Lesson 12.1: Numbers, Numbers, and More Numbers!

Property

The most basic numbers used in mathematics are the numbers we use to count objects in our world: $1, 2, 3, 4$, and so on. These are called the counting numbers or natural numbers. Natural numbers are all the positive integers we use for counting.

Natural Numbers (Counting Numbers): $1, 2, 3, 4, 5, \ldots$

Property

A rational number is a number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. All fractions, both positive and negative, are rational numbers. Since any integer can be written as a ratio of two integers, all integers are also rational numbers.

Examples

Property

A fraction $\frac{p}{q}$ is in lowest terms (or simplest form) when the greatest common divisor of $p$ and $q$ is 1, written as $\gcd(p,q) = 1$. This means $p$ and $q$ share no common factors other than 1.

Examples

Property

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.
Famous examples include $\pi$ and the square roots of numbers that are not perfect squares.

Examples

• The number $\pi$ is a famous irrational number, beginning with $3.14159...$ and continuing infinitely without repetition.
• The square root of 3, $\sqrt{3}$, is irrational because 3 is not a perfect square. Its decimal form is $1.7320508...$.
• A decimal that continues without a pattern, such as $67.121231234...$, is an irrational number.

Explanation

Irrational numbers cannot be written as a simple fraction. Their decimal representations are infinite and non-repeating, meaning they go on forever without any predictable pattern. Think of them as the 'wild' numbers on the number line.