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0.99999999 by Any Other Name


August 8, 2023

In mathematics, the expression of the number 1 can be through the number 1 and by the repeating decimal 0.9999 … . Mathematically, these are the same number. The vinculum (horizontal line above the repeating number to indicate that it is repeating) indicates the limit of the sequence could go on forever and mean the same thing as the whole number — it “converges” on 1.

A good friend of mine took exception to this on a purely visual basis. “Those are not the same numbers.” While he is technically correct — they certainly “look” different — there is a deeper lesson to be learned.

First, the amount of work that it takes to write these two different representations. The visual representation for 1 is a single stroke of the pen. It requires little work and there is little question about what it is. On the other hand, if you go to the trouble of writing a 0, then a decimal point, then a discrete (or never-ending) series of 9’s with a vinculum (see above), you have done a fair amount of writing already – depending on how many 9’s you felt you needed to write to get an accurate representation of what you feel best represents, well, 1.

0.999 … vs. 1

When I was in high school, I participated in a paper airplane making contest as a learning module of an engineering workshop. The proctor drew a circle on the blackboard, and we were given time and materials to construct a paper airplane. All vying to get our planes to fly directly to the center of the target, most of the class went to the work of designing elaborate miniature airframes, complete with a diverse array of vertical stabilizers, multiple wing configurations, considerations of the distribution of weights and countless other factors that likely went into the design of these tiny engineering marvels.

In the standard model of physics, the basic building blocks of all the things we are made up interact through four fundamental basic forces: gravity, electromagnetism, weak forces and strong forces. So, generally, there are only so many things you would need to worry about as you consider your design for a paper airplane, and certain things, when considered, are not necessarily directly relevant or applicable. They might add up to or mean the same thing but are the product of additional work that is at once unnecessary and has little bearing on the final outcome.

After the 30 minutes or so of work, the many and varied designs of the paper airplanes flew. All with different results, seemingly tied (up to that point) by the amount of work that went into the design. But that did not directly prove out. The winning entry required no work at all. There was a single entrant who did not put any work into their design — in fact seemed to be waiting for something — not disengaged, not focused on something else — just waiting.

Sure enough, this individual went last, crumpled up their single sheet of paper into a ball and threw it directly into the center of the chalkboard target for the winning result. The rest of the class set about to write as many 9’s as they could with their variously dressed forms of vinculum, attempting to respond to every component represented by the standard model and more. But the victor in this case chose to just write a 1, as they realized that it ultimately means the same thing — even if the visual representation is not the same.

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Joe Benesh

Joe Benesh is the President and CEO of The Ingenuity Company, located in Des Moines. The Ingenuity Company specializes in Strategic Planning, Diagramming, Organizational Design Thinking, and Leadership/Change Facilitation. He also teaches strategic planning at the University of Iowa in the MBA Program.