Problem 5 - Elementary and Intermediate Algebra

Question

Fill in the blanks. Positive and negative numbers are called _____ numbers.

Step-by-Step Solution

Answer

Integers.

Step 1: Understand the Concept

To solve this problem, we need to know the term that encompasses both positive and negative numbers in mathematics. This is a fundamental concept in number theory.

Step 2: Recall the Definition

In mathematics, numbers that include both positive numbers and negative numbers are referred to as 'integers.' Integers include zero, positive natural numbers, and negative numbers.

Step 3: Provide the Term

Based on the definition above, the appropriate term to fill in the blank is "integers." Therefore, positive and negative numbers are called integers.

Key Concepts

Positive Numbers

Positive numbers are the set of numbers that are greater than zero. These numbers are often used in everyday scenarios to represent quantities or values that have real-world applications such as counting objects, measuring distances, or representing temperatures that are above freezing. In mathematical notation, positive numbers are represented on the right of zero on a number line. They do not include zero.

There are a few characteristics worth noting regarding positive numbers:

• They can be whole numbers, fractions, or decimals.
• When you add two positive numbers, the result is always positive.
• Multiplying two positive numbers also results in a positive number.

Examples of positive numbers include 3, 4.5, and 100.

Negative Numbers

Negative numbers are the set of numbers that are less than zero. They appear to the left of zero on the number line and denote a deficit, decrease, or loss in a variety of real-world contexts. This can include falling temperatures, debt in financial situations, or depth measurements below a baseline.

There are some key points about negative numbers:

• They are always less than zero.
• Adding a negative number is equivalent to subtracting its absolute value.
• Multiplying two negative numbers results in a positive product, while multiplying a positive and a negative number results in a negative product.

Examples of negative numbers are -1, -2.5, and -50.

Number Theory

Number theory is a branch of pure mathematics that deals primarily with the properties and relationships of numbers, especially integers. It is a basic and foundational area of mathematics with a rich history and a wide array of applications in fields such as cryptography, computer science, and numerical analysis.

In number theory, integers consist of positive numbers, negative numbers, and zero. The study involves understanding concepts such as:

• Prime numbers, which are numbers greater than one with no divisors other than one and themselves.
• Factors and multiples, which involve the divisibility of one integer by another.
• Even and odd numbers, which are determined by their divisibility by two.

Fundamentally, number theory helps us grasp how numbers interact and provides insight into solving various mathematical problems.