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Real Numbers and Integers: Properties and Operations, Study notes of Calculus

Virginia Highlands Community CollegeCalculus

An introduction to the real numbers and integers, including their primitive terms, operations, and properties. It covers the concepts of addition, multiplication, less than, absolute value, and the properties of equality for real numbers. It also includes axioms for the real numbers and integers.

What you will learn

  • What is the meaning of the statement 'a is less than b' for real numbers?
  • How is the product of two real numbers calculated?
  • What is the definition of a real number?
  • How is the sum of two real numbers calculated?
  • What is the definition of an integer?

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The Real Numbers and the Integers
PRIMITIVE TERMS
To avoid circularity, we cannot give every term a rigorous mathematical definition; we have to
accept some things as undefined terms. For this course, we will take the following fundamental
notions as primitive undefined terms. You already know what these terms mean; but the only facts
about them that can be used in proofs are the ones expressed in the axioms listed below (and any
theorems that can be proved from the axioms).
•Real number: Intuitively, a real number represents a point on the number line, or a (signed)
distance left or right from the origin, or any quantity that has a finite or infinite decimal
representation. Real numbers include integers, positive and negative fractions, and irrational
numbers like √2, π, and e.
•Integer: An integer is a whole number (positive, negative, or zero).
•Zero: The number zero is denoted by 0.
•One: The number one is denoted by 1.
•Addition: The result of adding two real numbers aand bis denoted by a+b, and is called
the sum of aand b.
•Multiplication: The result of multiplying two real numbers aand bis denoted by ab or a·b
or a×b, and is called the product of aand b.
•Less than: To say that ais less than b, denoted by a < b, means intuitively that ais to the
left of bon the number line.
DEFINITIONS
In all the definitions below, aand brepresent arbitrary real numbers.
•The numbers 2through 10 are defined by 2 = 1+1, 3 = 2+1, etc. The decimal representations
for other numbers are defined by the usual rules of decimal notation: For example, 23 is defined
to be 2 ·10 + 3, etc.
•The additive inverse or negative of ais the number −athat satisfies a+ (−a) = 0, and
whose existence and uniqueness are guaranteed by Axiom 9.
•The difference between aand b, denoted by a−b, is the real number defined by a−b=
a+ (−b), and is said to be obtained by subtracting bfrom a.
•If a6= 0, the multiplicative inverse or reciprocal of ais the number a−1that satisfies
a·a−1= 1, and whose existence and uniqueness are guaranteed by Axiom 10.
•If b6= 0, the quotient of aand b, denoted by a/b, is the real number defined by a/b =ab−1,
and is said to be obtained by dividing aby b.
•A real number is said to be rational if it is equal to p/q for some integers pand qwith q6= 0.
•A real number is said to be irrational if it is not rational.
•The statement ais less than or equal to b, denoted by a≤b, means a < b or a=b.
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The Real Numbers and the Integers

PRIMITIVE TERMS

To avoid circularity, we cannot give every term a rigorous mathematical definition; we have to accept some things as undefined terms. For this course, we will take the following fundamental notions as primitive undefined terms. You already know what these terms mean; but the only facts about them that can be used in proofs are the ones expressed in the axioms listed below (and any theorems that can be proved from the axioms).

  • Real number: Intuitively, a real number represents a point on the number line, or a (signed) distance left or right from the origin, or any quantity that has a finite or infinite decimal representation. Real numbers include integers, positive and negative fractions, and irrational numbers like

2, π, and e.

  • Integer: An integer is a whole number (positive, negative, or zero).
  • Zero: The number zero is denoted by 0.
  • One: The number one is denoted by 1.
  • Addition: The result of adding two real numbers a and b is denoted by a + b, and is called the sum of a and b.
  • Multiplication: The result of multiplying two real numbers a and b is denoted by ab or a · b or a × b, and is called the product of a and b.
  • Less than: To say that a is less than b, denoted by a < b, means intuitively that a is to the left of b on the number line.

DEFINITIONS

In all the definitions below, a and b represent arbitrary real numbers.

