Chapter 1, Problem 1RCC

(A)

To determine

To define:

The definition of integers

Explanation of Solution

Given:

The integer number

Concept used:

  $21,4,0$ and $-2048$ are integers , while $9.75,5\frac{1}{2}$ and $\sqrt{2}$ are not

The set of integers consists of zero

Calculation:

An integer (from the latin integer meaning whole) is colloquially defined as a number that can be written without a fraction component

For example :- $21,4,0$ and $-2048$ are integers , while $9.75,5\frac{1}{2}$ and $\sqrt{2}$ are not

The set of integers consists of zero

It is denoted by $Z$

  $Z$ is the subset of the set of all rational numbers

  $Z$ is countably infinite

The integers from the smallest group and the smallest ring the natural numbers

(B)

To determine

To define:

The definition of a rational number

Explanation of Solution

Given:

The rational number

Concept used:

Anrational number is a number that can be expressed as the quotient or fraction $\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.$ of two integers a numerator p and a non-zero denominator q

Calculation:

In mathematics

Anrational number is a number that can be expressed as the quotient or fraction $\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.$ of two integers a numerator p and a non-zero denominator q

Since q may be equal to $1$

Every integeris a rational number

The set of all rational numbers , the rationals, field of rational , or the field of rational numbers is usually denoted by $Q$

The decimal expansion of a rational number always either terminates aftera finite number of digits

Any repeating or terminating decimal represents a rational number

(C)

To determine

To define:

The definition of airrational number

Explanation of Solution

Given:

The irrational number

Concept used:

The set of all irrrationalnumbers , the irrationals, field of irrational , or the field of irrational numbers is usually denoted by $Q^c$

Calculation:

In mathematics

An irrational number is a number that can notbe expressed as a fraction for any integers and irrational numbers have decimal expansions that neither terminate nor become periodic

Every transcendental number is irrational number

The irrational numbers are all the real numbers

The ratio of lengths of two line segments is an irrational number

Irrational numbers are the ratio $\pi$ of a circle circumference to its diameter

The set of all irrrationalnumbers , the irrationals, field of irrational , or the field of irrational numbers is usually denoted by $Q^c$

(C)

To determine

To define:

The definition of a real number

Explanation of Solution

Given:

The real number

Concept used:

The set of all real numbers, field of real , or the field of real numbers is usually denoted by $R$

Calculation:

In mathematics

A real number is a value of a continuous quantity that can represent a distance along a line

The real numbers include all the rational numbers

Such as the integers abd fraction and all irrational numbers

The set of real numbers is uncountable

That is both the set of all natural numbers and the set of all real numbers are infinite sets

The cardinality of the set of all real numbers is strictly greater than the cardinality of the set of all natural numbers

The set of all real numbers, field of real , or the field of real numbers is usually denoted by $R$