# Rational and Irrational Numbers Worksheets

This is one of those topics where we have students classify mathematical values or well know constants as they become more familiar with the numeracy at a more sophisticated level. As this may feel like an a tedious exercise, we can assure you that this is a requirement to use more advanced math as the your first step in many cases will be to classify a value so that you know how to approach it and ultimately find the solution. When you move on to using these types of values in many science related courses, it will be mandatory that understand the foundational concepts that we cover here. These worksheets will help students learn to identify if a number should be classified as a rational or irrational number.

### Aligned Standard: HSN-RN.B.3

- Radical 16 Step-by-Step Lesson- See if this number fits the mold. See if you can further simplify the number to make it completely rational.
- Guided Lesson - See if you can be an irrational number super hero or just a zero; pun totally intended.
- Guided Lesson Explanation - Make sure students read the topic, it explains all of the problems at once.
- Practice Worksheet - I like to do this as a timed activity with my students.
- Matching Worksheet - Is it an "A" or a "B"? This is like a true/false worksheet.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

Determine the classification of the value presented.

- Homework 1 - A rational number is a number that can be written as a ratio. That means it can be written as a fraction.
- Homework 2 - An irrational number can be written as a decimal, but not as a fraction.
- Homework 3 - See if we can further simplify the number to make it completely rational.

### Practice Worksheets

What does a farmer have to do with number classification? I wish I knew too.

- Practice 1 - An irrational number can be written as a decimal, but not as a fraction. They are made up of non-repeating numbers and seem like a series of endless digits.
- Practice 2 - A rational number is a number that can be written as a ratio. That means it can be written as a fraction. Both the numerator and denominator of the fraction are whole numbers.
- Practice 3 - Are the numbers that are presented to you rational or irrational numbers?

### Math Skill Quizzes

Find the final value and classify it all together.

- Quiz 1 - See if you can further simplify the number to make it completely rational.
- Quiz 2 - If the number terminate it is therefore a rational number.
- Quiz 3 - Start by stating what you know about rational and irrational numbers.

### Rational and Irrational Numbers: What's the Difference?

Are you often confused between rational and irrational numbers? The thing to remember is that a rational value can be written as a fraction, for instance the value 1/3. They can be written as a ratio between two integers. Working off of our previous example, the ratio is 1 to 3. An irrational number cannot be written as a fraction or a ratio between two integers. The symbol ℼ which we use to represent the ratio of the circumference of a circle to its diameter is an irrational number. Because when you try to express it as a fraction it repeats infinitely as a decimal and cannot be converted.

**Rational Numbers** - A number that can be expressed in a ratio of two values. They can be expressed in fractional form where the denominator is not equal to zero. When written as fractions the numerator and denominator are whole numbers and the denominator is not equal to zero. They are known to include perfect squares. The decimal expansion of rational numbers is either finite or recurring. Examples include whole numbers: 1, 50, and 1,000 | fractions: 1/4, 15/40, 1/100 | decimals: 0.1, 0.54

**Irrational Numbers** - These values that cannot be written as a ratio of two integers. They cannot be written in fractions. They include values that are non-terminating and non-repeating in nature. They are often expressions that include a root symbol. The decimal expansion of irrational numbers is non-finite and non-recurring. Examples include: √2, √11

Students often get to this portion of the curriculum and ask if why the concept of irrational numbers even exists. They often think it is just a bunch of malarky and hogwash. They make many calculations that we use in the areas of geometry, physics, and engineering possible. That is because they help use model situations where real number systems fail or have gaps. When ever you run into a problem that seems to have a non-terminating value they often will come in handy.

### Why Do We Care About Irrational Numbers?

We learn many different things in math that we do not understand the importance. I cannot think of a single topic, except maybe the power of i, that sees this level of criticism by students greater than irrational numbers and values. We lay the foundation of this concept is most high school math classes, but you really will not understand the significance of it until you actually apply it in science class. These imaginary values are used to model physical and theoretical phenomena. We are all comfortable with your standard numbers line, where each successive follow in all directions differs by a fixed unit. When following this pattern, you can always find a rational value for a previous value. For every day scientific experiments this does not hold true due to the theory of chaos and all the possible variables. Take for example my asphalt driveway. Asphalt is a substance that contracts when cool and expands when hot. Do you think that the exact precise measure of length of my driveway differs between the winter and summer months? You can surely bet it does. In fact, I have measured it to have a 2.4 inches in length difference between the months of August and January. Using imaginary numbers, we can model this value and make sense of it.