You’ve worked with fractions and also decimals, like 3.8 and

*
. These numbers have the right to be uncovered in between the integer numbers on a number line. There are other numbers that have the right to be found on a number line, as well. When you encompass all the numbers that can be put on a number line, you have actually the actual number line. Let"s dig deeper right into the number line and view what those numbers look prefer. Let’s take a closer look to see where these numbers fall on the number line.

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The fractivity , combined number

*
, and decimal 5.33… (or ) all represent the same number. This number belongs to a collection of numbers that mathematicians contact rational numbers. Rational numbers are numbers that have the right to be created as a proportion of two integers. Regardmuch less of the form provided,  is rational because this number have the right to be written as the proportion of 16 over 3, or .

Instances of rational numbers encompass the following.

0.5, as it can be written as

*
, as it can be created as
*

−1.6, as it have the right to be created as

*

4, as it can be composed as

*

-10, as it can be composed as

*

All of these numbers deserve to be written as the ratio of two integers.

You deserve to find these points on the number line.

In the following illustration, points are presented for 0.5 or , and for 2.75 or

*
.

*

As you have viewed, rational numbers have the right to be negative. Each positive rational number has actually an oppowebsite. The opposite of  is

*
, for instance.

Be cautious once placing negative numbers on a number line. The negative authorize indicates the number is to the left of 0, and the absolute worth of the number is the distance from 0. So to place −1.6 on a number line, you would certainly uncover a point that is |−1.6| or 1.6 systems to the left of 0. This is more than 1 unit ameans, however much less than 2.

*


Example

Problem

Place

*
 on a number line.

It"s advantageous to initially write this imappropriate fraction as a blended number: 23 separated by 5 is 4 via a remainder of 3, so

*
 is .

Because the number is negative, you have the right to think of it as moving

*
 units to the left of 0.  will certainly be in between −4 and also −5.

Answer

*


Which of the following points represents ?

*


Show/Hide Answer

A)

Incorrect. This point is just over 2 devices to the left of 0. The point need to be 1.25 devices to the left of 0. The correct answer is allude B.

B)

Correct. Negative numbers are to the left of 0, and  should be 1.25 units to the left. Point B is the just allude that’s even more than 1 unit and much less than 2 units to the left of 0.

C)

Incorrect. Notice that this allude is between 0 and the first unit mark to the left of 0, so it represents a number in between −1 and 0. The point for  have to be 1.25 units to the left of 0. You may have actually effectively discovered 1 unit to the left, but rather of continuing to the left one more 0.25 unit, you relocated best. The correct answer is point B.

D)

Incorrect. Negative numbers are to the left of 0, not to the ideal. The point for  should be 1.25 devices to the left of 0. The correct answer is suggest B.

E)

Incorrect. This allude is 1.25 systems to appropriate of 0, so it has actually the correct distance however in the wrong direction. Negative numbers are to the left of 0. The correct answer is point B.

Comparing Rational Numbers


When 2 whole numbers are graphed on a number line, the number to the right on the number line is constantly higher than the number on the left.

The same is true when comparing two integers or rational numbers. The number to the appropriate on the number line is constantly greater than the one on the left.

Here are some examples.


Numbers to Compare

Comparison

Symbolic Expression

−2 and −3

−2 is better than −3 because −2 is to the ideal of −3

−2 > −3 or −3 −2

2 and 3

3 is greater than 2 because 3 is to the ideal of 2

3 > 2 or 2

−3.5 and also −3.1

−3.1 is higher than −3.5 bereason −3.1 is to the right of −3.5 (view below)

−3.1 > −3.5 or

−3.5 −3.1


*

Which of the following are true?

i. −4.1 > 3.2

ii. −3.2 > −4.1

iii. 3.2 > 4.1

iv. −4.6

A) i and also iv

B) i and also ii

C) ii and also iii

D) ii and also iv

E) i, ii, and iii


Show/Hide Answer

A) i and also iv

Incorrect. −4.6 is to the left of −4.1, so −4.6 −4.1 or −4.1 −4.1 and −4.6

B) i and also ii

Incorrect. −3.2 is to the right of −4.1, so −3.2 > −4.1. However before, positive numbers such as 3.2 are always to the best of negative numbers such as −4.1, so 3.2 > −4.1 or −4.1 ii and iv, −3.2 > −4.1 and also −4.6

C) ii and also iii

Incorrect. −3.2 is to the right of −4.1, so −3.2 > −4.1. However before, 3.2 is to the left of 4.1, so 3.2 ii and iv, −3.2 > −4.1 and −4.6

D) ii and iv

Correct. −3.2 is to the best of −4.1, so −3.2 > −4.1. Also, −4.6 is to the left of −4.1, so −4.6

E) i, ii, and iii

Incorrect. −3.2 is to the best of −4.1, so −3.2 > −4.1. However, positive numbers such as 3.2 are always to the ideal of negative numbers such as −4.1, so 3.2 > −4.1 or −4.1 ii and iv, −3.2 > −4.1 and also −4.6


Irrational and also Real Numbers


Tright here are likewise numbers that are not rational. Irrational numbers cannot be written as the proportion of 2 integers.

