GMAT Quantitative Review

3.0 Math Review

3.1 Arithmetic

4. Real Numbers

All real numbers correspond to points on the number line and all points on the number line correspond to real numbers. All real numbers except zero are either positive or negative.


On a number line, numbers corresponding to points to the left of zero are negative and numbers corresponding to points to the right of zero are positive. For any two numbers on the number line, the number to the left is less than the number to the right; for example,

image, and image.

To say that the number n is between 1 and 4 on the number line means that image and image, that is, image. If n is “between 1 and 4, inclusive,” then image.

The distance between a number and zero on the number line is called the absolute value of the number. Thus 3 and −3 have the same absolute value, 3, since they are both three units from zero. The absolute value of 3 is denoted image. Examples of absolute values of numbers are


Note that the absolute value of any nonzero number is positive.

Here are some properties of real numbers that are used frequently. If x, y, and z are real numbers, then

1.  (1) image and image. For example, image, and image.

2.  (2) image and image. For example, image, and image.

3.  (3) image. For example, image.

4.  (4) If x and y are both positive, then image and xy are positive.

5.  (5) If x and y are both negative, then image is negative and xy is positive.

6.  (6) If x is positive and y is negative, then xy is negative.

7.  (7) If image, then image or image. For example, image implies image.

8.  (8) image. For example, if image and image, then image; and if image and image, then image.