Number systems: synthesis and challenges

Summary

This lesson closes the unit on number systems. It gathers into a single deductive chain what the lessons of the unit proved: the construction of the integers, the rational numbers and the complex numbers from the previous system, the axioms of a complete ordered field that are admitted for the real numbers, with completeness as their distinctive feature, and the consequences of each property for calculating with inequalities, decimals, radicals, equations and complex numbers. It explains how that chain rests on the arithmetic of the natural numbers from the previous unit and how it prepares the study of functions and polynomials in the next unit. It ends with an exit assessment, parallel to the entry assessment, and with ten challenges, harder than the problems of the lessons, each of which combines results from several of them.

Learning objectives

Set out in logical order the main definitions and theorems of the unit, indicating which earlier results each one depends on.
Explain which property each numerical extension gains and which it loses, and why completeness cannot be obtained by algebraic means.
Relate the results of the unit to the arithmetic of the natural numbers and to the study of algebraic functions and polynomials.
Solve non-routine problems that require combining the arithmetic of fractions, the order of the real numbers, calculation with radicals, linear systems and complex numbers.

What the unit achieved

The unit started from a shortcoming and from a principle. The shortcoming: in the natural numbers one cannot always subtract, divide or extract roots. The principle, stated in the lesson on number systems: each new system preserves the operations and the properties of the previous one and adds the elements needed for an inverse operation, previously restricted, to become defined. The result is the chain of inclusions that organizes the whole unit:

\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}

The constructions

Three of the four extensions were made with pairs of elements of the previous system, and two of them, that of the integers and that of the rational numbers, with one and the same procedure, which is worth recognizing as such. One declares when two pairs represent the same number and proves that this declaration behaves like an equality, that is, that it is an equivalence relation; its classes then form a partition, as was proved in the lesson on sets, relations and functions, and the new set is the quotient set; then the operations are defined with representatives, and one proves that the result does not depend on the representatives chosen. In the integers, the pair of natural numbers $(a, b)$ represents the subtraction $a - b$ , and two pairs are equivalent when $a + d = b + c$ ; in the rational numbers, the fraction $\frac{p}{q}$ represents a quotient, and two fractions are equivalent when $ps = qr$ ; in the complex numbers, the pair of real numbers $(a, b)$ is already the number, with a product defined so that $(0, 1)^2 = (-1, 0)$ . In all three cases, the previous system reappears inside the new one through a correspondence that respects the operations, and what in the intuitive presentation were conventions becomes a theorem: the rule of signs, the absence of zero divisors, cancellation in products, the uniqueness of the fraction in lowest terms and the equality $i^2 = -1$ .

Order and completeness

The fourth extension is of a different nature, and it was not made by construction. The rational numbers already allow addition, subtraction, multiplication and division, and they have an order compatible with those operations, dense and Archimedean; but the set of rational numbers whose square is less than $2$ is bounded and has no rational supremum. Since the construction of the real numbers from the rational numbers lies beyond the course, the lesson on number systems admitted them as a complete ordered field: a set that contains the rational numbers and satisfies the field axioms, the order axioms and the completeness axiom, according to which every nonempty set that is bounded above has a supremum. The first two groups are also satisfied by the rational numbers; the third is the one that characterizes the real numbers. From completeness were deduced, in the lesson on the decimal expansion of rational numbers, the Archimedean property of the real numbers and the value of every infinite decimal expansion, with the characterization of the rational numbers as the numbers with a terminating or repeating expansion; and in the lesson on the real line, the existence of the $n$ th root of every nonnegative real number, the density of the rational numbers in the line and the characterization of intervals as the sets without gaps. None of these properties is obtained with sums and products: completeness is a property of the order, and that is why the real numbers cannot be constructed by adding to the rational numbers the solutions of certain equations.

Calculation

The calculation techniques of the unit rest on these foundations. The lesson on powers with rational exponents and radicals used the uniqueness of the principal root to prove that $a^{\frac{p}{q}}$ does not depend on the fraction chosen and that the laws of exponents hold, and it established the laws of radicals with their sign hypotheses, among them $\sqrt{a^2} = \abs{a}$ . The lesson on first-degree equations and inequalities turned the properties of fields and of order into rules for transforming equations and inequalities without losing or adding solutions, and it classified systems of two equations by means of their determinant. Finally, the complex numbers completed the algebra: every quadratic equation with real coefficients has a solution in them, and when its discriminant is negative it has exactly two, conjugate to each other; and the geometry of the plane was translated into operations with the conjugate and the modulus. The price was order: in $\mathbb{C}$ there is none compatible with the operations, and the size of a number is measured by its modulus.