  • The numbers 2 through 10 are defined by 2 = 1+1, 3 = 2+1, etc. The decimal representations for other numbers are defined by the usual rules of decimal notation: For example, 23 is defined to be 2 · 10 + 3, etc.
  • The additive inverse or negative of a is the number −a that satisfies a + (−a) = 0, and whose existence and uniqueness are guaranteed by Axiom 9.
  • The difference between a and b, denoted by a − b, is the real number defined by a − b = a + (−b), and is said to be obtained by subtracting b from a.
  • If a 6 = 0, the multiplicative inverse or reciprocal of a is the number a−^1 that satisfies a · a−^1 = 1, and whose existence and uniqueness are guaranteed by Axiom 10.
  • If b 6 = 0, the quotient of a and b, denoted by a/b, is the real number defined by a/b = ab−^1 , and is said to be obtained by dividing a by b.
  • A real number is said to be rational if it is equal to p/q for some integers p and q with q 6 = 0.
  • A real number is said to be irrational if it is not rational.
  • The statement a is less than or equal to b, denoted by a ≤ b, means a < b or a = b.
  • The statement a is greater than b, denoted by a > b, means b < a.
  • The statement a is greater than or equal to b, denoted by a ≥ b, means a > b or a = b.
  • A real number a is said to be positive if a > 0. The set of all positive real numbers is denoted by R+, and the set of all positive integers by Z+.
  • A real number a is said to be negative if a < 0.
  • A real number a is said to be nonnegative if a ≥ 0.
  • A real number a is said to be nonpositive if a ≤ 0.
  • If a and b are two distinct real numbers, a real number c is said to be between a and b if either a < c < b or a > c > b.
  • For any real number a, the absolute value of a, denoted by |a|, is defined by

|a| =

a if a ≥ 0 , −a if a < 0.

  • If a is a real number and n is a positive integer, the nth power of a, denoted by an, is the product of n factors of a. The square of a is the number a^2 = a · a.
  • If a is a nonnegative real number, the square root of a, denoted by

a, is the unique nonnegative real number whose square is a (see Theorem 9 below).

  • If n and k are integers, we say that n is divisible by k if there is an integer m such that n = km.
  • An integer n is said to be even if it is divisible by 2, and odd if not.
  • If S is a set of real numbers, a real number b is said to be a maximum of S or a largest element of S if b is an element of S and, in addition, b ≥ x whenever x is any element of S. The terms minimum and smallest element are defined similarly.
  • If S is a set of real numbers, a real number b (not necessarily in S) is said to be an upper bound for S if b ≥ x for every x in S. It is said to be a least upper bound for S if every other upper bound b′^ for S satisfies b′^ ≥ b. The terms lower bound and greatest lower bound are defined similarly.

PROPERTIES OF EQUALITY

In modern mathematics, the relation “equals” can be used between any two “mathematical objects” of the same type, such as numbers, matrices, ordered pairs, sets, functions, etc. To say that a = b is simply to say that the symbols a and b represent the very same object. Thus the concept of “equality” really belongs to mathematical logic rather than to any particular branch of mathematics.

Equality always has the following fundamental properties, no matter what kinds of objects it is applied to. In the following statements, a, b, and c can represent any mathematical objects whatsoever. (In our applications, they will usually be real numbers.)

  1. (The least upper bound axiom) Every nonempty set of real numbers that has an upper bound has a least upper bound.

SELECTED THEOREMS

These theorems can be proved from the axioms in the order listed below.

