Rational and Irrational Numbers Worksheet, Difference, and Examples

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Key Points about Rational and Irrational Numbers

Rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot.
Rational and irrational numbers are both important in mathematics, and are used in different areas of study.
Understanding the difference between rational and irrational numbers is important for working with numbers in mathematics.

What are Rational and Irrational numbers?

Irrational Numbers are numbers that cannot be written as fractions. Irrational Numbers have two things special about their decimal forms. The first is that Irrational Numbers have decimals that do not terminate, meaning they never end. The second is that Irrational Numbers have decimals that will never repeat in pattern. This means that all integers, whole numbers, and natural numbers are not Irrational Numbers, they are instead Rational Numbers. The most common examples of Irrational Numbers are π, √2, √3, and e.

Rational and irrational numbers are two types of numbers that are used in mathematics. Rational numbers are numbers that can be expressed as a ratio of two integers, while irrational numbers cannot be expressed as a ratio of two integers. This means that irrational numbers cannot be written as a fraction and have decimal expansions that are non-repeating and non-terminating.

The difference between rational and irrational numbers is important to understand because it affects how we work with numbers in mathematics. Rational numbers are used in many different areas of math, including algebra, geometry, and calculus. Irrational numbers are also important in math, particularly in the study of geometry and trigonometry.

Common Core Standard: 8.NS.A
Basic Topics:
Related Topics: Square Roots, Cube Roots, Irrational Numbers, Powers of 10, Scientific Notation Intro, Converting Numbers to Scientific Notation, Converting Numbers from Scientific Notation, Adding and Subtracting in Scientific Notation, Multiplying in Scientific Notation, Dividing in Scientific Notation
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Difference Between Rational and Irrational Numbers

Rational and irrational numbers are two different types of real numbers. While both are real numbers, they have some fundamental differences that set them apart. This section will define both rational and irrational numbers and highlight the differences between them.

Rational Numbers Definition

A rational number is a real number that can be expressed as a ratio of two integers. In other words, it can be written in the form of p/q, where p and q are integers, and q is not equal to zero. For example, 3/4, 5/2, and -7/11 are all rational numbers.

Rational numbers can be positive, negative, or zero. They can also be expressed as terminating or repeating decimals. For example, 1/2 can be expressed as 0.5, which is a terminating decimal. On the other hand, 1/3 can be expressed as 0.333…, which is a repeating decimal.

Irrational Numbers Definition

An irrational number is a real number that cannot be expressed as a ratio of two integers. In other words, it cannot be written in the form of p/q, where p and q are integers, and q is not equal to zero. For example, √2, π, and e are all irrational numbers.

Irrational numbers can be positive or negative. They cannot be expressed as terminating or repeating decimals. Instead, they are expressed as non-repeating, non-terminating decimals. For example, √2 can be expressed as 1.41421356…, which goes on forever without repeating.

Differences Between Rational and Irrational Numbers

The key difference between rational and irrational numbers is that rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot. Another difference is that rational numbers can be expressed as either terminating or repeating decimals, while irrational numbers cannot.

Rational numbers and irrational numbers are both real numbers, which means they can be plotted on a number line. However, irrational numbers are not included in the set of rational numbers. This means that between any two rational numbers, there is always an irrational number.

In summary, rational and irrational numbers are two different types of real numbers. Rational numbers can be expressed as a ratio of two integers and can be expressed as terminating or repeating decimals. Irrational numbers cannot be expressed as a ratio of two integers and cannot be expressed as terminating or repeating decimals.

Rational and Irrational Numbers Chart

Rational and irrational numbers are two types of real numbers. A real number can be either rational or irrational, but not both. In this section, we will discuss the differences between these two types of numbers and provide a chart that illustrates the characteristics of each.

Chart

The chart below summarizes the differences between rational and irrational numbers:

Rational Numbers	Irrational Numbers
Can be expressed as a ratio of two integers	Cannot be expressed as a ratio of two integers
Terminating or repeating decimals	Non-terminating and non-repeating decimals
Can be plotted on a number line	Can be plotted on a number line
Examples: 1/2, 3, -4, 0.75	Examples: √2, π, e

Venn Diagrams

A Venn diagram is a visual representation of the relationships between different sets of data. In the case of rational and irrational numbers, a Venn diagram can be used to show how these two types of numbers are related.

The Venn diagram below shows the relationship between rational and irrational numbers:

Rational and Irrational Numbers Venn Diagram

As the diagram shows, all rational numbers are real numbers, but not all real numbers are rational. Irrational numbers are a subset of real numbers that cannot be expressed as a ratio of two integers.

Number Lines

A number line is a visual representation of the real number system. Rational and irrational numbers can be plotted on a number line to show their relative positions.

The number line below shows the positions of some rational and irrational numbers:

Rational and Irrational Numbers Number Line

On the number line, rational numbers are represented by points that are evenly spaced, while irrational numbers are represented by points that are not evenly spaced. Negative rational and irrational numbers are located to the left of zero, while positive rational and irrational numbers are located to the right of zero.

In conclusion, rational and irrational numbers are two types of real numbers with distinct characteristics. A chart, Venn diagram, and number line can be used to illustrate the differences between these two types of numbers.

Rational and Irrational Numbers Worksheet Solution

4 Simple Rational and Irrational Numbers Examples

Determine if the number can be written as a fraction.
If the number is a whole number it is a Rational Number.
If the number is a terminating decimal it is a Rational Number.
If the number is a repeating decimal it is a Rational Number.
If the number is a non-repeating decimal it is an Irrational Number.
If the number is a non-terminating decimal it is an Irrational Number.

Rational Numbers as Fractions

Rational numbers are numbers that can be expressed as a ratio of two integers. In other words, they can be written as fractions. For example, 3/4, -5/2, and 7/1 are all rational numbers.

Rational Numbers as Decimals

Rational numbers can also be expressed as decimals. If the decimal terminates or repeats, then the number is rational. For example, 0.5, 2.75, and 0.333… (repeating) are all rational numbers.

Irrational Numbers as Roots

Irrational numbers cannot be expressed as a ratio of two integers. One common type of irrational number is a root, such as the square root of 2 or the cube root of 7. These numbers cannot be expressed as a fraction or a terminating decimal.

Irrational Numbers as Decimals

Irrational numbers can also be expressed as decimals. However, unlike rational numbers, the decimal representation of an irrational number never terminates or repeats. For example, pi (π) and the square root of 3 are both irrational numbers with non-repeating, non-terminating decimal representations.

In summary, rational numbers can be expressed as a ratio of two integers or as a terminating or repeating decimal, while irrational numbers cannot be expressed as a ratio of two integers and have non-repeating, non-terminating decimal representations.

5 Quick Rational and Irrational Numbers Practice Problems

Rational Numbers in Real Life

Rational Numbers in Daily Life

Rational numbers are numbers that can be expressed as a fraction of two integers. They are present in many aspects of daily life, from cooking to shopping. For example, when measuring ingredients for a recipe, one might use 1/2 cup of flour or 3/4 teaspoon of salt. When calculating the total cost of items at a store, one might add up the prices of items that cost $1.99 or $2.50.

Rational numbers are also commonly used in measurements of length, width, and height. For example, a piece of paper might measure 8.5 inches by 11 inches, or a room might be 10 feet by 12 feet. In addition, rational numbers are used in calculations involving time, such as determining the number of hours worked or the length of a movie.

Irrational Numbers in Daily Life

Irrational numbers are numbers that cannot be expressed as a fraction of two integers. They are present in many aspects of daily life, from the shapes of objects to the patterns in nature. For example, the value of pi (approximately 3.14159) is an irrational number that is used in calculations involving circles, such as determining the circumference or area of a circle.

Another example of an irrational number is the golden ratio (approximately 1.61803398875), which is a mathematical concept that is found in many natural phenomena, such as the spiral patterns of seashells and the proportions of the human body. Irrational numbers are also used in calculations involving right triangles, such as determining the length of the hypotenuse or the angle measures.

Finally, irrational numbers are used in calculations involving infinity, such as determining the limit of a sequence or the area under a curve. While irrational numbers may seem abstract and theoretical, they are actually present in many aspects of daily life and are essential to our understanding of the world around us.

Is 0 a Rational Number?

Zero is a rational number because it can be expressed as a ratio of two integers, where the denominator is not zero. In other words, zero can be written as 0/1, which is a ratio of the integers 0 and 1. This means that zero is a rational number because it can be expressed as a fraction.

It is important to note that a rational number is any number that can be expressed as a fraction, where the numerator and denominator are integers and the denominator is not zero. Therefore, all integers are rational numbers, including zero.

One way to think about why zero is a rational number is to consider the definition of a rational number. A rational number is a number that can be expressed as a fraction, where the numerator and denominator are integers. Since zero is an integer, it can be the numerator or denominator in a fraction, making it a rational number.

Another way to think about this is to consider the properties of rational numbers. Rational numbers have the property of closure under addition, subtraction, multiplication, and division. This means that if you add, subtract, multiply, or divide two rational numbers, the result will always be a rational number. Since zero is a rational number, it also has this property of closure.

In summary, zero is a rational number because it can be expressed as a ratio of two integers, and it satisfies the properties of rational numbers.