What is gained and what is lost

The balance of the unit can be summarized as follows:

From $\mathbb{N}$ to $\mathbb{Z}$ one gains the opposite, and with it subtraction; one loses well-ordering, because the integers have no least element.
From $\mathbb{Z}$ to $\mathbb{Q}$ one gains the inverse of every nonzero number, and with it division; one loses the successor, because between two rational numbers there is always another.
From $\mathbb{Q}$ to $\mathbb{R}$ one gains completeness, and with it the roots of nonnegative numbers and the measure of every line segment; one loses the terminating or repeating expansion of all numbers.
From $\mathbb{R}$ to $\mathbb{C}$ one gains the solution of every quadratic equation and, by the fundamental theorem of algebra, of every nonconstant polynomial equation; one loses the order compatible with the operations.

The following figure gathers the three segments into a single chain. Each node of the column is a definition or a theorem, numbered according to the logical order and joined by an arc to the earlier links that its proof uses; around it float the supports, that is, the principles and tools to which the chain resorts again and again, each one joined by a solid line to the links that rest on it and by a dotted one to the link where it is established, except those that are admitted or come from the previous unit, which are drawn as rings. Clicking a support highlights all the links that use it, and each link refers to the lesson in which it is proved.

Supports of the chain

APrinciple of extensions. Each new number system preserves the operations and the properties of the previous one, and adds the elements needed for an inverse operation, previously restricted, to become defined; it is adopted as the guide of the whole unit. Admitted without proof in this unit.
BArithmetic of the natural numbers. The properties of addition, multiplication and order of the natural numbers, induction, well-ordering and the sum of progressions, proved in the unit on the natural numbers. Admitted without proof in this unit.
CDivision with remainder. For every integer $a$ and every natural number $b$ there are a unique quotient $q$ and a unique remainder $r$ such that $a = bq + r$ and $0 \leq r < b$ ; proved in the unit on the natural numbers, with the rules of the integers that the lesson on the integers proves. Admitted without proof in this unit.
DBézout and Euclid. If $\operatorname{gcd}(a, b) = 1$ , there exist integers $x$ and $y$ with $ax + by = 1$ , and a prime that divides a product divides one of the factors; proved in the unit on the natural numbers. Admitted without proof in this unit.
EConstruction by classes. One takes pairs from the previous system, proves that the condition identifying two pairs is an equivalence relation, takes as the new set the quotient, whose classes form a partition of the pairs, and defines the operations with representatives, proving that the result does not depend on the ones chosen. Established in: 2. Construction of the integers.
FAxioms of a complete ordered field. It is admitted that the real numbers form a complete ordered field containing the rational numbers: addition and multiplication satisfy the field axioms; the positive numbers satisfy trichotomy and are closed under addition and multiplication; and every nonempty set of real numbers that is bounded above has a supremum, which is the axiom that characterizes the real numbers. Admitted without proof in this unit.
GUniqueness of the root. Two nonnegative real numbers with the same $n$ th power are equal; that is why it suffices to exhibit a nonnegative number whose power is the given one to know that it is the principal root. Established in: 11. Uniqueness of the principal root.
HArchimedean property. For every real number there exists a natural number greater than it; consequently, a real number $d$ with $0 \leq d \leq \frac{1}{10^n}$ for every natural number $n$ is zero. Established in: 13. Archimedean property.

