You’ve functioned with fractions and also decimals, prefer 3.8 and

*
. This numbers deserve to be found in between the integer numbers on a number line. Over there are other numbers that can be found on a number line, too. Once you encompass all the numbers that have the right to be placed on a number line, you have actually the actual number line. Let"s destruction deeper into the number line and see what those numbers look like. Let’s take a closer look at to view where this numbers loss on the number line.

You are watching: Every real number is a rational number


The fraction , mixed number

*
, and decimal 5.33… (or ) all stand for the exact same number. This number belongs to a set of numbers the mathematicians contact rational numbers. Reasonable numbers room numbers that can be created as a ratio of two integers. Regardless of the form used,  is rational due to the fact that this number deserve to be created as the proportion of 16 end 3, or .

Examples of rational numbers encompass the following.

0.5, as it can be composed as

*
, together it deserve to be written as
*

−1.6, together it can be composed as

*

4, as it can be created as

*

-10, together it can be written as

*

All of these numbers have the right to be composed as the proportion of two integers.

You can locate this points on the number line.

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

*
.

*

As you have seen, reasonable numbers have the right to be negative. Each hopeful rational number has an opposite. Opposing of  is

*
, because that example.

Be mindful when place negative numbers ~ above a number line. The an unfavorable sign method the number is come the left the 0, and also the absolute worth of the number is the distance from 0. So to ar −1.6 top top a number line, friend would uncover a point that is |−1.6| or 1.6 devices to the left that 0. This is an ext than 1 unit away, yet less than 2.

*


Example

Problem

Place

*
 on a number line.

It"s helpful to an initial write this improper fraction as a combined number: 23 split by 5 is 4 through a remainder that 3, for this reason

*
 is .

Since the number is negative, you deserve to think of the as relocating

*
 units come the left the 0.  will be 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 that 0. The suggest should it is in 1.25 systems to the left that 0. The exactly answer is allude B.

B)

Correct. Negative numbers room to the left the 0, and  should it is in 1.25 systems to the left. Allude B is the only allude that’s an ext than 1 unit and also less than 2 units to the left that 0.

C)

Incorrect. Notice that this suggest is in between 0 and also the very first unit note to the left that 0, so it to represent a number in between −1 and also 0. The suggest for  should it is in 1.25 units to the left that 0. You may have correctly uncovered 1 unit come the left, however instead of continuing to the left an additional 0.25 unit, you relocated right. The exactly answer is suggest B.

D)

Incorrect. An unfavorable numbers room to the left of 0, not to the right. The allude for  should be 1.25 devices to the left of 0. The correct answer is point B.

E)

Incorrect. This allude is 1.25 devices to appropriate of 0, so it has the correct distance however in the dorn direction. An unfavorable numbers are to the left that 0. The correct answer is allude B.

Comparing reasonable Numbers


When 2 whole numbers space graphed on a number line, the number come the ideal on the number heat is always greater than the number on the left.

The very same is true as soon as comparing 2 integers or rational numbers. The number come the appropriate on the number line is always greater 보다 the one top top the left.

Here room some examples.


Numbers come Compare

Comparison

Symbolic Expression

−2 and −3

−2 is greater than −3 since −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 best of 2

3 > 2 or 2

−3.5 and −3.1

−3.1 is better than −3.5 because −3.1 is come the appropriate of −3.5 (see below)

−3.1 > −3.5 or

−3.5 −3.1


*

Which the 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 iii

D) ii and iv

E) i, ii, and iii


Show/Hide Answer

A) i and iv

Incorrect. −4.6 is come the left that −4.1, therefore −4.6 −4.1 or −4.1 −4.1 and −4.6

B) i and ii

Incorrect. −3.2 is come the right of −4.1, for this reason −3.2 > −4.1. However, hopeful numbers such as 3.2 are constantly to the right of an unfavorable numbers such together −4.1, for this reason 3.2 > −4.1 or −4.1 ii and iv, −3.2 > −4.1 and also −4.6

C) ii and iii

Incorrect. −3.2 is come the ideal of −4.1, therefore −3.2 > −4.1. However, 3.2 is come the left that 4.1, therefore 3.2 ii and iv, −3.2 > −4.1 and −4.6

D) ii and iv

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

E) i, ii, and iii

Incorrect. −3.2 is to the right of −4.1, therefore −3.2 > −4.1. However, positive numbers such together 3.2 are constantly to the ideal of negative numbers such together −4.1, for this reason 3.2 > −4.1 or −4.1 ii and also iv, −3.2 > −4.1 and also −4.6


Irrational and also Real Numbers


There are also numbers that are not rational. Irrational numbers cannot be created as the proportion of two integers.

