Numbers and Functions

1. What is a Number?

Understanding the concept of a number is fundamental to calculus and mathematical analysis. Numbers are classified into various types based on their properties and the operations they support.

1.1 Different Kinds of Numbers

Numbers can be grouped into several categories, each with distinct characteristics and uses in mathematics.

• Positive Integers: These are the counting numbers starting from 1 and increasing without bound: 1, 2, 3, 4, …

• Zero: The integer 0 serves as the additive identity in arithmetic.

• Negative Integers: These are the numbers less than zero: …, -4, -3, -2, -1.

• Integers: The set of all positive integers, zero, and negative integers together form the set of integers (or "whole numbers").

1.2 Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. They include all integers and fractions.

• Definition: A rational number is any number that can be written in the form , where and are integers and .

• Examples: , , , , , etc.

• Properties:

  • You can add, subtract, multiply, and divide any pair of rational numbers (except division by zero), and the result is always a rational number.

  • All integers are rational numbers (e.g., ).

1.3 Irrational Numbers

Not all numbers are rational. Some numbers, such as the square root of 2, cannot be expressed as a ratio of two integers. These are called irrational numbers.

• Example: is irrational because there is no fraction such that , i.e., .

• Approximation: By checking values between 1.4 and 1.5:

  x
  1.2	1.44
  1.3	1.69
  1.4	1.96 < 2
  1.5	2.25 > 2
  1.6	2.56

  This shows that is between 1.4 and 1.5.

• Algebraic Properties: Irrational numbers, like rational numbers, obey the usual algebraic rules (e.g., ).

1.4 Decimal Expansions

Numbers can be represented as decimal expansions. Rational numbers have either terminating or repeating decimal expansions, while irrational numbers have non-repeating, non-terminating decimals.

• Example:

• Application: Decimal representation is useful for computation and for understanding the nature of numbers.

Additional info: The concept of real numbers includes both rational and irrational numbers. In calculus, we work extensively with real numbers and their properties, especially as they relate to functions and limits.