Positive negative problems

Understanding the properties of negative and positive numbers can save you a lot of time on the GRE. For example, imagine someone asked:

What has a greater altitude, the bottom of the ocean or the top of the Eiffel Tower?

You don’t need to calculate the exact distance from sea level for either location. You just need one key idea:

• A positive altitude is greater than a negative altitude.

That’s enough to answer the question quickly.

Similarly, if you’re told that $x$ is negative and $y$ is positive, you can immediately conclude that $y>x$ without finding either value.

So here’s the main point: when a problem tells you a variable is greater than or less than zero (in other words, positive or negative), first figure out whether the expression you’re asked about must be positive or negative. Sometimes that sign information alone is enough to answer the question.

Here’s a simple example:

$a>0$ and $b<0$

What could $a-b$ equal?

A. $2$
B. $0$
C. $-1$
D. $-2$
E. $-10$

We know $a$ is positive. We also know $b$ is negative, and we are subtracting $b$. Subtracting a negative number is the same as adding a positive number:

• $a-b=a-(\text{negative})=a+(\text{positive})$

So $a-b$ is the sum of two positive numbers, which must be positive. The only positive answer choice is $2$.

If you didn’t already know that subtracting a negative number is the same as addition, use the rules below. These are the core rules you’ll use for positive/negative number property questions.

Addition and subtraction

Rules:

• Subtracting a negative number is the same as addition
• Adding a negative number is the same as subtraction

Here are a few examples with actual numbers:

• $1+1=2$
• $1-(-1)=2$
• $1-1=0$
• $-1+1=0$
• $1+(-1)=0$

Multiplication and division

Multiplication and division follow a different set of sign rules:

• (positive) $\times$ (positive) $=$ positive
• (negative) $\times$ (negative) $=$ positive
• (positive) $\times$ (negative) $=$ negative
• (negative) $\times$ (positive) $=$ negative

And some examples:

• $1\times 2=2$
• $-1\times -2=2$
• $1\times -2=-2$
• $-1\times 2=-2$

These sign results work the same way for division:

• $2÷1=2$
• $-2÷-1=2$
• $2÷-1=-2$
• $-2÷1=-2$

Notice the pattern:

• If the signs match, the result is positive.
• If the signs are different (one positive and one negative), the result is negative.

Bringing it all together: question walkthrough video

Here’s a video that walks through one of our practice questions and shows these ideas in action: