Chapter 5, Problem 1CRE

To determine

The set to which the given numbers belong and complete the table.

Answer to Problem 1CRE

Solution:

The table can be completed as shown below.

  $\begin{array}{ccccc}\text{Set}\backslash \text{Number}& -22& \pi & 6& -\sqrt{2}\\ \text{Real Number}& \text{Yes}& \text{Yes}& \text{Yes}& \text{Yes}\\ \text{Irrational Number}& & \text{Yes}& & \text{Yes}\\ \text{Rational number}& \text{Yes}& & \text{Yes}& \\ \text{Integers}& \text{Yes}& & \text{Yes}& \\ \text{Whole numbers}& & & \text{Yes}& \\ \text{Natural number}& & & \text{Yes}& \end{array}$

Given:

The given table is shown below.

  $\begin{array}{ccccc}\text{Set}\backslash \text{Number}& -22& \pi & 6& -\sqrt{2}\\ \text{Real Number}& & & & \\ \text{Irrational Number}& & & & \\ \text{Rational number}& & & & \\ \text{Integers}& & & & \\ \text{Whole numbers}& & & & \\ \text{Natural number}& & & & \end{array}$

Concept used:

A value that represents a quantity along a line called a real number. Real numbers include all the rational numbers and irrational numbers.

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.

A number that cannot be expressed as a fraction for any integers and have decimal expansions that neither terminate nor become periodic called irrational number.

A number that can be written without a fractional component is called integer number.

Whole numbers are positive numbers including zero, without any decimal or fractional parts.

A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number.

Explanation of Solution

Calculation:

The number $-22$ represents a quantity along a line so it is a real number, rational number and integer number.

The number $\pi$ represents a quantity along a line so it is a real number and have decimal expansions, so it is also irrational number.

The number 6 represents a quantity along a line, without a fractional component, positive number, so it is a real number, rational number, integer number, whole number and natural number.

The number $-\sqrt{2}$represents a quantity along a line so it is a real number and have decimal expansions, so it is also irrational number.

Conclusion:

The sets for the numbers to which they belong can be represented by completing the given table as shown below.

  $\begin{array}{ccccc}\text{Set}\backslash \text{Number}& -22& \pi & 6& -\sqrt{2}\\ \text{Real Number}& \text{Yes}& \text{Yes}& \text{Yes}& \text{Yes}\\ \text{Irrational Number}& & \text{Yes}& & \text{Yes}\\ \text{Rational number}& \text{Yes}& & \text{Yes}& \\ \text{Integers}& \text{Yes}& & \text{Yes}& \\ \text{Whole numbers}& & & \text{Yes}& \\ \text{Natural number}& & & \text{Yes}& \end{array}$