HSN.RN.B.3Common CoreMathNumber and QuantityGrades 9-12
HSN.RN.B.3: Sums and Products of Rational and Irrational Numbers
In plain English: HSN.RN.B.3 is the Common Core number and quantity standard that asks students to explain why sums and products of two rational numbers are rational, why a rational number plus an irrational number is irrational, and why a nonzero rational number times an irrational number is irrational. Students write short proofs, including proofs by contradiction. It is usually taught in Algebra I.
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.
Common Core State Standards for Mathematics · Domain: The Real Number System (RN) · Cluster: Use properties of rational and irrational numbers. Also written as HSN-RN.B.3 or N-RN.3 · Official standard
Students prove four facts about the real numbers: the sum of two rational numbers is rational, the product of two rational numbers is rational, a rational number plus an irrational number is irrational, and a nonzero rational number times an irrational number is irrational. The first two follow directly from writing rational numbers as fractions of integers. The last two are proofs by contradiction that use the first two.
The emphasis is on explanation. Checking one example on a calculator is not enough, and the lesson shows why: a calculator display always ends, so it cannot tell a rational number from an irrational one. Students also meet the reason for the word "nonzero" and see, as a labeled extension, that sums and products of two irrational numbers follow no rule.
Learning Objectives
By the end of this lesson, students will be able to:
Explain why the sum of two rational numbers is rational by writing a/b + c/d as a single fraction of integers
Explain why the product of two rational numbers is rational by writing (a/b)(c/d) as a single fraction of integers
Prove by contradiction that the sum of a rational number and an irrational number is irrational
Prove by contradiction that the product of a nonzero rational number and an irrational number is irrational, and explain why the rational number must be nonzero
Classify sums and products as rational or irrational and justify each classification
Prior Knowledge Required
Students should already be comfortable with:
Adding, subtracting, multiplying and dividing rational numbers 7.NS.A.2
The definition of irrational numbers and repeating decimals 8.NS.A.1
Locating irrational numbers such as √2 and π approximately on a number line 8.NS.A.2
Using variables to stand for any number in a general statement
Students sort eight numbers into two columns, rational and irrational, and then answer two questions in pairs:
Warm-Up Prompt
"Sort: 0.75, -8, √36, √10, 2/9, 0.111... (the 1 repeats), π, 1.010010001... (one more 0 each time). Then: is 3/4 + 1/6 rational? Is √10 + 1 rational? How could you be sure without a calculator?"
Rational: 0.75 = 3/4, -8 = -8/1, √36 = 6, 2/9 and 0.111... = 1/9. Irrational: √10, π and 1.010010001..., whose decimal never repeats. For the questions, 3/4 + 1/6 = 9/12 + 2/12 = 11/12 is rational because it can be written as a fraction. For √10 + 1, many students say "irrational" but cannot say why. A calculator shows 4.16227766, which stops only because the screen does. Tell students that today they will prove the answer, not guess it.
Direct Instruction20 minutes
Part 1: Definitions and the tools. A number is rational if it can be written as a/b with a and b integers and b ≠ 0. A real number that is not rational is irrational. The proofs use one fact about integers: sums and products of integers are integers, and a product of two nonzero integers is nonzero. Students may use known irrational numbers such as π and √n when n is a positive integer that is not a perfect square.
Part 2: Rational with rational (direct proof). Let r = a/b and s = c/d. Then r + s = (ad + bc)/(bd) and rs = (ac)/(bd). The numerators are integers, and bd is a nonzero integer, so both results are rational.
Part 3: Rational with irrational (proof by contradiction). Model the four-step structure:
Name the numbers: let r be rational and x be irrational.
Assume the opposite: suppose r + x is rational; call it q.
Solve for the irrational number: x = q + (-r). Both q and -r are rational, so by Part 2 their sum is rational.
Reach the contradiction: x would be rational, but x is irrational. So the assumption was false and r + x is irrational.
Repeat for the product: if r ≠ 0 and rx = q, then x = q · (1/r). Here 1/r is rational only because r ≠ 0, and then x would be rational, a contradiction.