  1. Properties of zero (a) a − a = 0. (b) 0 − a = −a. (c) 0 · a = 0. (d) If ab = 0, then a = 0 or b = 0.
  2. Properties of signs (a) −0 = 0. (b) −(−a) = a. (c) (−a)b = −(ab) = a(−b). (d) (−a)(−b) = ab. (e) −a = (−1)a.
  3. More distributive properties (a) −(a + b) = (−a) + (−b) = −a − b. (b) −(a − b) = b − a. (c) −(−a − b) = a + b. (d) a + a = 2a. (e) a(b − c) = ab − ac = (b − c)a. (f) (a + b)(c + d) = ac + ad + bc + bd. (g) (a + b)(c − d) = ac − ad + bc − bd = (c − d)(a + b). (h) (a − b)(c − d) = ac − ad − bc + bd.
  4. Properties of inverses (a) If a is nonzero, then so is a−^1. (b) 1−^1 = 1. (c) (a−^1 )−^1 = a if a is nonzero. (d) (−a)−^1 = −(a−^1 ) if a is nonzero. (e) (ab)−^1 = a−^1 b−^1 if a and b are nonzero. (f) (a/b)−^1 = b/a if a and b are nonzero.
  5. Properties of quotients (a) a/1 = a. (b) 1/a = a−^1 if a is nonzero. (c) a/a = 1 if a is nonzero. (d) (a/b)(c/d) = (ac)/(bd) if b and d are nonzero. (e) (a/b)/(c/d) = (ad)/(bc) if b, c, and d are nonzero. (f) (ac)/(bc) = a/b if b and c are nonzero. (g) a(b/c) = (ab)/c if c is nonzero. (h) (ab)/b = a if b is nonzero. (i) (−a)/b = −(a/b) = a/(−b) if b is nonzero. (j) (−a)/(−b) = a/b if b is nonzero. (k) a/b + c/d = (ad + bc)/(bd) if b and d are nonzero. (l) a/b − c/d = (ad − bc)/(bd) if b and d are nonzero.
  1. Transitivity of inequalities (a) If a < b and b < c, then a < c. (b) If a ≤ b and b < c, then a < c. (c) If a < b and b ≤ c, then a < c. (d) If a ≤ b and b ≤ c, then a ≤ c.
  2. Other Properties of inequalities (a) If a ≤ b and b ≤ a, then a = b. (b) If a < b, then −a > −b. (c) 0 < 1. (d) If a > 0, then a−^1 > 0. (e) If a < 0, then a−^1 < 0. (f) If a < b and a and b are both positive, then a−^1 > b−^1. (g) If a < b and c < d, then a + c < b + d. (h) If a ≤ b and c < d, then a + c < b + d. (i) If a ≤ b and c ≤ d, then a + c ≤ b + d. (j) If a < b and c > 0, then ac < bc. (k) If a < b and c < 0, then ac > bc. (l) If a ≤ b and c > 0, then ac ≤ bc. (m) If a ≤ b and c < 0, then ac ≥ bc. (n) If a < b and c < d, and a, b, c, d are nonnegative, then ac < bd. (o) If a ≤ b and c ≤ d, and a, b, c, d are nonnegative, then ac ≤ bd. (p) ab > 0 if and only if a and b are both positive or both negative. (q) ab < 0 if and only if one is positive and the other is negative. (r) There is no smallest positive real number. (s) (Density) If a and b are two distinct real numbers, then there are infinitely many rational numbers and infinitely many irrational numbers between a and b.
  3. Properties of squares (a) For every a, a^2 ≥ 0. (b) a^2 = 0 if and only if a = 0. (c) a^2 > 0 if and only if a > 0. (d) (−a)^2 = a^2. (e) (a−^1 )^2 = 1/a^2. (f) If a^2 = b^2 , then a = ±b. (g) If a < b and a and b are both nonnegative, then a^2 < b^2. (h) If a < b and a and b are both negative, then a^2 > b^2.
  4. Properties of Square Roots (a) If a is any nonnegative real number, there is a unique nonnegative real number

a such that

a

= a. (b) If a = b and a and b are both nonnegative, then

a =

b. (c) If a < b and a and b are both nonnegative, then

a <

b. (d) If a^2 = b and b is nonnegative, then a = ±

b.

  1. Properties of Absolute Values

(a) If a is any real number, then |a| ≥ 0. (b) |a| = 0 if and only if a = 0. (c) |a| > 0 if and only if a 6 = 0. (d) | − a| = |a|. (e) |a| =

a^2.