GRE Quantitative Comparison: Don’t Be a Zero, Be a Hero
When it comes to quantitative comparison questions, zero is a pretty important number, because it’s a weird number. It reacts differently from other numbers when placed in some of the situations. And zero isn’t the only weirdo out there.
Most of us equate “number” with “positive integer”, and for good reason. Most of the numbers we think about and use in daily life are positive integers. Most of our math rules were learned, at least at first, with positive integers.
The GRE knows this, and takes advantage of our assumption. That’s why it’s important to remember all the “other” numbers out there. In particular, when testing numbers to determine the possible values of a variable, there are a few categories of numbers you want to keep in mind.
If I’m going to think about picking numbers, I want to pick numbers that are as different as possible. I try to choose my numbers from a mixture of seven categories, which can be remembered with the word FROZEN:
FR: fractions (both positive and negative)
O: one and negative none
ZE: zero
N: negatives
So we’ve got positive and negative integers (the bigger the absolute value, the better), positive and negative one, positive and negative fractions, and zero. Don’t forget, zero is an integer too!
There are other categories of numbers to think about, particularly if they are mentioned in the problem: odd versus even, prime versus non-prime, etc. But the seven groups listed above account for most of the different ways that numbers behave when you “do math” to them. Because of that fact, picking numbers from different categories can be a fast way to understand the limits of a problem.
To illustrate my point, let’s think about the value of x raised to the power of y. What happens to the value of that expression as y gets bigger? Let’s simplify our lives even further by stipulating that y is a positive integer.
What first comes to mind is the idea that as we increase the value of the exponent, we increase the value of the expression. Well, if x is a positive integer, that’s true: the expression gets exponentially bigger as y increases. Unless x is the positive integer 1, in which case the expression stays the same size, regardless of the value of y. The same is true if x is equal to 0. If x is a positive proper fraction, the expression gets smaller as the value of y increases.
Things get even more confusing if x has a negative value. If x is -1, the value of the expression could be 1 or -1, depending whether y is odd or even. If x is a negative integer smaller than -1, the absolute value of the expression gets bigger as the value of y increases, vacillating from positive (when y is even) to negative (when y is odd). If x is a negative proper fraction (a number between 0 and -1), the absolute value of the expression gets smaller as the value of y increases, vacillating from positive (when y is even) to negative (when y is odd).
Can you see why this matters? It is easy to think of the positive integer example and conclude that “x to the y” will always be smaller than “x to the 2y.” You can test all the positive integers you want and find proof after proof that this statement is true, and yet you only have to test one number from one of the other six categories to see that it isn’t always true. Having FROZEN in your back pocket can help you quickly provide counter-examples in QC questions, which can save you precious seconds and buy you precious points.
Exponents aren’t the only example of these groups of numbers reacting differently when you “do math” to them. Every integer is divisible by 1, but no integer is divisible by 0. The number 0 is a multiple of every integer. The number 1 is neither prime nor composite. The number 0 is its own square root and its own square. So is the number 1. While 0 isn’t positive or negative, it is even, because it’s evenly divisible by 2. They’re weirdos. So you have to keep them in mind as their own narrow but important number categories, because they can be the one exception to a QC question where one quantity is otherwise always bigger.