Sum of two rational numbers
Show that 2/3 + 5/7 is rational, then explain why a/b + c/d is always rational (a, b, c, d integers, b and d not 0).
Equation: 2/3 + 5/7 = (14 + 15)/21 = 29/21, and in general a/b + c/d = (ad + bc)/(bd): an integer over a nonzero integer
Product of two rational numbers
Show that (-4/9)(15/8) is rational, then explain why (a/b)(c/d) is always rational.
Equation: (-4/9)(15/8) = -60/72 = -5/6, and in general (a/b)(c/d) = (ac)/(bd) with bd ≠ 0
Rational plus irrational, by contradiction
Prove that 4 + √5 is irrational, using the fact that √5 is irrational.
Equation: If 4 + √5 = q with q rational, then √5 = q + (-4), a sum of two rationals, so √5 would be rational. Contradiction.
Nonzero rational times irrational, by contradiction
Prove that (2/3)√7 is irrational, using the fact that √7 is irrational.
Equation: If (2/3)√7 = q with q rational, then √7 = (3/2)q, a product of two rationals, so √7 would be rational. Contradiction.
Why the word "nonzero" matters
Multiply the irrational number √11 by the rational number 0.
Equation: 0 · √11 = 0 = 0/1, which is rational. The proof in the previous example divides by the rational factor, which is impossible when it is 0.
Part 4: Pictures and warnings. Use Diagram 1: adding a rational number slides a point along the number line, and sliding back by the same rational amount returns it, so an irrational point can never land on a rational one. Use Diagram 2 to summarize the four facts in the standard. Point out that its last column (two irrational numbers) has no rule: that case is beyond the standard and only shows why the standard is stated the way it is. Finally, warn that a calculator cannot decide rationality. Every calculator decimal terminates, so every number on the screen looks rational.
Guided Practice15 minutes
Pairs complete four tasks. One partner writes the argument, the other checks that each step names a fact (sum of rationals, product of rationals, or the definition):
Show 3/8 + 5/12 is rational: 9/24 + 10/24 = 19/24.
Show (-6/7)(14/15) is rational: -84/105 = -4/5.
Prove √15 - 2 is irrational. Write it as √15 + (-2); if it equaled a rational q, then √15 = q + 2 would be rational.
Prove -(3/4)√10 is irrational. If it equaled a rational q, then √10 = (-4/3)q would be rational.
Listen for students who "prove" the rational cases with one example only. An example shows one sum is rational; the general form (ad + bc)/(bd) shows every sum is.
Independent Practice15 minutes
Students classify six numbers as rational or irrational and justify each with one of the four facts or by writing a fraction: 0.6 + 2/3 (rational, 19/15), 7 · (5/14) (rational, 5/2), 1/9 + √17 (irrational, rational plus irrational), -5√19 (irrational, nonzero rational times irrational), √64 · (3/8) (rational, √64 = 8 and the product is 3) and 0 · √21 (rational, it equals 0). Students then write one full proof by contradiction of their choice.
Closure5-10 minutes
Exit ticket: (1) Prove that 2/5 + √14 is irrational. (2) Give an example that shows why the standard needs the word "nonzero". (3) Write 0.25 + 5/8 as a single fraction and say which fact from the standard explains why the answer had to be rational. (Answer: 7/8, the sum of two rational numbers is rational.)
Differentiation Strategies
For Struggling Students
Give a proof frame with blanks for the assumption, the equation solved for the irrational number, the fact used and the contradiction
Start every rational proof with numbers, such as 2/3 + 5/7, before replacing them with a/b and c/d
Keep a card with the four facts of the standard and the words "sum of rationals" and "product of rationals" to cite in proofs
For Advanced Students
Ask students to prove that the difference of a rational and an irrational number is irrational using only the facts in the standard
Ask for two irrational numbers whose sum is rational and two whose product is rational, and to explain why this does not contradict the standard
Ask students to read and present a proof that √2 is irrational, then use it with the standard to prove that 5 - 3√2 is irrational
Assessment Guidance
What to Look For
Check that explanations of the rational cases use general fractions a/b and c/d with nonzero denominators, not only one numerical example. In the proofs by contradiction, look for three parts: the assumption that the result is rational, an equation solved for the irrational number, and a sentence naming the fact that makes that expression rational. Students should also say where "nonzero" is used: dividing by the rational factor. Do not accept "the calculator decimal does not repeat" as a reason.