Any square source of a number that is no a perfect square, for example , is irrational. Irrational numbers room most commonly written in among three ways: as a source (such together a square root), using a one-of-a-kind symbol (such as ), or as a nonrepeating, nonterminating decimal.

Numbers through a decimal part can either be terminating decimals or nonterminating decimals. Terminating means the number stop ultimately (although you can constantly write 0s at the end). Because that example, 1.3 is terminating, since there’s a critical digit. The decimal kind of  is 0.25. Terminating decimal are always rational.

Nonterminating decimals have digits (other 보다 0) that proceed forever. Because that example, consider the decimal kind of

*
, which is 0.3333…. The 3s continue indefinitely. Or the decimal type of
*
 , which is 0.090909…: the sequence “09” proceeds forever.

In enhancement to being nonterminating, these two numbers are also repeating decimals. Their decimal parts are make of a number or succession of numbers the repeats again and again. A nonrepeating decimal has actually digits that never kind a repeating pattern. The worth of, for example, is 1.414213562…. No matter how much you lug out the numbers, the number will never repeat a previous sequence.

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


Type that Decimal

Rational or Irrational

Examples

Terminating

Rational

0.25 (or )

1.3 (or

*
)

Nonterminating and also Repeating

Rational

0.66… (or

*
)

3.242424… (or)

*

Nonterminating and also Nonrepeating

Irrational

 (or 3.14159…)

*
(or 2.6457…)


*


Example

Problem

Is 82.91 reasonable or irrational?

Answer

−82.91 is rational.

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


The set the real numbers is made by combine the set of rational numbers and the collection of irrational numbers. The actual numbers include natural numbers or counting numbers, whole numbers, integers, rational numbers (fractions and repeating or end decimals), and irrational numbers. The collection of real numbers is all the number that have a ar 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 created as a ratio of two integers—rational numbers room terminating or repeating once written in decimal form

Irrational number numbers than cannot be composed as a ratio of 2 integers—irrational numbers space nonterminating and nonrepeating as soon as written in decimal form

Real numbers any number the is reasonable or irrational


Example

Problem

What to adjust of number does 32 belong to?

Answer

The number 32 belong to all these to adjust of numbers:

Natural numbers

Whole numbers

Integers

Rational numbers

Real numbers

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


Example

Problem

What to adjust of number does

*
 belong to?

Answer

 belongs to this sets of numbers:

Rational numbers

Real numbers

The number is rational due to the fact that it"s a repeating decimal. It"s equal to

*
 or
*
 or .


Example

Problem

What set of numbers does

*
 belong to?

Answer

*
 belongs to these sets that numbers:

Irrational numbers

Real numbers

The number is irrational since it can"t be written as a ratio of two integers. Square roots that aren"t perfect squares are always irrational.


Which that the adhering to sets does

*
 belong to?

whole numbers

integers

rational numbers

irrational numbers

real numbers

A) rational number only

B) irrational numbers only

C) rational and also real numbers

D) irrational and real numbers

E) integers, rational numbers, and real numbers

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


Show/Hide Answer

A) rational numbers only

Incorrect. The number is rational (it"s composed as a ratio of two integers) but it"s likewise real. Every rational number are additionally real numbers. The correct answer is rational and real numbers, because all rational number are additionally real.

B) irrational numbers just

Incorrect. Irrational numbers can"t be created as a ratio of two integers. The exactly answer is rational and also real numbers, because all rational number are likewise real.

C) rational and also real number

Correct. The number is between integers, so it can"t be an integer or a entirety number. It"s written as a ratio of 2 integers, for this reason it"s a reasonable number and not irrational. Every rational number are real numbers, therefore this number is rational and also real.

D) irrational and real number

Incorrect. Irrational numbers can"t be composed as a proportion of 2 integers. The correct answer is rational and real numbers, because all rational number are also real.

E) integers, rational numbers, and also real number

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

F) entirety numbers, integers, reasonable numbers, and also real number

Incorrect. The number is between integers, so that can"t be an integer or a whole number. The correct answer is rational and also real numbers.

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Summary


The collection of actual numbers is every numbers that deserve to be shown on a number line. This consists of natural or counting numbers, totality numbers, and integers. It additionally includes rational numbers, which are numbers that have the right to be composed as a proportion of 2 integers, and also irrational numbers, which can not be composed as a the proportion of 2 integers. As soon as comparing two numbers, the one with the higher value would appear on the number heat to the appropriate of the various other one.