Any square root of a number that is not a perfect square, for instance , is irrational. Irrational numbers are the majority of generally created in among three ways: as a root (such as a square root), utilizing a unique symbol (such as ), or as a nonrepeating, nonterminating decimal.

Numbers with a decimal component can either be terminating decimals or nonterminating decimals. Terminating indicates the digits stop inevitably (although you have the right to constantly create 0s at the end). For example, 1.3 is terminating, bereason there’s a last digit. The decimal create of  is 0.25. Terminating decimals are constantly rational.

Nonterminating decimals have digits (other than 0) that continue forever before. For instance, think about the decimal create of

*
, which is 0.3333…. The 3s proceed inabsolutely. Or the decimal develop of
*
 , which is 0.090909…: the sequence “09” proceeds forever.

In addition to being nonterminating, these two numbers are additionally repeating decimals. Their decimal components are made of a number or sequence of numbers that repeats aacquire and also aobtain. A nonrepeating decimal has digits that never before create a repeating pattern. The value of, for example, is 1.414213562…. No matter exactly how far you lug out the numbers, the digits will certainly never before repeat a previous sequence.

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


Type of Decimal

Rational or Irrational

Examples

Terminating

Rational

0.25 (or )

1.3 (or

*
)

Nonterminating and Repeating

Rational

0.66… (or

*
)

3.242424… (or)

*

Nonterminating and Nonrepeating

Irrational

 (or 3.14159…)

*
(or 2.6457…)


*


Example

Problem

Is 82.91 rational or irrational?

Answer

−82.91 is rational.

The number is rational, because it is a terminating decimal.


The set of real numbers is made by combining the collection of rational numbers and the set of irrational numbers. The actual numbers include organic numbers or counting numbers, totality numbers, integers, rational numbers (fractions and repeating or terminating decimals), and irrational numbers. The collection of actual numbers is all the numbers that have actually a area on the number line.

Sets of Numbers

Natural numbers 1, 2, 3, …

Whole numbers 0, 1, 2, 3, …

Integers …, −3, −2, −1, 0, 1, 2, 3, …

Rational numbers numbers that deserve to be composed as a proportion of 2 integers—rational numbers are terminating or repeating once composed in decimal form

Irrational numbers numbers than cannot be written as a proportion of 2 integers—irrational numbers are nonterminating and also nonrepeating once written in decimal form

Real numbers any number that is rational or irrational


Example

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

Eincredibly organic or counting number belongs to every one of these sets!


Example

Problem

What sets of numbers does

*
 belengthy to?

Answer

 belongs to these sets of numbers:

Rational numbers

Real numbers

The number is rational because it"s a repeating decimal. It"s equal to

*
 or
*
 or .


Example

Problem

What sets of numbers does

*
 belengthy to?

Answer

*
 belongs to these sets of numbers:

Irrational numbers

Real numbers

The number is irrational bereason it can not be composed as a ratio of 2 integers. Square roots that aren"t perfect squares are constantly irrational.


Which of the complying with sets does

*
 belengthy to?

whole numbers

integers

rational numbers

irrational numbers

actual numbers

A) rational numbers only

B) irrational numbers only

C) rational and actual numbers

D) irrational and actual numbers

E) integers, rational numbers, and actual numbers

F) entirety numbers, integers, rational numbers, and genuine numbers


Show/Hide Answer

A) rational numbers just

Incorrect. The number is rational (it"s created as a ratio of two integers) yet it"s also actual. All rational numbers are also real numbers. The correct answer is rational and real numbers, bereason all rational numbers are also actual.

B) irrational numbers just

Incorrect. Irrational numbers can not be created as a ratio of two integers. The correct answer is rational and actual numbers, because all rational numbers are additionally real.

C) rational and also genuine numbers

Correct. The number is in between integers, so it can not be an integer or a entirety number. It"s written as a proportion of 2 integers, so it"s a rational number and not irrational. All rational numbers are real numbers, so this number is rational and genuine.

D) irrational and genuine numbers

Incorrect. Irrational numbers can"t be written as a ratio of two integers. The correct answer is rational and real numbers, bereason all rational numbers are likewise genuine.

E) integers, rational numbers, and genuine numbers

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

F) totality numbers, integers, rational numbers, and also real numbers

Incorrect. The number is in between integers, so it can not be an integer or a whole number. The correct answer is rational and genuine numbers.

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Summary


The set of actual numbers is all numbers that can be displayed on a number line. This includes organic or counting numbers, whole numbers, and also integers. It additionally has rational numbers, which are numbers that can be created as a ratio of two integers, and also irrational numbers, which cannot be composed as a the proportion of 2 integers. When comparing 2 numbers, the one through the better worth would certainly appear on the number line to the best of the other one.