# What exactly is Base 10 arithmetic ?

Teacher: “Now we’re going to learn about base 10 arithmetic”.
Wise guy: “Is that where 3 + 4 = 12, or is it where 3 x 4 = 12 ?”.

I did a search on the net and found the term “base 10” all over the place. What does it mean?

An apparently annoying question:
“Does the 1 in 10 stand for the number 10’s in 10?”.

The interpretation of 10 in the system described as “Base 10” depends on the base of the system, so what is it? How do I find out?

We have here a logical problem. The term “Base 10” as a definition is self referential. It is more subtle than this definition of a straight line:

“A straight line is a line which is straight”.

The problem arises from the almost universal confusion between the two things:
1: The name of a number, in this case “ten” is supposedly implied
2: The symbols representing a number, in this case 10 in the base ten system”

So the answers to the questions “What is it? How do I find out?” above are “Unknown” and “You can’t”

Writing “Base 10” when you mean “Base ten” is probably the first step in making math meaningless.

Filed under abstract, arithmetic, confusion, definitions, language in math, math, teaching

### 3 responses to “What exactly is Base 10 arithmetic ?”

1. Joseph Nebus

It is a subtle point. But I remember as a kid that while I was sincerely thrilled by the novelty of working on other-base arithmetic, the rule I had that I had to write out the base in English every time was just too unwieldy. It’s not so bad if you’re working base ten or base two, but if for some mad reason you’re doing base eleven? Base sixteen? The notation will just crush the joy out of amusing numbers like 2E.

• howardat58

I am assuming that they made you write the base as a subscripte on every number. That would take the fun away. I would think that putting “Working in base eleven” once at the top of the page was enough, but teachers are strange people.

• Joseph Nebus

Yes, every number was what the rule dictated. I understand not wanting to confuse people who glanced at the work and didn’t know that the abnormal rules of a different base applied here. But it would have saved so much tedium to just put a header or a box or something to note “unusual circumstance until further notice”.