02
Classroom Activities
3 Activities
1
Always, Sometimes or Never
20 minGroups of 3-4
Each group sorts 10 statement cards into three piles: always true, sometimes true, never true. A card counts only with a reason: a general argument for "always" or "never", and two examples (one true, one false) for "sometimes".
The 10 Statement Cards
The sum of two rational numbers is rational. (Always)
The product of two rational numbers is rational. (Always)
The sum of a rational number and an irrational number is rational. (Never)
The product of a rational number and an irrational number is irrational. (Sometimes: false when the rational number is 0)
The product of a nonzero rational number and an irrational number is irrational. (Always)
The square of a rational number is rational. (Always: r · r is a product of two rationals)
Half of an irrational number is irrational. (Always: multiply by the nonzero rational 1/2)
A rational number minus an irrational number is irrational. (Always: r - x = r + (-1)x, and (-1)x is irrational)
Going further: the sum of two irrational numbers is irrational. (Sometimes)
Going further: the product of two irrational numbers is rational. (Sometimes)
Procedure
Groups take turns: one student reads a card and proposes a pile, the next student gives the reason, and the group agrees before moving on
Groups write the reason on the back of each card
Each group presents the card that caused the most discussion
Modification for Distance Learning
Put the 10 statements on a shared slide with three columns. Groups drag each statement into a column and type the reason in a text box beside it.
2
Proof Frames
20 minPairs
Pairs complete a proof frame twice: once for a sum and once for a product. The frame makes the structure of a proof by contradiction visible, so students can focus on which fact justifies each line.
The Frame
Let r = ____ (rational) and x = ____ (irrational).
Suppose ____ is rational, and call it q.
Then x = ____. This is rational because ____.
That contradicts ____. Therefore ____ is irrational.
Cases to Prove
Sum: π + 1/2 (then x = q + (-1/2), a sum of two rationals)
Product: (4/9)√30 (then x = (9/4)q, a product of two rationals)
General sum: r + x for any rational r and irrational x
General product: rx for any nonzero rational r and irrational x (the line "x = q · (1/r)" needs r ≠ 0)
Discussion Questions
Which line of the product frame breaks if r = 0?
Why does the sum proof need the fact about sums of two rational numbers?
Could you prove that π + 1/2 is irrational by computing its decimal? Why or why not?
3
The Calculator Trap
15 minPairs
Pairs use a calculator on four expressions, record the display and predict rational or irrational from the display alone. Then they decide with the standard and compare. The goal is to see that a finite display cannot settle the question.
Expressions
1/7 + 2/3 (display 0.8095238095; rational, it equals 17/21, and the decimal repeats with period 6)
√2 + 1/5 (display 1.614213562; irrational, rational plus irrational)
(5/8)π (display 1.963495408; irrational, nonzero rational times irrational)
1/17 (display 0.0588235294; rational, even though the display shows no repeat, because the period is 16 digits long)
Procedure
For each expression, record the full display and write a prediction
Then classify with a fraction or one of the four facts of the standard
Mark each prediction that was wrong or that the display could not decide
Discussion Questions
Why does every calculator display look like a rational number?
What would you need to see to know a decimal repeats forever?
Which is more convincing, the display or the proof? Why?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Rational Shift or Stretch Keeps √2 Irrational
Number lines drawn to scale. Adding 3/2 moves √2 ≈ 1.414 to √2 + 3/2 ≈ 2.914. If that point were rational, subtracting 3/2 would bring it back to √2 and make √2 rational. Multiplying by 3/2 stretches the distance from 0, sending √2 to (3/2)√2 ≈ 2.121, and multiplying by 2/3 undoes it. Multiplying by 0 cannot be undone, which is why the rational factor must be nonzero.
Diagram 2: What the Standard Says About Sums and Products
The first two columns are the four facts of HSN.RN.B.3. The last column, two irrational numbers, is marked as going further: the result can be rational or irrational, so no general rule exists there.