Segment 1 The constructions
- Construct the integers as classes of pairs of natural numbers and the rational numbers as classes of fractions, with well-defined operations.
- Turn into theorems the rule of signs, the absence of zero divisors and cancellation in products.
- Prove that every rational number has a unique fraction in lowest terms, and that the order of the rational numbers is dense and Archimedean.
- Construct the complex numbers as pairs of real numbers, with a product that gives $i^2 = -1$ .
1. Uniqueness of the identity and of the opposite. The additive identity is unique and each number has a unique opposite, so that $-(-c) = c$ ; likewise, each nonzero number has a unique inverse.
  Proved in: Number systems: from the natural to the complex numbers.
2. Construction of the integers. The relation that makes the pairs of natural numbers $(a, b)$ and $(c, d)$ equivalent when $a + d = b + c$ is an equivalence relation; its classes are the integers, with a well-defined addition and multiplication, commutative, associative and distributive, with the identities $C(1, 1)$ and $C(2, 1)$ and the opposite $-C(a, b) = C(b, a)$ .
  Rests on: B (Arithmetic of the natural numbers). Proved in: The integers: construction, signs and order.
3. Trichotomy of the integers. Identifying the natural number $n$ with the class $C(n + 1, 1)$ preserves addition and multiplication, and every integer is exactly one of these things: a natural number, zero or the opposite of a natural number.
  Uses: 2. Construction of the integers. Rests on: B (Arithmetic of the natural numbers), A (Principle of extensions). Proved in: The integers: construction, signs and order.
4. Rule of signs. For all integers, $a \cdot 0 = 0$ , $(-a) b = -(ab)$ and $(-a)(-b) = ab$ hold; moreover, if $ab = 0$ , one of the factors is zero, and a nonzero factor common to two equal products cancels.
  Uses: 1. Uniqueness of the identity and of the opposite, 2. Construction of the integers, 3. Trichotomy of the integers. Proved in: The integers: construction, signs and order.
5. Order of the integers. If $a < b$ means that $b - a$ is natural, adding the same integer preserves the order, multiplying by a positive number preserves it and by a negative number reverses it; and division with remainder of a negative dividend reduces to that of a natural dividend.
  Uses: 1. Uniqueness of the identity and of the opposite, 3. Trichotomy of the integers, 4. Rule of signs. Rests on: C (Division with remainder). Proved in: The integers: construction, signs and order.
6. Construction of the rational numbers. The relation that makes the fractions $\frac{p}{q}$ and $\frac{r}{s}$ equivalent when $ps = qr$ is an equivalence relation; its classes are the rational numbers, with a well-defined addition and multiplication, and every nonzero rational number $\frac{p}{q}$ has $\frac{q}{p}$ as its inverse.
  Uses: 1. Uniqueness of the identity and of the opposite, 4. Rule of signs. Rests on: E (Construction by classes), A (Principle of extensions). Proved in: The rational numbers: fractions and their operations.
7. Fraction in lowest terms. Every rational number is represented by a fraction in lowest terms with a natural denominator, and by only one.
  Uses: 4. Rule of signs, 6. Construction of the rational numbers. Rests on: D (Bézout and Euclid). Proved in: The rational numbers: fractions and their operations.
8. Order of the rational numbers. With natural denominators, $\frac{p}{q} < \frac{r}{s}$ when $ps < qr$ ; this order does not depend on the fractions chosen, extends that of the integers and is compatible with the operations, and it is dense and Archimedean: between two rational numbers there are infinitely many, and every rational number is exceeded by a natural number.
  Uses: 5. Order of the integers, 6. Construction of the rational numbers. Proved in: The rational numbers: fractions and their operations.
9. Construction of the complex numbers. With $(a, b) + (c, d) = (a + c, b + d)$ and $(a, b)(c, d) = (ac - bd, ad + bc)$ , the pairs of real numbers form the complex numbers: their operations are commutative, associative and distributive, the pairs $(a, 0)$ operate like the real numbers, $i = (0, 1)$ satisfies $i^2 = -1$ and each complex number is written in a unique way as $a + bi$ .
  Rests on: A (Principle of extensions), F (Axioms of a complete ordered field). Proved in: The complex numbers: operations and equations.
Segment 2 Order and completeness
- Deduce from the order axioms the rules for operating with inequalities and the uniqueness of the principal root.
- Prove that the rational numbers are not complete, because the set of rational numbers with square less than $2$ has no rational supremum.
