GRE Exam: A Simple Guide To Number Properties - Positives and Negatives

All About Zero
A number’s position in relation to zero determines its sign. If a number is greater than zero, then it is positive. If a number is less than zero, it is negative.

Generally, when you are told or asked whether a term is greater than zero or less than zero, you should interpret it using properties of positives and negatives. For example, if you are told that xy > 0, think of this information as “xy is positive,” and proceed to use the rules covered in this chapter. As with odds and evens, you will need to know certain properties of positive and negative numbers.

Multiplication and Division
When multiplying or dividing two terms, the result will be positive if the terms are the same sign (both positive or both negative) and negative if the two terms are different signs (one positive and one negative).

Example: If , then which of the following must be true?
y > 0
x > 0
xy > 0
y > x
x > y

SOLUTION: If is negative, then the numerator and denominator must have different signs. Since the numerator is positive, y – x must be negative. Algebraically y – x < 0. Add x to both sides: y < x. Of the choices, the only one matching what you have deduced is E.

Many positive/negative questions will raise the terms to an exponent.

For such questions, it’s important to remember a property in quadratics: When a variable is raised to an even exponent, the result will always be positive. When a variable is raised to an odd exponent, the sign of the result will always be the same as the sign of the base.

Example: If a4b3c7 > 0, then which of the following must be true? (Indicate all the apply.) (A) a > 0 (B) b > 0 (C) bc > 0 (D) b/c > 0 (E) ab > 0 (F) abc > 0

SOLUTION: You are not told the sign of any of the unknowns, but since a is raised to an even exponent, you know that a2 is positive. Thus you have: (+) × (b3) × (c7) > 0. Since + × + = +, it must be true that b3 × c7 is positive. Since b and c are each raised to odd exponents, the signs of b3 and c7 will be the same as the signs of b and c, respectively. Thus you know that bc > 0. If bc > 0, then b and c must have the same sign, meaning that their product and their quotient are positive. The correct answer is C and D.

Many test-takers mistakenly assume that since a4 > 0, a must be greater than zero. However, remember that even exponents hide the sign of the base. Whether the base is positive or negative, the result of a variable raised to an even exponent will always be positive. Thus F is not necessarily true.

Quantitative Comparison Strategy: Positives and Negatives
The GRE loves testing properties of positives and negatives in Quantitative Comparison questions. One important building block of success for the savvy test-taker is to identify situations in which these properties are being tested.

Situation 1 : You are told that an unknown is greater than or less than zero.
If the stem says that x > 0, then you know that x is positive. If it says that xy < 0, then you know that xy is negative. Once you recognize that these properties are being tested, you can then start using the rules that have been covered.

Situation 2: One of the quantities has a value of zero.
If you are comparing an unknown to zero, your ultimate goal is to determine the sign of that unknown.

Remember that you can determine its sign without knowing its actual value.


SOLUTION: Since you are comparing (x16 + 1) to zero, you should focus on determining the sign of Quantity A. Since the exponent on x16 is even, x16 > 0. Thus x16 + 1 > 0. The correct answer is Quantity A.

Situation 3: One of the quantities has an even exponent, and the other quantity has an odd exponent.
Since the result of an even exponent hides the sign of the base, and the result of an odd exponent preserves the sign of the base, these properties are fertile ground to test your knowledge of positives and negatives.

SOLUTION: In both quantities, the base is negative. Since the exponent in Quantity A is even, the value in Quantity A must be positive. Since the exponent in Quantity B is odd, the value in Quantity B must be negative. The correct answer is A.

Finally, when testing numbers for Quantitative Comparison questions testing positives and negatives, you should always test at least one negative case and one positive case.

You want to play devil’s advocate, and the best way to do so is by choosing numbers with different signs.

SOLUTION: The even exponents in the quantities should clue you in that the question might be addressing properties of positives and negatives.

Let’s thus choose positive and negative cases for x and y:

Case 1: x = 3 and y = 2. Quantity A: 32 = 9. Quantity B: 22 = 4. Quantity A is greater. The answer is A or D.
Case 2: x = 3 and y = –2. Quantity A: 32 = 9. Quantity B: –22 = 4. Quantity A is still greater. Though you might be tempted to stop here and select A, notice that you have not looked at situations where both values are negative.
Case 3: x = –5 and y = –6. Quantity A: –52 = 25. Quantity B: –62 = 36. In this case, Quantity B is greater. A relationship cannot be determined, so the answer is D.