04
Homework Assignment
~30 min
HSN.RN.B.3 Homework: Rational and Irrational Numbers
Directions: Show all work. You may use the fact that π is irrational and that √n is irrational when n is a positive integer that is not a perfect square. In every explanation, name the fact you use: sum of two rationals, product of two rationals, rational plus irrational, or nonzero rational times irrational.
Part 1: Rational with Rational (Problems 1-2)
Write 5/6 + (-3/8) as a single fraction of integers. Then explain, using a/b and c/d, why the sum of any two rational numbers is rational. Where in your explanation do you use the fact that b and d are not 0?
Write (7/10)(-4/21) as a fraction in lowest terms. Then write 1.25 and 0.444... (the 4 repeats) as fractions and find their product. Explain why both products had to be rational before you computed them.
Part 2: Rational with Irrational (Problems 3-4)
Prove that 3/5 + √6 is irrational. Write the proof in complete sentences, using proof by contradiction.
Prove that -(5/3)√2 is irrational. Then explain why the same proof does not work for 0 · √2, and say whether 0 · √2 is rational.
Part 3: Deciding and Explaining (Problems 5-6)
Classify each number as rational or irrational and justify: (a) 2.5 + 1/8 (b) (3/7)π (c) (1/4) · √5 · 8 (d) √49 + 1/3. For each rational one, write it as a fraction.
Let x be any irrational number. (a) Explain why x + 1/2 is irrational. (b) Explain why 6x + 3 is irrational, citing two facts from the standard in order. (c) Use your answer to (b) to decide whether 6π + 3 is rational or irrational.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Rational Cases
General fraction form with nonzero denominator explained
Correct computation, general reason missing
Examples only, or incorrect
Proofs by Contradiction
Assumption, equation solved for the irrational number, fact named, contradiction stated
One of the four parts missing
No valid structure
Nonzero Condition
Explains that the proof divides by the rational factor
Gives the 0 example without the reason
Missing or incorrect
Classification
All four parts of Problem 5 correct with reasons
Correct answers, some reasons missing
Two or more incorrect
05
Quiz: 20 Questions
Interactive, with answers
Instructions
You may use the fact that π is irrational and that √n is irrational when n is a positive integer that is not a perfect square. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which fraction equals 3/4 + 2/5?
Answer: A
3/4 + 2/5 = 15/20 + 8/20 = 23/20, a ratio of integers, so the sum is rational. Choice B adds numerators and denominators separately. Choice C multiplies the fractions instead of adding them.
Question 2 of 20 · Multiple Choice
Why is the sum of two rational numbers a/b and c/d (b, d ≠ 0) always rational?
Answer: C
Sums and products of integers are integers, so ad + bc is an integer, and bd is a nonzero integer. That is exactly the definition of a rational number. Choice B describes adding fractions incorrectly, and choice D checks examples, which cannot show "always".
Question 3 of 20 · Multiple Choice
What is (-5/6)(9/10) in lowest terms?
Answer: B
(-5/6)(9/10) = -45/60 = -3/4. The product of two rationals is rational: (ac)/(bd). Choice A adds the numerators (-5 + 9) and the denominators (6 + 10). Choice C multiplies the numerators but keeps only one denominator. Choice D drops the negative sign.
Question 4 of 20 · Multiple Choice
In the proof that the product of two rational numbers is rational, (a/b)(c/d) = (ac)/(bd). Which fact makes bd a valid denominator?
Answer: D
A rational number needs an integer denominator that is not 0. Since b ≠ 0 and d ≠ 0, their product bd is a nonzero integer. Choice A is too strong (denominators can be negative), and choice B is about the numerators, which may be 0.
Question 5 of 20 · Multiple Choice
Which number is irrational?
Answer: C
√7 is irrational and 0.25 = 1/4 is rational, so their sum is irrational. The others are traps: √16 = 4, √9 = 3 and √25 = 5 are rational, so choices A, B and D are sums of two rational numbers and are rational (13/3, 5 and 9/2).
