1.5. Real numbers

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The set of real numbers consists of all rational numbers and all irrational numbers. The real numbers include all integers, fractions, and decimals. The set of real numbers can be represented by a number line called the real number line. Arithmetic Figure 2 below is a number line.

Every real number corresponds to a point on the number line, and every point on the number line corresponds to a real number. On the number line, all numbers to the left of 0 are negative and all numbers to the right of 0 are positive. As shown in Arithmetic Figure 2, the negative numbers -0.4, -1, -3/2, -2,, and -3 are to the left of 0, and the positive numbers 1,2 1, 2, 2, 2.6, and 3 are to the right of 0. Only the number 0 is neither negative nor positive.

A real number x is less than a real number y if x is to the left of y on the number line, which is written as x < y. A real number y is greater than a real number x if y is to the right of x on the number line, which is written as y > x. For example, the number line in Arithmetic Figure 2 shows the following three relationships.

Relationship 1: 

Relationship 2: 

Relationship 3: 

A real number x is less than or equal to a real number y if x is to the left of, or corresponds to the same point as, y on the number line, which is written as x ≤ y. A real number y is greater than or equal to a real number x if y is to the right of, or corresponds to the same point as, x on the number line, which is written as y ≥ x.

To say that a real number x is between 2 and 3 on the number line means that x > 2 and x < 3, which can also be written as 2 < x <3. The set of all real numbers that are between 2 and 3 is called an interval, and 2 < x < 3 is often used to represent that interval. Note that the endpoints of the interval, 2 and 3, are not included in the interval. Sometimes one or both of the endpoints are to be included in an interval. The following inequalities represent four types of intervals, depending on whether or not the endpoints are included.

Interval type 1:  2 < x < 3

Interval type 2:  2 ≤ x <3

Interval type 3:  2 < x ≤3

Interval type 4:  2 ≤ x ≤3

There are also four types of intervals with only one endpoint, each of which consists of all real numbers to the right or to the left of the endpoint and include or do not include the endpoint. The following inequalities represent these types of intervals.

Interval type 1:  x < 4

Interval type 2:  x ≤ 4

Interval type 3:  x > 4

Interval type 4:  x ≥ 4

The entire real number line is also considered to be an interval.

Absolute Value

The distance between a number x and 0 on the number line is called the absolute value of x, written as |x|. Therefore, |3|=3 and |-3|=3 because each of the numbers 3 and -3 is a distance of 3 from 0. Note that if x is positive, then|x| =x; if x is negative, then |x| = -x; and lastly, |0|=0. It follows that the absolute value of any nonzero number is positive. Here are three examples.

Example 1.5.1: | √5| = √5

Example 1.5.2: |-23| = -(-23) = 23

Example 1.5.3: |-10.2| = 10.2

Properties of Real Numbers

Here are twelve general properties of real numbers that are used frequently. In each property, r, s, and t are real numbers.

Property 1: r + s = s + r and rs = sr

  • Example A: 8 + 2 = 2 + 8 = 10
  • Example B: (-3)(17) = (17)(-3) = -51

Property 2: (r + s) + t = r + (s+t) and (rs)t = r(st).

  • Example A: (7 + 3) + 8 = 7 +(3+8) = 18
  • Example B:  (7√2)√2= 7√2√2 = (7)(2) = 14

Property 3:  r (s + t) = rs +rt

  • Example:  5(3 + 16) = (5)(3) + (5)(16) = 95
  • Property 4:  r + 0 = r,  (r)(0) = 0, and (r)(1) = r.

Property 5:  If rs = 0 then either r = 0 or s = 0 or both.

  • Example: If  -2s = 0, then s = 0.

Property 6: Division by 0 is undefined.

  • Example A:  5 ÷ 0 is undefined.
  • Example B:  -7/0 is undefined.
  • Example C:  0/0 is undefined.

Property 7: If both r and s are positive, then both r + s and rs are positive.

Property 8: If both r and s are negative, then r + s is negative and rs is positive.

Property 9: If r is positive and s is negative, then rs is negative.

Property 10: |r + s| ≤ |r| + |s|. This is known as the triangle inequality.

  • Example: If r = 5 and s = -2, then |5 + (-2)| = |5-2| = |3| = 3 and |5| + |-2| = 5 + 2 = 7. Therefore, |5 +(-2)| ≤ |5| + |-2|.

Property 11:  |r||s| = |rs|

  • Example:  |5||-2| = |(5)(-2)| = |-10| = 10

Property 12:  If r > 1 then r2 > r. If 0 < s <1 then s2 < s

  • Example: 52 = 25 > 5, but