Sum of Rational and Irrational Is Irrational

Standard(s) for Mathematical Practice (MP)

MP 3 Construct viable arguments and critique the reasoning of others.
MP 7 Look for and make use of structure.
MP 8 Look for and express regularity in repeated reasoning.

Grade Level

Grades 8–9

Content Domain

The Real Number System (Number and Quantity Conceptual Category)

Standard(s) for Mathematical Content

N-RN.B.3

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Math Topic Keywords

  • rational numbers
  • irrational numbers
  • proof
  • proof by contradiction
  • indirect proof

The mathematics task is intended to be a problem or question that encourages the use of mathematical practices. The dialogue is meant to show how students might engage in the mathematical practices as they work on the task. Before reading the dialogue, work on the mathematics task. Next reflect on the mathematical practices you engaged in while working on the task. Finally read the student dialogue.

Mathematics Task

Is $\sqrt{2}+\frac{1}{2}$ a rational number?

Student Dialogue

Students have previously learned the difference between rational and irrational numbers and have seen examples of each. They are now investigating what sums of different types of numbers will produce.

(1) Chris What’s a rational number again? Isn’t it a fraction?
(2) Lee Yeah, it’s a number that can be written as $\frac{a}{b}$ where $a$ and $b$ are integers and $b$ is not equal to zero.
(3) Chris Ok. So $\sqrt{2}$ is irrational and $\frac{1}{2}$ is rational. What does that tell us about the whole thing, $\sqrt{2}+\frac{1}{2}$?
(4) Matei Well, we basically want to figure out if the following is true: $$ \begin{align*} \sqrt{2}+\frac{1}{2}=\text{rational number} \end{align*} $$ Let’s try rewriting it as: $$ \begin{align*} \sqrt{2}=\text{rational number}-\frac{1}{2} \end{align*} $$
(5) Chris Well… wait! That doesn’t look right! Isn’t the difference of two rational numbers also rational? How can an irrational number equal a rational number?!?! $$ \begin{align*} \underbrace{\sqrt{2}}_{\text{irrational number}}=\underbrace{\text{rational number}-\frac{1}{2}}_{\text{rational number}} \end{align*} $$
(6) Lee You’re right—that doesn’t make sense! So what does that mean?
(7) Matei Well, we assumed $\sqrt{2}+\frac{1}{2}=\text{rational number}$. I guess that must be false. The only other option is for it to be irrational. I wonder though…what would have happened if we started with something different?
(8) Chris Like what?
(9) Matei Well, let’s try $\sqrt{2}-\frac{1}{2}$.
(10) Lee So let’s see if $\sqrt{2}-\frac{1}{2}=\text{rational number}$. We can rewrite that as $\sqrt{2}=\text{rational number}+\frac{1}{2}$.
(11) Chris But we have the same problem as before. An irrational number can’t be equal to a rational number. So that means $\sqrt{2}-\frac{1}{2}$ is irrational, too.
(12) Matei What if we tried $\sqrt{2}+2$?
(13) Lee Well, we can say $\sqrt{2}+2=\text{rational number}$ or $\sqrt{2}=\text{rational number}-2$.
(14) Chris That doesn’t work either. $\sqrt{2}+2$ has to be irrational.
(15) Matei Let’s see if $\sqrt{7}+2$ is rational.
(16) Chris We’re just going to end up with $\sqrt{7}=\text{rational number}-2$ so $\sqrt{7}+2$ is irrational.
(17) Lee Hmm… I’m starting to see a pattern here. Every time we have: $$ \begin{align*} \text{irrational number}\pm\text{rational number}=\text{rational number} \end{align*} $$ we end up rewriting that as: $$ \begin{align*} \text{irrational number}=\text{rational number}\mp\text{rational number} \end{align*} $$
(18) Chris You’re right! And that ends up giving us something that doesn’t make sense. The right-hand side is always a rational number, and that can’t be equal to an irrational number.
(19) Matei Well, I guess that can only mean one thing: $$ \begin{align*} \text{irrational number}\pm\text{rational number}\neq\text{rational number} \end{align*} $$ So the sum of an irrational and a rational number must be irrational.

The teacher reflection questions are intended to prompt thinking about 1) the mathematical practices, 2) the mathematical content and extensions, 3) student thinking, and 4) teaching practices. Begin by reflecting on each of the questions, referring to the student dialogue as needed. After you had some time to do your own reflection, read the possible responses and comment at the bottom of the page with questions or thoughts provoked by the reflection questions and/or responses. Please note some of the mathematics extension tasks are meant for teacher exploration, to prompt engagement in the mathematical practices, and may not be appropriate for student use.

Teacher Reflection Questions

  1. What evidence do you see of the students engaging in the Standards for Mathematical Practice?

    Possible Response

  2. The students in this dialogue are able to recognize that the sum of two rational numbers is always rational (line 5 of the dialogue). If students were not able to see this for themselves, how could you help them come to this realization?

    Possible Response

  3. What challenges might students encounter if they had tried to show that $\sqrt{2}+\frac{1}{2}$ is irrational? How would you help students overcome that challenge?

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  4. If students have trouble understanding what is meant by a contradiction, what other examples of contradictions can you offer that students might be more familiar with?

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  5. In line 3, Chris says “$\sqrt{2}$ is irrational.” Prove that this is true.

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  6. What is the product of a rational and irrational number? Prove your conjecture

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  7. Make a conjecture about the sum of two irrational numbers and support your reasoning.

    Possible Response

  8. Write down 20 irrational numbers. What do you notice?

    Possible Response

The mathematical overview is intended to 1) explain how students in the dialogue engaged in the mathematical practices and 2) further explore the mathematical content and its extensions. Read the mathematical overview and comment at the bottom of this page with questions or thoughts it provoked.

