Properties of Zero

by on April 14th, 2013

Today’s article comes from Veritas Prep GMAT instructor, Ron Awad.

Most people see the GMAT as an obstacle to be surmounted in an effort to get into the business school of their choice. Some people see it as an unfortunate barrier to their future plans. Personally, I like to think of it as an opportunity to test your reasoning skills against an unseen test maker. Your goal is to stay one step ahead of the test and predict the traps that will be laid out for you as you answer questions.

In this metaphor, you are the protagonist trying to avoid pitfalls and maximize your score, but these pitfalls come in predictable and recurring ways to try and trap you. It is important to note that predictable does not mean easy, only that you can expect it to happen. Some traps are therefore completely predictable and you can expect to see instances of them on questions in every GMAT you are  likely to ever take.

At the risk of mixing metaphors, I have been contemplating the idea of the GMAT as a videogame, specifically a platformer like the original Mario Bros franchise. The GMAT has all the hallmarks of a great game: a likable protagonist (you!), a looming antagonist (the GMAC), puzzles and obstacles to overcome (Reading Comprehension, Data Sufficiency, etc) and helpful friends along the way (Veritas Prep, including yours truly as Yoshi). If the GMAT were a game, the last boss would undoubtedly be the number zero. No other concept on the GMAT traps students more than forgetting about the possibility of zero.

The number zero can be used in myriad ways to mess up students and change seemingly innocuous questions into head-scratchers, so let’s review some of the basic properties of zero:

  1. Zero is even (not odd, not neutral)
  2. Zero is neither positive nor negative (the only number with this property)
  3. Zero is an integer (and must be considered when question limits choices to integers)
  4. Zero is a multiple of all numbers (x*0 = 0, so a multiple of any x)
  5. Zero is not a prime number (neither is 1; smallest prime number is 2)
  6. Zero is neither black nor red (pertains to roulette only)

There are actually dozens of questions that I could use to illustrate the zero trap, but I figured I would go with the shortest GMAT question I have ever seen, clocking in at a whopping 35 characters including spacing but excluding answer choices:

Is x^y > 0

  1. y = 2
  2. x is an integer

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
(C) Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient to answer the question asked
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed

Now, there are other properties of zero, but the first five listed are the most commonly tested on the GMAT. Keep these in mind and you should be able to answer most GMAT questions without falling into traps. On this specific data sufficiency question, the question is asking us if x^y is positive, with no restrictions whatsoever on the values of x or y. Will either of the two statements be sufficient? To answer this, let’s look at each statement a little closer.

Statement 1 tells you that y =2, in other words, whatever x is will be multiplied by itself exactly once. Does this guarantee that x^2 is positive? The answer is: almost. Any positive number squared will remain positive, and any negative number squared will also give a positive number. As such, almost any number you can think of will be squared when positive, be it 2, 0.5 or -37.  The only number that will buck that trend is zero. Zero x zero = zero (0^2 = 0). We just said zero isn’t positive or negative, so this equation holds for all the real numbers ( ) in this and a million other galaxies, except for zero. As such, statement #1 is (just barely) insufficient.

Statement 2 is much more straight forward. It only limits the value of x to an integer. This is clearly insufficient, if only because it still allows for zero as a value for x. Moreover, it also allows for all kinds of options such as negative x’s and multiple y’s. Example: (-2)^2 is positive (+4) and (-2)^3 is negative (-8). Statement 2 is insufficient on its own. Furthermore, statement 1 already accounted for any value of x that was not zero, so combining these two statements does not solve this problem any further.

The correct answer is (E). 

What is interesting to note about this question is how close the answer is to being A. Had we only managed to discount the number zero somehow (i.e. if the question had asked ≥ 0 or if we’d been using Roman numerals), statement 1 would have been sufficient on its own. However, much like the Romans discovered, when you forget to account for the number zero, your system may be usurped by a more complete system and your GMAT score (or empire) might fall.

For more GMAT tips like this one, make sure to visit the Veritas Prep blog!

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