Is 3.5 A Rational Number

Rational and Real Numbers

Learning Objective(south)

· Identify the subset(due south) of the real numbers that a given number belongs to.

· Locate points on a number line.

· Compare rational numbers.

· Identify rational and irrational numbers.

Introduction

You've worked with fractions and decimals, like 3.viii and . These numbers can be constitute between the integer numbers on a number line. There are other numbers that tin can be found on a number line, too. When yous include all the numbers that can be put on a number line, yous have the real number line. Let's dig deeper into the number line and come across what those numbers look like. Allow'south take a closer look to see where these numbers fall on the number line.

Rational Numbers

The fraction , mixed number , and decimal 5.33… (or ) all represent the aforementioned number. This number belongs to a set of numbers that mathematicians call rational numbers. Rational numbers are numbers that can be written as a ratio of two integers. Regardless of the form used, is rational because this number can exist written equally the ratio of 16 over iii, or .

Examples of rational numbers include the following.

0.five, every bit information technology can be written as

, as it can be written as

−1.six, as it can be written as

four, as it can be written as

-10, every bit it can be written as

All of these numbers tin can be written equally the ratio of ii integers.

You can locate these points on the number line.

In the following illustration, points are shown for 0.v or , and for 2.75 or .

As yous take seen, rational numbers can exist negative. Each positive rational number has an opposite. The opposite of is , for instance.

Be conscientious when placing negative numbers on a number line. The negative sign means the number is to the left of 0, and the absolute value of the number is the distance from 0. So to place −1.6 on a number line, yous would find a point that is |−one.6| or 1.6 units to the left of 0. This is more than than 1 unit of measurement abroad, but less than 2.

Example
Problem	Identify on a number line.
	Information technology's helpful to first write this improper fraction equally a mixed number: 23 divided past 5 is 4 with a remainder of 3, then is .
	Since the number is negative, you can think of it as moving units to the left of 0. will be between − 4 and − 5.
Answer

Which of the following points represents ?

Prove/Hide Reply

Wrong. This point is just over two units to the left of 0. The betoken should be i.25 units to the left of 0. The correct answer is point B.

Right. Negative numbers are to the left of 0, and should exist 1.25 units to the left. Betoken B is the only point that'south more ane unit and less than 2 units to the left of 0.

Wrong. Notice that this point is between 0 and the commencement unit marking to the left of 0, so it represents a number between −1 and 0. The bespeak for should be 1.25 units to the left of 0. You may have correctly found 1 unit to the left, simply instead of standing to the left another 0.25 unit, you moved right. The correct answer is point B.

Wrong. Negative numbers are to the left of 0, not to the right. The indicate for should be 1.25 units to the left of 0. The correct respond is point B.

Wrong. This betoken is 1.25 units to right of 0, so information technology has the right distance only in the wrong management. Negative numbers are to the left of 0. The correct respond is betoken B.

Comparison Rational Numbers

When two whole numbers are graphed on a number line, the number to the right on the number line is always greater than the number on the left.

The same is truthful when comparing two integers or rational numbers. The number to the right on the number line is always greater than the one on the left.

Hither are some examples.

Numbers to Compare	Comparison	Symbolic Expression
− two and − iii	− ii is greater than − three because − 2 is to the right of − 3	− 2 > − iii or − 3 < − 2
2 and 3	3 is greater than 2 because three is to the right of two	iii > 2 or 2 < three
− three.5 and − 3.1	− iii.1 is greater than − three.5 considering − 3.one is to the right of − 3.5 (meet below)	− 3.1 > − 3.v or − 3.5 < − 3.1

Which of the following are true?

i. −4.1 > three.2

two. −3.2 > −four.1

iii. 3.2 > four.1

iv. −iv.6 < −iv.1

A) i and four

B) i and ii

C) 2 and iii

D) 2 and iv

E) i, 2, and iii

Show/Hide Answer

A) i and 4

Incorrect. −four.half dozen is to the left of −4.one, so −4.6 < −4.1. Nonetheless, positive numbers such as 3.two are e'er to the right of negative numbers such every bit −4.i, so three.2 > −four.1 or −four.i < three.ii. The correct answer is ii and iv, −3.two > −4.i and −iv.vi < −4.one.

B) i and ii

Incorrect. −3.2 is to the right of −4.1, so −3.2 > −iv.i. However, positive numbers such as 3.2 are always to the right of negative numbers such every bit −4.1, so 3.2 > −4.1 or −iv.1 < 3.2. The correct answer is two and iv , −iii.two > −4.1 and −4.6 < −4.i.

C) ii and iii

Incorrect. −3.2 is to the correct of −4.ane, so −3.2 > −4.ane. Even so, 3.2 is to the left of 4.one, then 3.2 < 4.1. The correct answer is two and iv , −3.2 > −4.1 and −4.6 < −4.1.

D) ii and iv

Right. −3.2 is to the right of −4.1, so −iii.2 > −4.ane. Also, −4.6 is to the left of −4.1, so −4.half-dozen < −iv.one.