- Deduce from completeness the Archimedean property, the value of every infinite decimal expansion and the characterization of the rational numbers by their terminating or repeating expansions.
- Deduce from completeness the density of the rational numbers in the line, the existence of roots and the characterization of intervals, and operate with the absolute value.
1. Order of the real numbers. If the positive numbers satisfy trichotomy and are closed under addition and multiplication, the order of the real numbers is transitive, it is preserved by adding and by multiplying by a positive number, it is reversed by multiplying by a negative number, and taking the inverses of two positive numbers reverses their order.
  Uses: 4. Rule of signs. Rests on: F (Axioms of a complete ordered field). Proved in: Number systems: from the natural to the complex numbers.
2. Uniqueness of the principal root. If $p$ and $r$ are nonnegative and $p^n = r^n$ , then $p = r$ ; if $n$ is odd, each real number has at most one $n$ th root, which is negative when the number is.
  Uses: 4. Rule of signs, 10. Order of the real numbers. Rests on: B (Arithmetic of the natural numbers). Proved in: Number systems: from the natural to the complex numbers.
3. The rational numbers are not complete. The square root of $2$ is not rational; hence the set of rational numbers whose square is less than $2$ is bounded above, but none of its rational upper bounds is the least.
  Uses: 8. Order of the rational numbers, 11. Uniqueness of the principal root. Rests on: D (Bézout and Euclid). Proved in: Number systems: from the natural to the complex numbers.
4. Archimedean property. For every real number $x$ there exists a natural number greater than $x$ ; consequently, if $0 \leq d \leq \frac{1}{10^n}$ for every natural number $n$ , then $d = 0$ .
  Uses: 10. Order of the real numbers. Rests on: F (Axioms of a complete ordered field), B (Arithmetic of the natural numbers). Proved in: The decimal expansion of rational numbers.
5. Value of a decimal expansion. The long division of $p$ by $q$ gives digits whose truncations differ from $\frac{p}{q}$ by less than $\frac{1}{10^n}$ ; and every infinite decimal expansion represents a unique real number, which for the long division is $\frac{p}{q}$ ; a positive real number has two expansions only when one is terminating and the other ends in nines.
  Uses: 6. Construction of the rational numbers, 10. Order of the real numbers. Rests on: B (Arithmetic of the natural numbers), C (Division with remainder), F (Axioms of a complete ordered field), H (Archimedean property). Proved in: The decimal expansion of rational numbers.
6. Terminating expansions. The decimal expansion of a fraction in lowest terms $\frac{p}{q}$ is terminating if and only if $q = 2^a 5^b$ with $a$ and $b$ nonnegative integers.
  Uses: 7. Fraction in lowest terms, 14. Value of a decimal expansion. Rests on: D (Bézout and Euclid). Proved in: The decimal expansion of rational numbers.
7. Expansions of the rational numbers. The decimal expansion of every positive rational number $\frac{p}{q}$ is terminating or repeating, and the length of its period is at most $q - 1$ .
  Uses: 14. Value of a decimal expansion. Rests on: B (Arithmetic of the natural numbers), C (Division with remainder). Proved in: The decimal expansion of rational numbers.
8. Generating fraction. Every repeating decimal expansion represents a rational number, and the purely periodic one with period $A$ , of $k$ digits, represents $\frac{A}{10^k - 1}$ ; hence a real number is rational if and only if its expansion is terminating or repeating.
  Uses: 14. Value of a decimal expansion, 16. Expansions of the rational numbers. Rests on: B (Arithmetic of the natural numbers), H (Archimedean property). Proved in: The decimal expansion of rational numbers.
9. Density of the rational numbers. Between two distinct real numbers there is always a rational number, and therefore infinitely many.
  Uses: 10. Order of the real numbers. Rests on: B (Arithmetic of the natural numbers), H (Archimedean property). Proved in: The real line: order, intervals and absolute value.
10. Existence of the root. For every natural number $n \geq 2$ and every real number $a \geq 0$ there exists a real number $s \geq 0$ such that $s^n = a$ , which is the principal $n$ th root of $a$ .
  Uses: 10. Order of the real numbers. Rests on: F (Axioms of a complete ordered field), G (Uniqueness of the root). Proved in: The real line: order, intervals and absolute value.
11. Characterization of intervals. A set of real numbers with at least two elements is an interval if and only if it contains all the numbers lying between any two of its elements.
  Uses: 10. Order of the real numbers. Rests on: F (Axioms of a complete ordered field). Proved in: The real line: order, intervals and absolute value.
