What are Integers? Why are they useful?

StevensED Mathematics presents a post about Integers.

Integers are a foundational concept in mathematics. As one studies math or uses numbers day-to-day, it is likely that they use integers. As an extremely versatile subset of the real number system, integers are used in many different areas of study. In this blog post, we'll dive deep into what integers are, their properties, and how they are used in real-world applications.

What are Integers? Integers are whole or discrete numbers that can be positive, negative, or zero. They are a subset of the real number system and are used to represent quantities such as counting, temperature, distance, and more. Some examples of integers include -3, 0, and 5.

Properties of Integers Integers have several important properties that are useful in mathematical operations. Some of these properties include:

1. Closure: The sum or product of two integers is always an integer.

2. Commutativity: The order in which integers are added or multiplied does not affect the result. For example, 2+3 = 3+2.

3. Associativity: The grouping of integers does not affect the result of addition or multiplication. For example, (2+3)+4 = 2+(3+4).

4. Identity: The integer 0 is the additive identity, and the integer 1 is the multiplicative identity.

5. Inverses: Every integer has an additive inverse (the opposite of the number) and a multiplicative inverse (the reciprocal of the number) except for 0.

Applications of Integers. Integers have numerous real-world applications in a diverse array of disciplines. Some practical applications include their use as a discrete numeral system in finances, temperature, and distance. Here are some examples:

1. Temperature: Temperature is often measured using integers on the conventional temperature scales: Celcius, Fahrenheit, and Kelvin. 0 degrees Celsius is the freezing point of water, and 100 degrees Celsius is the boiling point of water. In Fahrenheit, 32 degrees is the freezing point of water, and 212 degrees is the boiling point. All of these fixed scalar values are quantified or represented by integers.

2. Finance: Integers are used in finance to represent money. Positive integers represent deposits or gains, while negative integers represent withdrawals or losses.

3. Distance: Integers are used to represent distance in many contexts, such as measuring the distance between two cities or tracking the number of steps taken in a day.

4. Sports: Integers are used in sports to represent scores or points. In basketball, for example, each basket is tallied in whole numbers, all of which are integers, and the team with the most points at the end of the game wins.

Integers are a basal concept in mathematics that have many applications in the real world. By understanding the properties and uses of integers, we can better appreciate their importance in our daily lives, even if they are used without much thought. Whether you're a student just learning about integers or an experienced mathematician, it's essential to have a solid understanding of this foundational concept.