Identifying Rational and Irrational Numbers FAQ

What is the difference between rational and irrational numbers?

Rational numbers can be expressed as a fraction or ratio of two integers, while irrational numbers cannot be expressed as a fraction or ratio of two integers. Irrational numbers are numbers that go on forever, such as pi and the square root of 2.

How do you know if it’s rational or irrational?

To determine if a number is rational or irrational, you can try to express it as a fraction. If the number can be expressed as a fraction, it’s a rational number. If it can’t be expressed as a fraction, it’s an irrational number.

What are 5 examples of irrational numbers?

Some examples of irrational numbers include pi, the square root of 2, the square root of 3, the golden ratio, and e.

Give examples of rational and irrational numbers.

Examples of rational numbers include 3/4, -2, and 1.5. Examples of irrational numbers include pi, the square root of 2, and the square root of 3.

Are negative numbers rational?

Yes, negative numbers can be rational. For example, -3 can be expressed as the fraction -3/1.

Is 3.14 a rational number?

No, 3.14 is not a rational number because it cannot be expressed as a fraction or ratio of two integers.

What is rational and irrational numbers with examples?

Rational numbers are numbers that can be expressed as a fraction or ratio of two integers, while irrational numbers cannot be expressed as a fraction or ratio of two integers. Examples of rational numbers include 1/2, 3/4, and -5. Examples of irrational numbers include pi, the square root of 2, and the golden ratio.

Rational and Irrational Numbers Worksheet Video Explanation

Watch our free video on how to solve Rational and Irrational Numbers. This video shows how to solve problems that are on our free Rational and Irrational Numbers worksheet that you can get by submitting your email above.

Watch the free Rational and Irrational Numbers video on YouTube here: Rational and Irrational Numbers Worksheet

This video is about our rational and irrational numbers worksheet with answers. You can get the rational or irrational worksheet we use in this video for free by clicking on the link in the description below.

Before we do a couple practice problems on our rational and irrational numbers worksheet, I want to go over what exactly a rational number is and what exactly an irrational number is. Now a rational number is any number that you can write as a ratio of two numbers. In other words, any number that you can write as a fraction.

The first type of rational numbers are whole numbers. The reason whole numbers are rational is because every whole number can be written as a fraction. For example if we have the whole number of six, all whole numbers technically have this one underneath of them, because it’s like saying 6 divided by one. We typically do not write this one because obviously 6 divided by one is just six so you do not have to write it because it doesn’t change the number. When you see a whole number, it doesn’t matter what the whole number is, there is a divided by one or a fraction one underneath of it.

The second type of rational number our terminating decimals. Now a terminating decimal is any decimal that stops or ends. If you look at one point seven five, this clearly has an end point. It’s right here after the 5. You can write this as a fraction because you can rewrite 1.75 as a mixed number. It would be 1 and 75 over 100 which would be the unsimplified version but it just proves that you can write it as a fraction.

The third type of rational numbers are repeating decimals. Most commonly you will see point three repeating which is one-third or 0.6 repeating which would be 2/3. It’s any decimal that repeats the same pattern over and over again.

Irrational numbers practice problems are numbers that cannot be written as fractions. The easiest way to remember what an irrational number is, is that it’s any non repeating and non terminating decimal. The most common examples of irrational numbers examples are pi, because it goes on forever. Unless a square root is a perfect square, it will be an irrational number. You can see that both of these decimals do not follow a pattern and the dots indicate that they go on forever so they never end. These are what makes a number irrational.

Moving on to some rational number practice problems on a rational and irrational numbers worksheet. Now the directions for this part of the worksheet just say to identify whether it’s a rational number or an irrational number number. Three gives us negative 8 which is a whole number. Now we know negative 8 can be re-written as negative 8 over 1 and negative 8 over 1 is a fraction, which means it is rational number. Four gives us the cube root of 64. In order to simplify this we have to find what number times what number times what number equals 64. In this case that number is 4. Because it’s 4 times 4 times 4, 4 is a whole number which means the cube root of 64 is rational number.

Eight gives us 4 times the square root of 2. Now we already know that the square root of 2 is an irrational number. This is equal to 1.4142 and it goes on forever. When you multiply four times the square root of 2, you will get five point six five six eight zero and it goes on forever. Our decimal never repeats and because of the dots that means it goes on forever. That means this will be an irrational number.

The square root of 77 is not a perfect square and when you do the square root of 77 in a calculator you get eight point seven seven four nine six that goes on forever. This is neither repeating or terminating so it’s non-repeating, non-terminating which means it is irrational. Try all the practice problems by downloading the free rational vs irrational numbers worksheet above.