Question 6 of 20 · Multiple Choice
To prove 6 + √2 is irrational, a student assumes 6 + √2 = q for a rational q. What is the best next step?
Answer: A
Solving for √2 gives √2 = q + (-6), a sum of two rational numbers, which is rational. That contradicts the fact that √2 is irrational. Choice B relies on a decimal, which cannot prove anything. Choice C squares incorrectly: (6 + √2)² = 38 + 12√2.
Question 7 of 20 · Multiple Choice
Which number is irrational?
Answer: B
π is irrational and -2 is a nonzero rational number, so -2π is irrational. Choice A equals 0, choice C equals 2, and choice D equals 6 · (1/6) = 1, all rational. Choice A is the case the word "nonzero" rules out.
Question 8 of 20 · Multiple Choice
Which product is rational?
Answer: D
0 times any number is 0, and 0 = 0/1 is rational. The other three are nonzero rational numbers times irrational numbers, so they are irrational. This is why the standard is stated for a nonzero rational factor.
Question 9 of 20 · Multiple Choice
To prove (5/2)√3 is irrational, a student assumes (5/2)√3 = q for a rational q. Which line correctly continues the proof?
Answer: A
Multiply both sides by 2/5, which is allowed because 5/2 ≠ 0: √3 = (2/5)q. A product of two rationals is rational, which contradicts √3 being irrational. Choice B subtracts instead of dividing. Choice D confuses a rounded decimal with the exact value.
Question 10 of 20 · Multiple Choice
Write 0.333... (the 3 repeats) + 0.2 as a fraction.
Answer: C
0.333... = 1/3 and 0.2 = 1/5, so the sum is 1/3 + 1/5 = 5/15 + 3/15 = 8/15. Its decimal 0.5333... repeats, as the decimal of every rational number does. Choice A cuts off the repeating decimal at 0.33. Choice B adds numerators and denominators. Choice D forgets that repeating decimals are rational.
Question 11 of 20 · Multiple Choice
Extension (beyond the standard): which pair of irrational numbers has a rational sum?
Answer: B
(3 + √5) + (4 - √5) = 7, which is rational. This shows the standard says nothing about two irrational numbers: their sum can be rational. Choice C gives 3π, a nonzero rational times π, and choice D gives 3√5; both are irrational by the product fact.
Question 12 of 20 · Multiple Choice
A calculator shows 2/7 + √3 ≈ 2.017765093. A student says the number is rational because the decimal ends. What is the best response?
Answer: D
The calculator shows only 10 digits of a decimal that goes on forever. The sum of a rational number and an irrational number is irrational. Choice C reaches the right conclusion for the wrong reason: a few displayed digits cannot show that a decimal never repeats.
Question 13 of 20 · Multiple Choice
If x is irrational, which number must also be irrational?
Answer: C
x/4 = (1/4)x, a nonzero rational times an irrational number, so it is irrational. Choice B equals 0 and choice D equals 1. Choice A can be rational: if x = √2, then x · x = 2.
Question 14 of 20 · Multiple Choice
If p and q are rational numbers, which expression is always rational?
Answer: B
p² = p · p and 3q are products of rationals, and their sum is a sum of rationals, so p² + 3q is rational. Choices A and D contain square roots that may be irrational (for example q = 2), and choice C is irrational whenever p ≠ 0.
Question 15 of 20 · Short Answer
Show that 7/12 + 5/18 is rational by writing it as a single fraction in lowest terms.
The least common denominator is 36: 21/36 + 10/36 = 31/36, an integer over a nonzero integer, so the sum is rational.
Question 16 of 20 · Short Answer
Write (2/9)(-3/8) as a fraction in lowest terms and explain why the result had to be rational.
(2/9)(-3/8) = -6/72 = -1/12. In general (a/b)(c/d) = (ac)/(bd), and ac and bd are integers with bd ≠ 0, so every product of two rational numbers is rational.
Question 17 of 20 · Short Answer
Prove that 1/4 + √11 is irrational. (You may use the fact that √11 is irrational.)