Eastward) i, ii, and three

Wrong. −3.two is to the right of −4.1, then −3.2 > −4.i. Nonetheless, positive numbers such as iii.2 are always to the correct of negative numbers such every bit −4.1, and so 3.2 > −4.i or −4.1 < 3.2. Besides, 3.two is to the left of 4.1, then iii.2 < 4.1. The correct reply is ii and iv , −3.2 > −four.ane and −4.vi < −4.1.

Irrational and Real Numbers

There are besides numbers that are not rational. Irrational numbers cannot be written as the ratio of ii integers.

Whatever square root of a number that is not a perfect square, for example , is irrational. Irrational numbers are most ordinarily written in i of 3 ways: as a root (such as a square root), using a special symbol (such every bit ), or equally a nonrepeating, nonterminating decimal.

Numbers with a decimal function can either be terminating decimals or nonterminating decimals. Terminating means the digits stop eventually (although y'all can ever write 0s at the end). For example, 1.3 is terminating, because there'south a final digit. The decimal form of is 0.25. Terminating decimals are ever rational.

Nonterminating decimals have digits (other than 0) that continue forever. For example, consider the decimal grade of , which is 0.3333…. The 3s continue indefinitely. Or the decimal form of , which is 0.090909…: the sequence "09" continues forever.

In improver to being nonterminating, these two numbers are too repeating decimals. Their decimal parts are fabricated of a number or sequence of numbers that repeats again and again. A nonrepeating decimal has digits that never grade a repeating pattern. The value of, for example, is 1.414213562…. No thing how far you acquit out the numbers, the digits volition never repeat a previous sequence.

If a number is terminating or repeating, it must be rational; if it is both nonterminating and nonrepeating, the number is irrational.

Blazon of Decimal	Rational or Irrational	Examples
Terminating	Rational	0.25 (or ) i.iii (or )
Nonterminating and Repeating	Rational	0.66… (or ) 3.242424… (or)
Nonterminating and Nonrepeating	Irrational	(or 3.14159…) (or 2.6457…)

Case
Problem	Is −82.91 rational or irrational?
Answer	−82.91 is r ational.	The number is rational, considering it is a terminating decimal.

The set up of real numbers is fabricated past combining the set of rational numbers and the fix of irrational numbers. The real numbers include natural numbers or counting numbers, whole numbers, integers, rational numbers (fractions and repeating or terminating decimals), and irrational numbers. The fix of real numbers is all the numbers that have a location on the number line.

Sets of Numbers

Natural numbers 1, 2, iii, …

Whole numbers 0, i, 2, 3, …

Integers …, −3, −2, −ane, 0, ane, two, iii, …

Rational numbers numbers that can be written as a ratio of two integers—rational numbers are terminating or repeating when written in decimal form

Irrational numbers numbers than cannot be written as a ratio of two integers—irrational numbers are nonterminating and nonrepeating when written in decimal grade

Existent numbers any number that is rational or irrational

Case

Problem

What sets of numbers does 32 belong to?

Answer

The number 32 belongs to all these sets of numbers:

Natural numbers

Whole numbers

Integers

Rational numbers

Real numbers

Every natural or counting number belongs to all of these sets!

Example

Trouble

What sets of numbers does belong to?

Respond

belongs to these sets of numbers :

Rational numbers

Real numbers

The number is rational considering it'southward a repeating decimal. It'due south equal to or or _.

Instance

Problem

What sets of numbers does belong to?

Answer

belongs to these sets of numbers:

Irrational numbers

Real numbers

The number is irrational because information technology can't be written equally a ratio of two integers. Square roots that aren't perfect squares are ever irrational.

Which of the following sets does vest to?

whole numbers

integers

rational numbers

irrational numbers

real numbers

A) rational numbers merely

B) irrational numbers only

C) rational and real numbers

D) irrational and real numbers

E) integers, rational numbers, and existent numbers

F) whole numbers, integers, rational numbers, and existent numbers

Show/Hibernate Answer

A) rational numbers only

Incorrect. The number is rational (it's written equally a ratio of two integers) only it's too real. All rational numbers are also existent numbers. The correct answer is rational and real numbers, because all rational numbers are also real.

B) irrational numbers only

Incorrect. Irrational numbers can't be written as a ratio of two integers. The correct answer is rational and real numbers, because all rational numbers are also existent.

C) rational and real numbers

Right. The number is between integers, so it can't be an integer or a whole number. It'south written as a ratio of two integers, so it's a rational number and not irrational. All rational numbers are real numbers, so this number is rational and existent.

D) irrational and real numbers

Incorrect. Irrational numbers tin't be written as a ratio of 2 integers. The correct respond is rational and real numbers, because all rational numbers are also existent.

E) integers, rational numbers, and real numbers

Incorrect. The number is between integers, not an integer itself. The right answer is rational and real numbers.

F) whole numbers, integers, rational numbers, and real numbers

Incorrect. The number is between integers, so it tin't exist an integer or a whole number. The right respond is rational and real numbers.

Summary

The set of real numbers is all numbers that tin be shown on a number line. This includes natural or counting numbers, whole numbers, and integers. Information technology also includes rational numbers, which are numbers that can be written equally a ratio of two integers, and irrational numbers, which cannot be written equally a the ratio of two integers. When comparing two numbers, the one with the greater value would appear on the number line to the right of the other one.