Start with Nothing, End with Nothing

Source: Pixabay/Geralt CC0 Public Domain

Singer Billy Preston understood Integers, or so it seems, based on his song “Nothing From Nothing Leaves Nothing.”

Integers, as I’ve mentioned in earlier posts, consist of all the Whole Numbers going all the way to Positive Infinity, all their Opposites (the Negative Numbers) going all the way to Negative Infinity, and the number 0 (zero), which is its own opposite.

A horizontal Number Line is used to help students visualize this, with 0 (zero) in the center as the starting point. Positive Integers are to the right of 0 (zero) and Negative Integers are to the left.

Based on their Absolute Values, the distance between Positive Infinity and Negative Infinity would appear to be two times infinity (2 x ∞). Since infinity isn’t really a number, rather it is the concept that numbers never end because no matter how big a number you can conceive of you can always add 1 to it, the expression (2 x ∞) has no real meaning.

What is important to remember about Integers, however, is the fact that they add up to precisely 0 (zero). If you take all the Positive Integers and add them to all the Negative Integers, the sum is nothing, nada, zero.

Allow me to illustrate. If you add (-1) to (+1) the sum is (0). Eliminate the verbiage and you get:

(-1) + 1 = 0.

(+1) and (-1) are Opposites. For Math purposes, Opposites are defined as two numbers the same distance from, but on opposites sides of, 0 (zero).[i]

The Mathematical term for one number that is the Opposite of another is, Additive Inverse.[ii] In my previous example, (+1) is the Additive Inverse of (-1) because when you add (+1) to (-1) the sum is (0).

The concept of Opposites and Additive Inverse can also be demonstrated Algebraically. Algebra, at its most basic, is simply Math where Variables – usually lower case letters, are used to indicated unknown numbers. Algebraically, I could write my example this way:

                (-n) + n = 0

This will hold true not just for any Integer n, but for any number n.

Students need to have a firm understanding of the concepts of Opposites and Additive Inverse before we can move on to the addition of Integers, and beyond that to the addition of rational numbers.

As always, I remain,

The Exhausted Educator

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[i] Larson, Ron, and Laurie Boswell. "Chapter 1, Lesson 2." Big Ideas Math: A Common Core Curriculum ; Red. Erie, PA: Big Ideas Learning, 2014. 10. Print.

[ii] ibid