Suppose 1/4 + √11 = q for some rational q. Then √11 = q + (-1/4). Both q and -1/4 are rational, so their sum is rational, and √11 would be rational. That contradicts the fact that √11 is irrational, so 1/4 + √11 is irrational.
Question 18 of 20 · Short Answer
Prove that -6√13 is irrational. (You may use the fact that √13 is irrational.)
Suppose -6√13 = q for some rational q. Since -6 ≠ 0, divide by -6: √13 = (-1/6)q, a product of two rational numbers, which is rational. That contradicts √13 being irrational, so -6√13 is irrational.
Question 19 of 20 · Short Answer
Is 2√3 + 5 rational or irrational? Justify your answer with two facts from the standard, in order.
Irrational. First, 2√3 is a nonzero rational number times an irrational number, so it is irrational. Second, 2√3 + 5 is an irrational number plus a rational number, so it is irrational.
Question 20 of 20 · Short Answer
Explain why the average (r + s)/2 of two rational numbers r and s is rational. Then find the average of 2/5 and 5/6.
r + s is rational (sum of two rationals), and (r + s)/2 = (1/2)(r + s) is a product of two rationals, so it is rational. For 2/5 and 5/6: (12/30 + 25/30)/2 = (37/30)/2 = 37/60.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSN.RN.B.3 mean?
HSN.RN.B.3 means students can explain four facts: rational + rational is rational, rational × rational is rational, rational + irrational is irrational, and nonzero rational × irrational is irrational. The key word is "explain": students give general arguments, not only examples.
Is HSN.RN.B.3 Algebra 1 or Algebra 2?
It is usually taught in Algebra I, often near the start of the year alongside work with radicals, or in the first unit of an integrated Math 1 course. Algebra II courses sometimes revisit it before complex numbers.
How do you prove that the sum of two rational numbers is rational?
Write the two numbers as a/b and c/d with integer numerators and nonzero integer denominators. Then a/b + c/d = (ad + bc)/(bd). The numerator is an integer, and the denominator is a nonzero integer, so the sum fits the definition of a rational number. The product works the same way: (a/b)(c/d) = (ac)/(bd).
Why is a rational number plus an irrational number always irrational?
Because the other outcome leads to a contradiction. If r + x were a rational number q, then x = q - r would be a difference of two rational numbers, which is rational. But x is irrational. So r + x cannot be rational. For example, if 10 + π were rational, π would be too.
Why does HSN.RN.B.3 say "nonzero" rational number?
Because zero times any number is 0, which is rational. The proof for products divides by the rational factor to isolate the irrational number, and dividing by 0 is impossible. For every other rational factor, the product with an irrational number is irrational.
Is the sum of two irrational numbers always irrational?
No. π and -π are both irrational, and their sum is 0. Other pairs, such as √2 and √3, have irrational sums. Products behave the same way: √5 · √5 = 5 is rational. That is why the standard gives rules only for cases that involve at least one rational number.
Do students need to prove that √2 is irrational for this standard?
No. The standard asks students to reason from known irrational numbers, so they may take facts such as "√2 is irrational" or "π is irrational" as given. The classic proof that √2 is irrational makes a good extension for advanced students.
Can a calculator tell whether a number is irrational?
No. A calculator shows a finite number of digits, and every finite decimal is rational. A rational number can even look irrational on a screen: 1/19 has a repeating block of 18 digits, longer than most displays. Only a proof settles the question.
What is proof by contradiction, and why is it used here?
A proof by contradiction assumes the opposite of what you want to show and reaches an impossible conclusion. It fits this standard because "irrational" means "not rational": it is easier to assume a number is rational, write it as a fraction and follow the consequences than to prove directly that no fraction works.
How does HSN.RN.B.3 connect to later math?
The idea that a set is closed under an operation returns with polynomials (HSA.APR.A.1), which are closed under addition, subtraction and multiplication. The facts also explain why solutions of quadratic equations such as 3 ± √2 are irrational (HSA.REI.B.4), and the proof style prepares students for proofs in geometry.
07
Related Standards
6 standards
These standards connect to HSN.RN.B.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.NS.A.2Prerequisite
Apply and extend multiplication and division of fractions to all rational numbers
