Understanding Integers in Mathematics

Study Notes

What Are Integers?

In mathematics, integers refer to whole numbers inclusive of zero, positive numbers such as one, two, three, etc., and negative numbers like -1, -2, -3, etc. They can also represent counts, quantities, distances, time intervals, and so on.

The term "integer" is derived from the Latin word "integrum," which means complete or entire. Thus, when we say a number is integer, it signifies that the value is neither fractional nor decimal; rather, it represents a quantity without fractions or decimals attached to its digits. For example, (4) and (-7) are both integers because they don't have any fractional component.

Historically, the concept of integers was developed by ancient civilizations, including Sumerians who used them around 3000 BC and Babylonians who employed base-sixty place-value system around 400 BC. Later, Indians introduced negative numbers into math around 8th century AD with their rule called 'Negate Whatever Is Negated', meaning if you negate something else once more, it gives back the original thing. This contributed significantly to the modern understanding of integers.

Since integers do not involve fractions or other irrational numbers, operations involving only integers produce results that are either another integer or an undefined result. For instance, dividing 5 by 2 yields 2 as an answer, while dividing 9 by 3 produces 3. However, some mathematical operations involving integers may lead to non-integer outcomes like division by zero, root extraction of negative values, etc.. These cases highlight special instances where algebra deals with rational expressions instead of just plain numbers.

Description

Explore the concept of integers in mathematics, encompassing whole numbers, positive and negative numbers, as well as their historical significance. Learn about the origins, properties, and operations involving integers, and how they are used to represent counts, quantities, distances, and time intervals.