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The Set of Real Numbers

MAT1501 - Fundamental Mathematics · Number Skills

The Set of Real Numbers

This topic introduces you to the set of real numbers, their classifications, and how they are represented. Understanding real numbers is essential because they form the basis for most mathematical concepts you will encounter in your studies and everyday life.

Key idea: After studying this topic, you should be able to:

  • Recognise different types of numbers.
  • Understand the place value of digits in the decimal system.
  • Multiply and divide decimals by powers of ten.
  • Convert comparison statements into mathematical inequalities.
  • Manipulate inequalities correctly.
  • Describe and represent intervals using interval notation.

1.1 Number System

The number system includes various types of numbers. The most basic are:

  • Natural Numbers (N): These are the counting numbers, starting from 1, 2, 3, and so on.
  • Whole Numbers (N0): This set includes all natural numbers plus zero (0).
  • Integers (Z): This set includes all whole numbers and their negative counterparts, such as -3, -2, -1, 0, 1, 2, 3, etc.

We can represent these numbers on a number line:

... -3 -2 -1 0 1 2 3 ...

1.2 Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. For example:

  • -5 can be written as -5/1.
  • 0 can be written as 0/1.
  • 2/3 is a rational number.

Rational numbers can also be represented as decimals. They can be:

  • Terminating Decimals: For example, 0.5 or 1.25.
  • Repeating Decimals: For example, 1/3 = 0.333... (which can be written as 0.3 with a bar over the 3).

1.3 Irrational Numbers

Irrational numbers cannot be expressed as fractions. Their decimal representations are non-terminating and non-repeating. Examples include:

  • √2 (the square root of 2)
  • π (pi, approximately 3.14)

1.4 Real Numbers

The set of real numbers (R) includes all rational and irrational numbers. This means every point on the number line corresponds to a real number. The classification of real numbers can be summarised as follows:

Summary of Number Sets:

  • Natural Numbers (N): 1, 2, 3, ...
  • Whole Numbers (N0): 0, 1, 2, 3, ...
  • Integers (Z): ..., -3, -2, -1, 0, 1, 2, 3, ...
  • Rational Numbers (Q): Any number that can be expressed as p/q where p and q are integers and q ≠ 0.
  • Irrational Numbers: Numbers that cannot be expressed as a fraction.
  • Real Numbers (R): The set of all rational and irrational numbers.

1.5 Ordering Real Numbers

Ordering real numbers involves comparing their values. If a number lies to the left of another on the number line, it is smaller. If it lies to the right, it is larger. For example:

-2 < 0 < 3

This shows that -2 is less than 0, and 0 is less than 3.

1.6 Intervals

Intervals are used to describe sets of real numbers. They can be represented using interval notation:

  • Open Interval (a, b): Includes all numbers between a and b, but not a and b themselves.
  • Closed Interval [a, b]: Includes all numbers between a and b, including a and b.
  • Half-Open Interval [a, b): Includes a but not b.
  • Half-Open Interval (a, b]: Includes b but not a.

For example, the interval (2, 5] includes all numbers greater than 2 and up to and including 5.

Watch out: Be careful with the notation. Using (a, b] means that a is not included, while [a, b) means that b is not included.

1.7 Summary

In summary, the set of real numbers includes natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Understanding these concepts is crucial for solving mathematical problems.

Summary List:

  • Natural Numbers (N): Counting numbers starting from 1.
  • Whole Numbers (N0): Natural numbers plus zero.
  • Integers (Z): Whole numbers and their negatives.
  • Rational Numbers (Q): Can be expressed as fractions.
  • Irrational Numbers: Cannot be expressed as fractions.
  • Real Numbers (R): All rational and irrational numbers.

Check your understanding

  1. What are the main types of numbers in the real number system?
  2. How do you represent the interval of all real numbers greater than 3?
  3. What is the difference between rational and irrational numbers?
  4. Explain how to order the numbers -1, 0, and 2 on the number line.