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David Terr
Ph.D. Math, UC Berkeley

 

Home >> Pre-Calculus >> P. Prerequisites

>> P.2. Exponents

 

P.1. Real Numbers

Real numbers are the numbers we are all familiar with. These include the integers, namely the set of all natural numbers or counting numbers (1,2,3,...) as well as their negatives and zero. Real numbers also include the set of rational numbers or fractions, such as 1/2, 2/3, 5/4, -7/5, etc. Finally, the set of real numbers includes all irrational numbers, i.e. numbers such as π which cannot be expressed as fractions.

The best way to visualize the set of real numbers is the number line. This is simply a line with the real numbers marked on it in order, going from left to right, as shown in Figure P.1.1.

number line

Figure P.1.1: The Number Line

 

Every point on the number line corresponds to a unique real number. Furthermore, the number line illustrates a very useful property of real numbers, namely their order.

We say one real number is greater than another if the first number lies to the right of the second on the number line. On the other hand, if the first number lies to the left of the second number, we say the first number is less than the second number. Finally, if both numbers correspond to the same point on the number line, we say the numbers are equal.

Note that the number line provides a convenient way to divide the real numbers into two types, positive and negative. A real number is said to be positive if it is greater than zero, i.e. if it lies to the right of zero on the number line. On the other hand, if it lies to the left of zero so that it is less than zero, it is said to be negative. Zero is the only real number which is neither positive nor negative. Negative numbers are written with a minus sign to their left.

 

Example 1: Draw a number line with the numbers -2 and -4. Which is greater?

Solution: The number line is shown below with the dots corresponding to the numbers -2 and -4. As can be seen, -2 lies to the right of -4, so -2 is greater than -4.

number line with -2 and -4 labeled

 

Example 2: Draw a number line with the numbers 3/2 and 5/3. Which is greater?

Solution: The number line is shown below with the dots corresponding to the numbers 3/2 = 1 1/2 and 5/3 = 1 2/3. As can be seen, 5/3 lies to the right of 3/2, so 5/3 is greater than 3/2.

number line with 3/2 and 5/3 marked

 

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