Question

The numbers in the set \(\left\\{x \mid x=\frac{a}{b},\right.\) where \(a, b\) are integers and \(b \neq 0\\}\) are called ________ numbers.

Short answer

Rational numbers

Step-by-step solution

- Identify the form of the numbers

The numbers are given in the form of fractions, \(x = \frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\).

- Categorize these numbers

If numbers can be expressed as the fraction \( \frac{a}{b} \), where both \(a\) and \(b\) are integers and \(b eq 0\), they belong to a specific category of numbers in mathematics.

- Define the category

Numbers that can be written in the form of \( \frac{a}{b} \), where \(a\) and \(b\) are integers and \(b eq 0\), are called rational numbers.

Key concepts

fractions

Fractions are a way to represent parts of a whole. A fraction is composed of two integers: a numerator and a denominator. The general form is \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. It is important to remember that the denominator cannot be zero, as division by zero is undefined.
Fractions can represent numbers greater than one, between zero and one, or even negative numbers. For instance:

• \( \frac{1}{2} \) represents half.
• \( \frac{3}{2} \) represents one and a half.
• \( \frac{-1}{2} \) represents negative half.

Fractions are also crucial when converting between mixed numbers and improper fractions. Mixed numbers are composed of a whole number and a fraction, like \(1 \frac{1}{2} \), which can be converted to the improper fraction \( \frac{3}{2} \).

integers

Integers are the set of whole numbers and their opposites. This set includes positive numbers, negative numbers, and zero. Mathematically, integers can be represented as \(\text{..., -3, -2, -1, 0, 1, 2, 3, ...}\).
A key characteristic of integers is that decimals and fractions are not included in this category. This means numbers like 0.5 or \( \frac{3}{4} \) are not considered integers.
Integers play an essential role in various mathematical operations:

• Adding and subtracting integers
• Multiplying and dividing integers
• Understanding integer properties such as the additive inverse (for any integer \(a\), \(a + (-a) = 0\))

These properties make the integers a foundational concept in math.

number categories

Understanding the various categories of numbers is crucial in math. Each category has distinct properties and uses.
Some main categories include:

• Natural Numbers: These numbers include all positive whole numbers starting from 1 (i.e., 1, 2, 3, ...).
• Whole Numbers: Like natural numbers but including zero (i.e., 0, 1, 2, 3, ...).
• Integers: As discussed, these include all whole numbers and their negatives (i.e., ..., -3, -2, -1, 0, 1, 2, 3, ...).
• Rational Numbers: Numbers that can be written as a fraction \(\frac{a}{b}\): where both \(a\) and \(b\) are integers and \(b eq 0\). Rational numbers include fractions, whole numbers, and integers.
• Irrational Numbers: Numbers that cannot be written as a simple fraction and have non-repeating, non-terminating decimals (e.g., \(\text{√2, π}\)).
• Real Numbers: All rational and irrational numbers combined, covering any number that can be found on the number line.

Learning about these categories helps in understanding the vast landscape of numbers and their applications in different mathematical contexts.