12. Absolute value. The equalities $\sqrt{x^2} = \abs{x}$ and $\abs{xy} = \abs{x} \abs{y}$ hold; the condition $\abs{x - a} < r$ is equivalent to $a - r < x < a + r$ , and the triangle inequality $\abs{x + y} \leq \abs{x} + \abs{y}$ holds.
  Uses: 4. Rule of signs, 10. Order of the real numbers. Rests on: G (Uniqueness of the root). Proved in: The real line: order, intervals and absolute value.
Segment 3 Calculation
- Define powers with rational exponents, prove that they do not depend on the fraction chosen and deduce the laws of exponents.
- Establish the laws of radicals with their sign hypotheses, among them $\sqrt{a^2} = \abs{a}$ .
- Transform equations and inequalities without losing or adding solutions, and classify systems of two equations by means of their determinant.
- Operate with the conjugate and the modulus, solve in $\mathbb{C}$ every quadratic equation with real coefficients and recognize that $\mathbb{C}$ admits no order compatible with its operations.
1. Laws of exponents. For $a > 0$ , the power $a^{\frac{p}{q}}$ , the principal $q$ th root of $a^p$ , does not depend on the fraction chosen, and $a^x a^y = a^{x + y}$ , $(a^x)^y = a^{xy}$ and $(ab)^x = a^x b^x$ hold; if $a > 1$ , it increases with the exponent.
  Uses: 6. Construction of the rational numbers, 19. Existence of the root. Rests on: G (Uniqueness of the root). Proved in: Powers with rational exponents and radicals.
2. Laws of radicals. For nonnegative radicands, the root of a product is the product of the roots and the root of a root is the root whose index is the product; $\sqrt[n]{a^n}$ is $\abs{a}$ if $n$ is even, and $\sqrt{a + b} \leq \sqrt{a} + \sqrt{b}$ , with equality only if $ab = 0$ .
  Uses: 19. Existence of the root, 21. Absolute value. Rests on: G (Uniqueness of the root). Proved in: Powers with rational exponents and radicals.
3. Equivalent equations. Adding the same expression to both sides or multiplying them by a nonzero number preserves the solutions of an equation; in an inequality, multiplying by a positive number preserves the direction and by a negative number reverses it; and squaring can add the solutions of $A = -B$ .
  Uses: 1. Uniqueness of the identity and of the opposite, 10. Order of the real numbers. Proved in: First-degree equations and inequalities, and linear systems.
4. Systems of two equations. If the determinant $D$ of the system is not zero, the unique solution is $x = \frac{D_x}{D}$ , $y = \frac{D_y}{D}$ ; if $D = 0$ and each equation has some nonzero coefficient, the system has infinitely many solutions when $D_x = D_y = 0$ , and none otherwise.
  Uses: 24. Equivalent equations. Proved in: First-degree equations and inequalities, and linear systems.
5. Conjugate and modulus. The conjugate of a sum or of a product is the sum or the product of the conjugates, $z \conj{z} = \abs{z}^2$ , $\abs{zw} = \abs{z} \abs{w}$ and $\abs{z + w} \leq \abs{z} + \abs{w}$ .
  Uses: 9. Construction of the complex numbers, 21. Absolute value. Rests on: G (Uniqueness of the root). Proved in: The complex numbers: operations and equations.
6. Inverse of a complex number. Every nonzero complex number $z$ has a unique inverse, $\frac{\conj{z}}{\abs{z}^2}$ ; therefore one can divide by every nonzero complex number, and a product of complex numbers is zero only if one of its factors is zero.
  Uses: 1. Uniqueness of the identity and of the opposite, 26. Conjugate and modulus. Proved in: The complex numbers: operations and equations.
7. Quadratic equation. If $a$ , $b$ and $c$ are real, $a \neq 0$ and $\Delta = b^2 - 4ac < 0$ , the equation $az^2 + bz + c = 0$ has in $\mathbb{C}$ exactly two solutions, $\frac{-b \pm i \sqrt{-\Delta}}{2a}$ , which are conjugate and not real.
  Uses: 19. Existence of the root, 27. Inverse of a complex number. Proved in: The complex numbers: operations and equations.
8. Conjugate roots. If a polynomial with real coefficients vanishes at $z_0$ , it also vanishes at $\conj{z_0}$ ; so its non-real roots come in pairs of conjugates.
  Uses: 26. Conjugate and modulus. Proved in: The complex numbers: operations and equations.
9. The complex numbers cannot be ordered. In $\mathbb{C}$ there is no order compatible with the operations: the square of every nonzero number would be positive, and then $1$ and $i^2 = -1$ would be positive, and so would their sum, which is $0$ .
  Uses: 4. Rule of signs, 9. Construction of the complex numbers. Proved in: Number systems: from the natural to the complex numbers.

Towards algebraic functions. The chain stops where numbers cease to be studied in isolation: the next unit studies rules that assign numbers to numbers, with domains that are intervals, nonnegative radicands, sign tables and the conjugate roots theorem for factoring polynomials with real coefficients.