BackNumbers and Functions: Foundations for Calculus
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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 how they are constructed.
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: 1, 2, 3, 4, …
Zero: The integer 0 serves as the additive identity.
Negative Integers: These are the numbers less than zero: …, -4, -3, -2, -1.
Integers: The set of all positive integers, zero, and negative integers. Integers are sometimes called "whole numbers."
Rational Numbers
Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. These include fractions and can be written as:
By definition, any integer is also a rational number (for example, and ).
Properties of Rational Numbers:
You can add, subtract, multiply, and divide any pair of rational numbers (except division by zero), and the result will always be a rational number.
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: The square root of 2 () is irrational because there is no fraction such that , i.e., .
Table: Approximating
x | |
|---|---|
1.2 | 1.44 |
1.3 | 1.69 |
1.4 | 1.96 < 2 |
1.5 | 2.25 > 2 |
1.6 | 2.56 |
This table shows that is between 1.4 and 1.5, but it cannot be written exactly as a fraction.
Infinite Decimal Expansions
To address the existence of irrational numbers, mathematicians often define real numbers as infinite decimal expansions. This approach allows us to represent both rational and irrational numbers in a unified way.
Example: Certain fractions can be represented as decimal fractions:
In general, every rational number has a terminating or repeating decimal expansion, while irrational numbers have non-repeating, non-terminating decimal expansions.
Additional info: The concept of real numbers as infinite decimal expansions is foundational for calculus, as it allows for precise definitions of limits, continuity, and other key ideas.
