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When it comes to quantitative comparison questions, zero is a pretty important number, because it\u2019s a weird number. It reacts differently from other numbers when placed in some of the situations. And zero isn\u2019t the only weirdo out there.<\/p>\n
Most of us equate \u201cnumber\u201d with \u201cpositive integer\u201d, 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.<\/p>\n
The GRE knows this, and takes advantage of our assumption. That\u2019s why it\u2019s important to remember all the \u201cother\u201d 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.<\/p>\n
If I\u2019m 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:<\/p>\n
FR: fractions (both positive and negative)
\nO: one and negative none
\nZE: zero
\nN: negatives<\/p>\n
So we\u2019ve got positive and negative integers (the bigger the absolute value, the better), positive and negative one, positive and negative fractions, and zero. Don\u2019t forget, zero is an integer too!<\/p>\n
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 \u201cdo math\u201d to them. Because of that fact, picking numbers from different categories can be a fast way to understand the limits of a problem.<\/p>\n
To illustrate my point, let\u2019s 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\u2019s simplify our lives even further by stipulating that y is a positive integer.<\/p>\n
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\u2019s 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.
\n
\nThings 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).<\/p>\n
Can you see why this matters? It is easy to think of the positive integer example and conclude that \u201cx to the y\u201d will always be smaller than \u201cx to the 2y.\u201d 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\u2019t 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.<\/p>\n
Exponents aren\u2019t the only example of these groups of numbers reacting differently when you \u201cdo math\u201d 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\u2019t positive or negative, it is even, because it\u2019s evenly divisible by 2. They\u2019re 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.<\/p>\n","protected":false},"excerpt":{"rendered":"
When it comes to quantitative comparison questions, zero is a pretty important number, because it\u2019s a weird number. It reacts differently from other numbers when placed in some of the situations. And zero isn\u2019t the only weirdo out there. Most of us equate \u201cnumber\u201d with \u201cpositive integer\u201d, and for good reason. Most of the numbers […]<\/p>\n","protected":false},"author":53,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[133,161,229,269,272,350],"yst_prominent_words":[],"class_list":["post-6508","post","type-post","status-publish","format-standard","hentry","category-math-gre-strategies","tag-gre","tag-gre-quant","tag-math","tag-qc","tag-quantitative-comparisons","tag-zero"],"_links":{"self":[{"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/posts\/6508","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/users\/53"}],"replies":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/comments?post=6508"}],"version-history":[{"count":1,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/posts\/6508\/revisions"}],"predecessor-version":[{"id":6826,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/posts\/6508\/revisions\/6826"}],"wp:attachment":[{"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/media?parent=6508"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/categories?post=6508"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/tags?post=6508"},{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/yst_prominent_words?post=6508"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}