Positive negative problems | Arithmetic & algebra | Quantitative reasoning

Understanding the properties of negative and positive numbers can be a huge time saver in the GRE. Imagine if someone asked you the question:

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

You don’t have to solve for the distance from sea level for either location to answer this question. Understanding that a positive altitude is greater than a negative altitude is all you need to know. This can save a lot of time that you would have spent measuring!

Similarly, if I told you that $x$ is negative and $y$ is positive, you can automatically know that $y$ is greater than $x$ without solving for either or finding the difference.

So here’s the main point: when a problem tells you that a variable is greater than or less than zero (in other words, positive or negative) you should determine if the final answer is positive or negative. That may be enough information 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 that $a$ is positive. We also know that we are subtracting a negative $b$ . Subtracting a negative number is the same as addition. In other words, we’re adding two positive numbers. So, which of the answers could be the sum of two positive numbers?

(spoiler)

Answer: A. $2$

The sum of two positive numbers will always be a positive number!

It’s okay if you didn’t know that subtracting a negative number is the same as addition. Here are a few rules you should memorize to approach 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 to help show this:

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

Multiplication and division

Multiplication and division have a different set of rules to memorize:

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

And some examples:

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

All of these results work the same for division as they do for multiplication:

$2 \div 1 = 2$
$- 2 \div - 1 = 2$
$2 \div - 1 = - 2$
$- 2 \div 1 = - 2$

Notice how if the sign of the two numbers matches, the answer will be positive, and if they are opposites (i.e. one positive and one negative), the answer is negative.

Bringing it all together: question walkthrough video

Here’s a video going through one of our practice questions to demonstrate these ideas in action: