limiting values.

They conform to the fundamental laws I–IX. This is immediately obvious with respect to the commutative, associative, and distributive laws, the corresponding terms of the two sequences ${\alpha }_1{\beta }_1$, ${\alpha }_2{\beta }_2,\dots$; ${\beta }_1{\alpha }_1$, ${\beta }_2{\alpha }_2,\dots$, for instance, being identically equal, by the commutative law for rationals.

But again division as just defined is determinate. For division can be indeterminate only when a product may vanish without either factor vanishing (cf. § 24); whereas ${\alpha }_1{\beta }_1$, ${\alpha }_2{\beta }_2,\dots$ can define 0, or its terms after the $n$th fall below any assignable number whatsoever, only when the same is true of one of the sequences ${\alpha }_1$, ${\alpha }_2,\dots$; ${\beta }_1$, ${\beta }_2,\dots$¹¹

It only remains to prove, therefore, that the sequences (1), (2), (3), (4) are qualified to define numbers (§ 27).

(1) and (2) Since the sequences ${\alpha }_1$, ${\alpha }_2,\dots$; ${\beta }_1$, ${\beta }_2,\dots$ are, by hypothesis, such as define numbers, corresponding terms in the two, ${\alpha }_{\mu }$, ${\beta }_{\mu }$ may be found, such that

for all values of $\nu$, and that however small $\delta$ may be.

Therefore each of the sequences ${\alpha }_1+{\beta }_1$, ${\alpha }_2+{\beta }_2,\dots$; ${\alpha }_1-{\beta }_1$, ${\alpha }_2-{\beta }_2,\dots$ is regular.

(3) Let ${\alpha }_{\mu }$ and ${\beta }_{\mu }$ be chosen as before.

Then ${\alpha }_{\mu +\nu }{\beta }_{\mu +\nu }-{\alpha }_{\mu }{\beta }_{\mu }$,

since it is identically equal to

Therefore the sequence ${\alpha }_1{\beta }_1,{\alpha }_2{\beta }_2,\dots$ is regular.

By choosing ${\alpha }_{\mu }$ and ${\beta }_{\mu }$ as before the numerator of this fraction, and therefore the fraction itself, may be made less than any assignable number; and that for all values of $\nu$.

Therefore the sequence $\frac{{\alpha }_1}{{\beta }_1},\frac{{\alpha }_2}{{\beta }_2},\dots$ is regular.

30. Equality. Greater and Lesser Inequality. Of two numbers, $a$ and $b$, defined by regular sequences ${\alpha }_1,{\alpha }_2,\dots ,$; ${\beta }_1,{\beta }_2,\dots$, the first is greater than, equal to or less than the second, according as the number defined by ${\alpha }_1-{\beta }_1,{\alpha }_1-{\beta }_2,\dots$ is greater than, equal to or less than $0$.

This definition is to be justified exactly as the definitions of the fundamental operations on numbers defined by regular sequences were justified in § 29.

From this definition, and the definition of $0$ in § 28, it immediately follows that

COR. Two numbers which differ by less than any assignable number are equal.

31. The Number Defined by a Regular Sequence is its Limiting Value. The difference between a number $a$ and the term ${\alpha }_{\mu }$ of the sequence by which it is defined may be made less than any assignable number by taking $\mu$ great enough.

For it is only a restatement of the definition of a regular sequence ${\alpha }_1,{\alpha }_2,\dots$ to say that the sequence

In other words, the limit of $a-{\alpha }_{\mu }$ as $\mu$ is indefinitely increased is 0, or . Hence

The number defined by a regular sequence is the limit to which the $\mu$th term of this sequence approaches as $\mu$ is indefinitely increased.¹²

The definitions (1), (2), (3), (4) of § 29 may, therefore, be stated in the form:

For limit (${\alpha }_{\mu }$) the more complete symbol is also used, read “the limit which ${\alpha }_{\mu }$ approaches as $\mu$ approaches infinity”; the phrase “approaches infinity” meaning only, “becomes greater than any assignable number.”

32. Division by Zero. (1) The sequence $\frac{{\alpha }_1}{{\beta }_1},\frac{{\alpha }_2}{{\beta }_2},\dots$ cannot define a number when the number defined by ${\beta }_1,{\beta }_2,\dots$ is 0, unless the number defined by ${\alpha }_1,{\alpha }_2,\dots$ be also 0. In this case it may; $\frac{{\alpha }_{\mu }}{{\beta }_{\mu }}$ may approach a definite limit as $\mu$ increases, however small ${\alpha }_{\mu }$ and ${\beta }_{\mu }$ become. But this number is not to be regarded as the mere quotient $\frac{0}{0}$. Its value is not at all determined by the fact that the numbers defined by ${\alpha }_1,{\alpha }_2,\dots$; ${\beta }_1,{\beta }_2,\dots$ are 0; for there is an indefinite number of different sequences which define 0, and by properly choosing ${\alpha }_1,{\alpha }_2,\dots$; ${\beta }_1,{\beta }_2,\dots$ from among them, the terms of the sequence $\frac{{\alpha }_1}{{\beta }_1},\frac{{\alpha }_2}{{\beta }_2},\dots$ may be made to take any value whatsoever.

(2) The sequence $\frac{{\alpha }_1}{{\beta }_1},\frac{{\alpha }_2}{{\beta }_2},\dots$ is not regular when ${\beta }_1,{\beta }_2,\dots$ defines 0 and ${\alpha }_1,{\alpha }_2,\dots$ defines a number different from 0.

No term $\frac{{\alpha }_{\mu }}{{\beta }_{\mu }}$ can be found which differs from the terms following it by less than any assignable number; but rather, by taking $\mu$ great enough, $\frac{{\alpha }_{\mu }}{{\beta }_{\mu }}$ can be made greater than any assignable number whatsoever.

Though not regular and though they do not define numbers, such sequences are found useful in the higher mathematics. They may be said to define infinity. Their usefulness is due to their determinate form, which makes it possible to bring them into combination with other sequences of like character or even with regular sequences.

Thus the quotient of any regular sequence ${\gamma }_1,{\gamma }_2,\dots$ by $\frac{{\alpha }_1}{{\beta }_1},\frac{{\alpha }_2}{{\beta }_2},\dots$ is a regular sequence and defines 0; and the quotient of $\frac{{\alpha }_1}{{\beta }_1},\frac{{\alpha }_2}{{\beta }_2},\dots$ by a similar sequence $\frac{{\gamma }_1}{{\delta }_1},\frac{{\gamma }_2}{{\delta }_2},\dots$ may also be regular and serve—if ${\alpha }_i$, ${\beta }_i$, ${\gamma }_i$, ${\delta }_i$ ($i=1,2,\dots$) be properly chosen—to define any number whatsoever.

The term $\frac{{\alpha }_{\mu }}{{\beta }_{\mu }}$ “approaches infinity” (i. e. increases without limit) as $\mu$ is indefinitely increased, in a definite or determinate manner; so that the infinity which $\frac{{\alpha }_1}{{\beta }_1},\frac{{\alpha }_2}{{\beta }_2},\dots$ defines is not indeterminate like the mere symbol $\frac{a}{0}$ of § 22.

But here again it is to be said that this determinateness is not due to the mere fact that ${\beta }_1,{\beta }_2\dots$ defines 0, which is all that the unqualified symbol $\frac{a}{0}$ expresses. For there is an indefinite number of different sequences which like ${\beta }_1,{\beta }_2,\dots$ define 0, and $\frac{a}{0}$ is a symbol for the quotient of $a$ by any one of them.

33. The System defined by Regular Sequences of Rationals, Closed and Continuous. A regular sequence of irrationals

For ${\beta }_1,{\beta }_2,\dots$ being any sequence of rationals which defines 0, construct a sequence of rationals ${\alpha }_1,{\alpha }_2,\dots$ such that $a_1-{\alpha }_1$ is numerically less than ${\beta }_1$ (§ 30), and in the same sense $a_2-{\alpha }_2<{\beta }_2$, $a_3-{\alpha }_3<{\beta }_3$ etc. Then limit (§§ 28, 31), or limit .

This theorem justifies the use of regular sequences of irrationals for defining numbers, and so makes possible a simple expression of the results of some very complex operations. Thus $a^m$, where $m$ is irrational, is a number; the number, namely, which the sequence $a^{{\alpha }_1},a^{{\alpha }_2},\dots$ defines, when ${\alpha }_1,{\alpha }_2,\dots$ is any sequence of rationals defining $m$.

But the importance of the theorem in the present discussion lies in its declaration that the number-system defined by regular sequences of rationals contains all numbers which result from the operations of regular sequence-building in general. It is a closed system with respect to the four fundamental operations and this new operation, exactly as the rational numbers constitute a closed system with respect to the four fundamental operations only (cf. § 25).

The system of numbers defined by regular sequences of rationals—real numbers, as they are called—therefore possesses the following two properties: (1) between every two unequal, real numbers there are other real numbers; (2) a variable which runs through any regular sequence of real numbers, rational or irrational, will approach a real number as limit. We indicate all this by saying that the system of real numbers is continuous.