Mathematical Overview

Mathematical Practice Evidence

Construct viable arguments and critique the reasoning of others.

According to MP 3, “They [students] make conjectures and build a logical progression of statements to explore the truth of their conjectures.” In this dialogue, students make the conjecture that $\sqrt{2}+\frac{1}{2}$ is rational (line 4) and explore to see if that conjecture is true. In the process, they find a contradiction (lines 5–7), thereby “distinguish[ing] correct logic or reasoning from that which is flawed.” Even though the students do not identify their process as such, they are in fact using the same logic as that found in a proof by contradiction (indirect proof).

Look for and express regularity in repeated reasoning.

Students in this dialogue “notice if calculations are repeated, and look both for general methods and for shortcuts.” They test several numerical cases (lines 7–16) before they make a generalization about the sum of an irrational and rational number (lines 17–19). Working through several concrete examples allows students to see that the contradiction they found when testing to see if $\sqrt{2}+\frac{1}{2}$ is rational was not limited to that case alone. By seeing that the conjecture was false for several examples, they are able to develop a new, more general conjecture—that the sum of an irrational and rational number is irrational—and begin to understand why that is true.

Look for and make use of structure.

In the dialogue, students “can see complicated things… as single objects.” In line 5, Chris is able to see that the expression "$\text{rational number}-\frac{1}{2}$” is a rational number by knowing something about the structure of rational numbers and the way they can be added/subtracted together, even though the student never fully explains the thinking used in the dialogue. Chris’s ability to identify expressions as rational is also based on the use of “clear definitions” (MP 6) for a rational number as a fraction of two integers with a non-zero denominator.

Commentary on the Mathematics

Indirect proofs

Indirect proofs can be challenging at first because of what may appear as backwards logic. Getting a contradiction (a false statement) in the process of trying to prove something and using that contradiction to draw a conclusion may seem bizarre to students. This type of proof is also not always purposeful. Sometimes in the process of trying to prove a statement, you may come across a contradiction that tells you something about the original statement you are trying to prove. Indirect proofs are very beneficial when trying to prove things that are difficult to quantify either due to their inability to be expressed in a generic form, as in the case of irrational numbers, or due to their inability to be measured, as in the case of an infinite number of objects (e.g., prime numbers). In these cases, making an assumption that your number is rational or that prime numbers are finite leads us to a contradiction that helps prove our solution is irrational or our set is infinite.

Student discussion questions and related mathematics tasks are supplementary materials intended for optional classroom use with students. If you choose to use this task and dialogue with your students these discussion questions are intended to stimulate discussion and further exploration of the mathematics. Related mathematics tasks are intended to provide students an opportunity to engage in the mathematical practices as they connect to content that is similar to or an extension of that found in the dialogue. Please note, responses are to be read by teachers only in preparation for using the questions and tasks with their class.

Student Materials

Student Discussion Questions

  1. What is a rational number? What is an irrational number?

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  2. In line 5, Chris identifies $\text{rational number}-\frac{1}{2}$ as a rational number. How do you think the student knows that it is a rational number?

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  3. How do students realize that $\sqrt{2}+\frac{1}{2}$ is an irrational number?

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  4. How do students come up with the conclusion that the sum of any rational and irrational number is always irrational?

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Related Mathematics Tasks

On odd and even integers

  1. For any integer $n$, what expression can be written to represent all even numbers? What expression can be written to represent all odd numbers? Explain how you came up with your expressions.

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  2. Explain why the sum of an even and an even is always even.

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  3. Explain why the sum of an odd and an odd is always even.

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  4. Explain why the sum of an even and an odd is always odd.

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  5. Explain why the product of an odd and even is always even.

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  6. Explain why the product of odd and odd is always odd.

    Possible Response

On rational and irrational numbers

  1. Find a pair of irrational numbers whose sum is rational. Find a pair of irrational numbers whose sum is irrational.

    Possible Response

  2. What can you say about the product of two irrational numbers? Support your answer.

    Possible Response

  3. Is $\sqrt{2}\cdot\frac{1}{2}$ a rational number? Explain why or why not.

    Possible Response

  4. What is the product of a rational and irrational number? Prove your conjecture.

    Possible Response

 
 
  • MP 3: Construct viable arguments and critique the reasoning of others.

    MP 3: Construct viable arguments and critique the reasoning of others.

    Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their  conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though  they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

  • MP 7: Look for and make use of structure.

    MP 7: Look for and make use of structure.

    Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(xy)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

  • MP 8: Look for and express regularity in repeated reasoning.

    MP 8: Look for and express regularity in repeated reasoning.

    Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Comments

Submitted by marcsmith on
a.) As I began this problem I pictured and drew out a number line. I then approximated the location of sqrt(2) and thought about adding .5 and finding the sums location on the number line. I kept looking for structure (MP7) picturing rational distances being added to sqrt(2) until I believed that there was no such rational distance that could be added in order to produce a rational number. I then tried to formally state my thoughts and look for a contradiction. I was lead to the same conclusion as the students in line 5 that the difference of rational numbers is rational which would contradict the sqrt(2) being irrational. b.) In line 11 the students are demonstrating mastery of MP8 (Look for and express regularity in repeated reasoning.) Even though there is no formal proof they are continually evaluating the reasonableness of their intermediate results and connecting the ideas from before. c.) Question 6: What is the product of a rational and an irrational number? Prove your conjecture. The product is going to be irrational I first thought of vectors of an irrational length. The scalar multiples of these vectors will again have an irrational length or on a number line be of the same conjugacy class. More formally. Proof by contradiction. Say r is a rational number and x is an irrational number. Suppose r(x) = q with q rational. Rewrite x = q/r which is a contradiction.