Proof by contradiction: how to be so wrong you end up being right

math SAT
By Roel R.

a tribute o my frinds since grade school-4Mathematical proofs are what make math objective: while you could find a few examples that "prove" a mathematical statement, it is often more important to write a rigorous proof that holds true in all cases. Mathematicians have a few methods in their toolkit to tackle different proofs. In this post, we will learn how to write a proof by contradiction.

In order to prove that some statement p is true, we will assume that the opposite of p (denoted as p) is true. We then show that, by assuming p is true, we arrive a contradiction, thus proving that p is false (i.e. p is true). This can be a tricky process to understand, so let us clarify with examples.

Example 1: Prove that √3 is irrational.

We will begin by letting p be the statement “√3 is irrational". Since we will be proving this statement by contradiction, we will assume that the opposite of p, p, is true i.e. “√3 is rational". By definition, if √3 is rational, then there must be some pair of integers a and b such that:

Screen Shot 2021-01-26 at 4.38.33 PM

We obtain following expressions after squaring both sides of the equation (1):

Screen Shot 2021-01-26 at 4.39.38 PMIf 3b2 = a2, then a2 (and consequently a) is divisible by 3. Therefore, we can say that a =3k for some integer k (which is the mathematical definition of being divisible by 3). Thus, we have that:

Screen Shot 2021-01-26 at 4.44.59 PM

Equation (3) tells us that b2 (and consequently b) is divisible by 3. Yet, according to our statements in equation (1), a and b have no common factors. Can a and b have no common factors and also have a common factor of 3? No - this is a contradiction! Therefore, our initial statement p: “√3 is rational" is false, which proves that p: “√3 is irrational" is true. We have thus completed our proof by contradiction.

Example 2: Using proof by contradiction on the SAT/ACT

Proof by contradiction can also be used to tackle certain SAT/ACT math problems. Let’s look at a problem from the calculator section of an official SAT practice test:

Screen Shot 2021-01-26 at 4.50.37 PM

We can solve this problem by checking each answer choice and using our proof by contradiction method. Let’s start with checking the first answer choice.

First, we let p be the statement: “−1 is NOT a solution to the inequality 3x−5 4x−3". Now, for the sake of contradiction, let’s assume the opposite, p, is true: “−1 is a solution to the inequality 3x−5 4x−3". We will then substitute x=−1 into the above inequality to check if p is true or not.

Screen Shot 2021-01-26 at 4.57.06 PM

Note that we get that −8 −7 which is obviously a contradiction. Therefore, since p:    “−1 is a solution to the inequality 3x−5 4x−3" is false, then it follows that p: “−1 is NOT a solution to the inequality 3x−5 4x−3" is true

Hence, the correct answer is (A). 

I recommend double checking the other answer choices just to be sure. Spoiler alert: none of the other choices give us a contradiction when we assume they are solutions to the inequality.

Final Thoughts

Proof by contradiction is one of the most powerful proof methods that mathematicians use routinely in their work. Notice that this methodology can be applied regardless of the difficulty of the problem. I often have my SAT/ACT students use this method on the multiple choice sections when they are stumped. This is one of the rare instances in life where being “wrong" is ultimately a good thing.

 

Mathematics – from high school math to graduate school math – is one of our most frequently requested subjects. Teaching math is notoriously difficult and we maintain a staff of mathematicians who are committed to the art of teaching. There is no course or standardized test that we do not have extensive experience teaching. We work with students who loathe math and students who love it, students who haven’t done math in a decade and students who work on mathematical problems every day. Many of our students work with tutors to address courses or exams – such as Geometry, Linear Algebra, Differential Equations – but we also work with students looking to explore more advanced or unconventional topics (like the mathematics of poker, or algrabraic topology, for example).

Contact us!

Check out some of our previous blog posts related to mathematics below!

The Intermediate Value Theorem explained by everyday life

An Insider's Tip to Prepping for the SAT Math Section: Plug in Numbers

How To Multiply Matrices Quickly and Correctly in Six Easy Steps

Comments

topicTopics
academics MCAT study skills SAT medical school admissions expository writing English college admissions GRE GMAT LSAT MD/PhD admissions chemistry math physics ACT writing biology language learning strategy law school admissions graduate admissions MBA admissions creative writing homework help MD test anxiety AP exams interview prep summer activities history philosophy career advice premed academic advice ESL economics grammar personal statements study schedules admissions coaching law statistics & probability PSAT computer science organic chemistry psychology SSAT covid-19 CARS legal studies logic games USMLE calculus parents reading comprehension 1L Latin Spanish dental admissions DAT engineering excel political science French Linguistics Tutoring Approaches chinese research DO MBA coursework Social Advocacy case coaching classics genetics kinematics secondary applications skills verbal reasoning ISEE academic integrity algebra business business skills careers diversity statement geometry medical school mental health social sciences trigonometry 2L 3L Anki EMT FlexMed Fourier Series Greek IB exams Italian MD/PhD programs STEM Sentence Correction Zoom amino acids analysis essay architecture art history artificial intelligence astrophysics athletics biochemistry capital markets cell biology central limit theorem chemical engineering chromatography climate change clinical experience curriculum data science dental school finance first generation student functions gap year harmonics health policy history of medicine history of science information sessions integrated reasoning international students investing investment banking mba meiosis mitosis music music theory neurology phrase structure rules plagiarism presentations pseudocode sociology software software engineering teaching tech industry transfer typology virtual interviews work and activities writing circles