GMAT Quantitative Review
3.0 Math Review
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,
, and .
To say that the number n is between 1 and 4 on the number line means that and , that is, . If n is “between 1 and 4, inclusive,” then .
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 . 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) and . For example, , and .
2. (2) and . For example, , and .
3. (3) . For example, .
4. (4) If x and y are both positive, then and xy are positive.
5. (5) If x and y are both negative, then is negative and xy is positive.
6. (6) If x is positive and y is negative, then xy is negative.
7. (7) If , then or . For example, implies .
8. (8) . For example, if and , then ; and if and , then .